problem solving in inequality

Solving Inequality Word Questions

(You might like to read Introduction to Inequalities and Solving Inequalities first.)

In Algebra we have "inequality" questions like:

Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. What are the possible number of goals Alex scored?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if needed
• Assign letters for the values
• Find or work out formulas

We should also write down what is actually being asked for , so we know where we are going and when we have arrived!

The best way to learn this is by example, so let's try our first example:

Assign Letters:

• the number of goals Alex scored: A
• the number of goals Sam scored: S

We know that Alex scored 3 more goals than Sam did, so: A = S + 3

And we know that together they scored less than 9 goals: S + A < 9

We are being asked for how many goals Alex might have scored: A

Sam scored less than 3 goals, which means that Sam could have scored 0, 1 or 2 goals.

Alex scored 3 more goals than Sam did, so Alex could have scored 3, 4, or 5 goals .

• When S = 0, then A = 3 and S + A = 3, and 3 < 9 is correct
• When S = 1, then A = 4 and S + A = 5, and 5 < 9 is correct
• When S = 2, then A = 5 and S + A = 7, and 7 < 9 is correct
• (But when S = 3, then A = 6 and S + A = 9, and 9 < 9 is incorrect)

Lots More Examples!

Example: Of 8 pups, there are more girls than boys. How many girl pups could there be?

• the number of girls: g
• the number of boys: b

We know that there are 8 pups, so: g + b = 8, which can be rearranged to

We also know there are more girls than boys, so:

We are being asked for the number of girl pups: g

So there could be 5, 6, 7 or 8 girl pups.

Could there be 8 girl pups? Then there would be no boys at all, and the question isn't clear on that point (sometimes questions are like that).

• When g = 8, then b = 0 and g > b is correct (but is b = 0 allowed?)
• When g = 7, then b = 1 and g > b is correct
• When g = 6, then b = 2 and g > b is correct
• When g = 5, then b = 3 and g > b is correct
• (But if g = 4, then b = 4 and g > b is incorrect)

A speedy example:

Example: Joe enters a race where he has to cycle and run. He cycles a distance of 25 km, and then runs for 20 km. His average running speed is half of his average cycling speed. Joe completes the race in less than 2½ hours, what can we say about his average speeds?

• Average running speed: s
• So average cycling speed: 2s
• Speed = Distance Time
• Which can be rearranged to: Time = Distance Speed

We are being asked for his average speeds: s and 2s

The race is divided into two parts:

• Distance = 25 km
• Average speed = 2s km/h
• So Time = Distance Average Speed = 25 2s hours
• Distance = 20 km
• Average speed = s km/h
• So Time = Distance Average Speed = 20 s hours

Joe completes the race in less than 2½ hours

• The total time < 2½
• 25 2s + 20 s < 2½

So his average speed running is greater than 13 km/h and his average speed cycling is greater than 26 km/h

In this example we get to use two inequalities at once:

Example: The velocity v m/s of a ball thrown directly up in the air is given by v = 20 − 10t , where t is the time in seconds. At what times will the velocity be between 10 m/s and 15 m/s?

• velocity in m/s: v
• the time in seconds: t
• v = 20 − 10t

We are being asked for the time t when v is between 5 and 15 m/s:

So the velocity is between 10 m/s and 15 m/s between 0.5 and 1 second after.

And a reasonably hard example to finish with:

Example: A rectangular room fits at least 7 tables that each have 1 square meter of surface area. The perimeter of the room is 16 m. What could the width and length of the room be?

Make a sketch: we don't know the size of the tables, only their area, they may fit perfectly or not!

• the length of the room: L
• the width of the room: W

The formula for the perimeter is 2(W + L) , and we know it is 16 m

• 2(W + L) = 16
• L = 8 − W

We also know the area of a rectangle is the width times the length: Area = W × L

And the area must be greater than or equal to 7:

• W × L ≥ 7

We are being asked for the possible values of W and L

Let's solve:

So the width must be between 1 m and 7 m (inclusive) and the length is 8−width .

• Say W = 1, then L = 8−1 = 7, and A = 1 x 7 = 7 m 2 (fits exactly 7 tables)
• Say W = 0.9 (less than 1), then L = 7.1, and A = 0.9 x 7.1 = 6.39 m 2 (7 won't fit)
• Say W = 1.1 (just above 1), then L = 6.9, and A = 1.1 x 6.9 = 7.59 m 2 (7 fit easily)
• Likewise for W around 7 m

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Inequality Word Problems

Inequalities are common in our everyday life. They help us express relationships between quantities that are unequal. Writing and solving word problems involving them helps develop our problem-solving approach, understanding, logical reasoning, and analytical skills.

Here are the 4 main keywords commonly used to write mathematical expressions involving inequalities.   

• At least →  ‘greater than or equal to’
• More than → ‘greater than’
• No more than or at most → ‘less than or equal to’
• Less than → ‘less than’

To participate in the annual sports day, Mr. Adams would like to have nine students in each group. But fewer than 54 students are in class today, so Mr. Adams is unable to make as many full groups as he wants. How many full groups can Mr. Adams make? Write the inequality that describes the situation.

Let ‘x’ be the total number of groups Mr. Adams can make. Since each group has 9 students, the total number of students in ‘x’ groups is 9x As we know, fewer than 54 students are in a class today. Thus, the inequality that represents the situation is: 9x < 54 On dividing both sides by 9, the maximum number of groups Mr. Adams can make is  x < 6 Thus, Mr. Adams can make a maximum of 6 full groups.

 Bruce needs at least \$561 to buy a new tablet. He has already saved \$121 and earns \$44 per month as a part-timer in a company. Write the inequality and determine how long he has to work to buy the tablet.

Let ‘x’ be the number of months Bruce needs to work. As we know,  The amount already saved by Bruce is \$121 He earns \$44 per month The cost of the tablet is at least \$561 After ‘x’ months of work, Bruce will have \$(121 + 44x) Now, the inequality representing the situation is: 121 + 44x ≥ 561 On subtracting 121 from both sides, 121 + 44x – 121 ≥ 561 – 121 ⇒ 44x ≥ 440 On dividing both sides by 44, x ≥ 10 Thus, Bruce needs to work for at least 10 months to buy the new tablet.

A store is offering a \$26 discount on all women’s clothes. Ava is looking at clothes originally priced between \$199 and \$299. How much can she expect to spend after the discount?

Let ‘x’ be the original price of the clothes Ava chooses. As we know, the original price range is 199 ≤ x ≤ 299, and the discount is \$26 Now, Ava pays \$(x – 26) after the discount. Thus, the inequality is: 199 – 26 ≤ x – 26 ≤ 299 – 26 ⇒ 173 ≤ x – 26 ≤ 273 Thus, she can expect to spend between \$173 and \$273 after the discount.

A florist makes a profit of \$6.25 per plant. If the store wants to profit at least \$4225, how many plants does it need to sell?

Let ‘P’ be the profit, ‘p’ be the profit per plant, and ‘n’ be the number of plants.  As we know, the store wants a profit of at least \$4225, and the florist makes a profit of \$6.25 per plant. Here, P ≥ 4225 and p = 6.25 …..(i) Also, P = p × n Substituting the values of (i), we get 6.25 × n ≥ 4225 On dividing both sides by 6.25, we get ${n\geq \dfrac{4225}{6\cdot 25}}$ ⇒ ${n\geq 676}$ Thus, the store needs to sell at least 676 plants to make a profit of \$4225.

Daniel had \$1200 in his savings account at the start of the year, but he withdraws \$60 each month to spend on transportation. He wants to have at least \$300 in the account at the end of the year. How many months can Daniel withdraw money from the account?

As we know, Daniel had \$1200 in his savings account at the start of the year, but he withdrew \$60 for transportation each month.  Thus, after ‘n’ months, he will have \$(1200−60n) left in his account. Also, Daniel wants to have at least \$300 in the account at the end of the year.  Here, the inequality is: 1200 – 60n ≥ 300 ⇒ 1200 – 60n – 1200 ≥ 300 – 1200 (by subtraction property) ⇒ -60n ≥ -900 ⇒ 60n ≤ 900 (by inversion property) ⇒ n ≤ ${\dfrac{900}{60}}$ ⇒ n ≤ 15 Thus, Daniel can withdraw money from the account for at most 15 months.

Anne is a model trying to lose weight for an upcoming beauty pageant. She currently weighs 165 lb. If she cuts 2 lb per week, how long will it take her to weigh less than 155 lb?

Let ‘t’ be the number of weeks to weigh less than 155 lb. As we know, Anne initially weighs 165 lb After ‘t’ weeks of cutting 2 lb per week, her weight will be 165 – 2t Now, Anne’s weight will be less than 155 lb Here, the inequality from the given word problem is: 165 – 2t < 155 On subtracting 165 from both sides, we get 165 – 2t – 165 < 155 – 165 ⇒ – 2t < -10 On dividing by -2, the inequality sign is reversed. ${\dfrac{-2t}{-2} >\dfrac{-10}{-2}}$ ⇒ t > 5

Rory and Cinder are on the same debate team. In one topic, Rory scored 5 points more than Cinder, but they scored less than 19 together. What are the possible points Rory scored?

Let Rory’s score be ‘r,’ and Cinder’s score be ‘c.’ As we know, Rory scored 5 points more than Cinder. Thus, Rory’s score is r = c + 5 …..(i) Also, their scores sum up to less than 19 points. Thus, the inequality is: r + c < 19 …..(ii) Substituting (i) in (ii), we get (c + 5) + c < 19 ⇒ 2c + 5 < 19 On subtracting 5 from both sides, we get 2c + 5 – 5 < 19 – 5 ⇒ 2c < 14 On dividing both sides by 2, we get ${\dfrac{2c}{2} >\dfrac{14}{2}}$ ⇒ c < 7 means Cinder’s score is less than 7 points. Now, from (i), r = c + 5 ⇒ c = r – 5 Thus, c < 7 ⇒ r – 5 < 7 On adding 5 to both sides, we get r – 5 + 5 < 7 + 5 ⇒ r < 12 means Rory’s score is less than 12 points. Hence, Rory’s scores can be 6, 7, 8, 9, 10, or 11 points.

An average carton of juice cans contains 74 pieces, but the number can vary by 4. Find out the maximum and minimum number of cans that can be present in a carton.

Let ‘c’ be the number of juice cans in a carton. As we know, the average number of cans in a carton is 74, and it varies by 4 cans. Thus, the required inequality is |c – 74| ≤ 4 ⇒ -4 ≤ c – 74 ≤ 4 On adding 74 to each side, we get -4 + 74 ≤ c – 74 + 74 ≤ 4 + 74 ⇒ 70 ≤ c ≤ 78 Hence, the minimum number of cans in a carton is 70, and the maximum number is 78.

 Layla rehearses singing for at least 12 hours per week, for three-fourths of an hour each session. If she has already sung 3 hours this week, how many more sessions remain for her to exceed her weekly practice goal?

Let ‘p’ be Layla’s total hours of practice in a week, and ‘s’ be the number of sessions she needs to complete. As we know, Layla has already rehearsed 3 hours, then her remaining rehearsal time is (p – 3) Each session lasts for three-fourths of an hour. Thus, we have the inequality: ${\dfrac{3}{4}s >p-3}$ …..(i) As we know, Layla rehearses for at least 12 hours, which means p ≥ 12 …..(ii) From (i), ${\dfrac{3}{4}s >p-3}$ ⇒ ${s >\dfrac{4}{3}\left( p-3\right)}$ From (ii), substituting the value p = 12 in (i), we get ${s >\dfrac{4}{3}\left( 12-3\right)}$ ⇒ ${s >\dfrac{4}{3}\cdot 9}$ ⇒ s > 12 Thus, Layla must complete more than 12 sessions to exceed her weekly rehearsal goal.

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Inequalities

In Mathematics, equations are not always about being balanced on both sides with an 'equal to' symbol. Sometimes it can be about 'not an equal to' relationship like something is greater than the other or less than. In mathematics, inequality refers to a relationship that makes a non-equal comparison between two numbers or other mathematical expressions. These mathematical expressions come under algebra and are called inequalities.

Let us learn the rules of inequalities, and how to solve and graph them.

What is an Inequality?

Inequalities  are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

Olivia is selected in the 12U Softball. How old is Olivia? You don't know the age of Olivia, because it doesn't say "equals". But you do know her age should be less than or equal to 12, so it can be written as Olivia's Age ≤ 12. This is a practical scenario related to inequalities.

Inequality Meaning

The meaning of inequality is to say that two things are NOT equal. One of the things may be less than, greater than, less than or equal to, or greater than or equal to the other things.

• p ≠ q means that p is not equal to q
• p < q means that p is less than q
• p > q means that p is greater than q
• p ≤ q means that p is less than or equal to q
• p ≥ q means that p is greater than or equal to q

There are different types of inequalities. Some of the important inequalities are:

• Polynomial inequalities
• Absolute value inequalities
• Rational inequalities

Rules of Inequalities

The rules of inequalities are special. Here are some listed with inequalities examples.

Inequalities Rule 1

When inequalities are linked up you can jump over the middle inequality.

• If, p < q and q < d, then p < d
• If, p > q and q > d, then p > d

Example: If Oggy is older than Mia and Mia is older than Cherry, then Oggy must be older than Cherry.

Inequalities Rule 2

Swapping of numbers p and q results in:

• If, p > q, then q < p
• If, p < q, then q > p

Example: Oggy is older than Mia, so Mia is younger than Oggy.

Inequalities Rule 3

Adding the number d to both sides of inequality: If p < q, then p + d < q + d

Example: Oggy has less money than Mia. If both Oggy and Mia get $5 more, then Oggy will still have less money than Mia.

• If p < q, then p − d < q − d
• If p > q, then p + d > q + d, and
• If p > q, then p − d > q − d

So, the addition and subtraction of the same value to both p and q will not change the inequality.

Inequalities Rule 4

If you multiply numbers p and q by a positive number , there is no change in inequality. If you multiply both p and q by a negative number , the inequality swaps: p<q becomes q<p after multiplying by (-2)

Here are the rules:

• If p < q, and d is positive, then pd < qd
• If p < q, and d is negative, then pd > qd (inequality swaps)

Positive case example: Oggy's score of 5 is lower than Mia's score of 9 (p < q). If Oggy and Mia double their scores '×2', Oggy's score will still be lower than Mia's score, 2p < 2q. If the scores turn minuses, then scores will be −p > −q.

Inequalities Rule 5

Putting minuses in front of p and q changes the direction of the inequality.

• If p < q then −p > −q
• If p > q, then −p < −q
• It is the same as multiplying by (-1) and changes direction.

Inequalities Rule 6

Taking the reciprocal 1/value of both p and q changes the direction of the inequality. When p and q are both positive or both negative:

• If, p < q, then 1/p > 1/q
• If p > q, then 1/p < 1/q

Inequalities Rule 7

A square of a number is always greater than or equal to zero p 2  ≥ 0.

Example: (4) 2 = 16, (−4) 2 = 16, (0) 2 = 0

Inequalities Rule 8

Taking a square root will not change the inequality. If p ≤ q, then √p ≤ √q (for p, q ≥ 0).

Example: p=2, q=7 2 ≤ 7, then √2 ≤ √7

The rules of inequalities are summarized in the following table.

Solving Inequalities

Here are the steps for  solving inequalities :

• Step - 1: Write the inequality as an equation.
• Step - 2: Solve the equation for one or more values.
• Step - 3: Represent all the values on the number line.
• Step - 4: Also, represent all excluded values on the number line using open circles.
• Step - 5: Identify the intervals.
• Step - 6: Take a random number from each interval, substitute it in the inequality and check whether the inequality is satisfied.
• Step - 7: Intervals that are satisfied are the solutions.

But for solving simple inequalities (linear), we usually apply algebraic operations like addition , subtraction , multiplication , and division . Consider the following example:

2x + 3 > 3x + 4

Subtracting 3x and 3 from both sides,

2x - 3x > 4 - 3

Multiplying both sides by -1,

Notice that we have changed the ">" symbol into "<" symbol. Why? This is because we have multiplied both sides of the inequality by a negative number. The process of solving inequalities mentioned above works for a simple linear inequality. But to solve any other complex inequality, we have to use the following process.

Let us use this procedure to solve inequalities of different types.

Graphing Inequalities

While graphing inequalities , we have to keep the following things in mind.

• If the endpoint is included (i.e., in case of ≤ or ≥) use a closed circle.
• If the endpoint is NOT included (i.e., in case of < or >), use an open circle.
• Use open circle at either ∞ or -∞.
• Draw a line from the endpoint that extends to the right side if the variable is greater than the number.
• Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.

Writing Inequalities in Interval Notation

While writing the solution of an inequality in the interval notation , we have to keep the following things in mind.

• If the endpoint is included (i.e., in case of ≤ or ≥) use the closed brackets '[' or ']'
• If the endpoint is not included (i.e., in case of < or >), use the open brackets '(' or ')'
• Use always open bracket at either ∞ or -∞.

Here are some examples to understand the same:

Graphing Inequalities with Two Variables

For graphing  inequalities with two variables , you will have to plot the "equals" line and then, shade the appropriate area. There are three steps:

• Write the equation such as "y" is on the left and everything else on the right.
• Plot the "y=" line (draw a solid line for y≤ or y≥, and a dashed line for y< or y>)
• Shade the region above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).

Let us try some example: This is a graph of a linear inequality: y ≤ x + 4

You can see, y = x + 4 line and the shaded area (in yellow) is where y is less than or equal to x + 4. Let us now see how to solve different types of inequalities and how to graph the solution in each case.

Solving Polynomial Inequalities

The polynomial inequalities are inequalities that can be expressed as a polynomial on one side and 0 on the other side of the inequality. There are different types of polynomial inequalities but the important ones are:

• Linear Inequalities
• Quadratic Inequalities

Solving Linear Inequalities

A linear inequality is an inequality that can be expressed with a linear expression on one side and a 0 on the other side. Solving linear inequalities is as same as solving linear equations , but just the rules of solving inequalities (that was explained before) should be taken care of. Let us see some examples.

Solving One Step Inequalities

Consider an inequality 2x < 6 (which is a linear inequality with one variable ). To solve this, just one step is sufficient which is dividing both sides by 2. Then we get x < 3. Therefore, the solution of the inequality is x < 3 (or) (-∞, 3).

Solving Two Step Inequalities

Consider an inequality -2x + 3 > 6. To solve this, we need two steps . The first step is subtracting 3 from both sides, which gives -2x > 3. Then we need to divide both sides by -2 and it results in x < -3/2 (note that we have changed the sign of the inequality). So the solution of the inequality is x < -3/2 (or) (-∞, -3/2).

Solving Compound Inequalities

Compound inequalities refer to the set of inequalities with either "and" or "or" in between them. For solving inequalities, in this case, just solve each inequality independently and then find the final solution according to the following rules:

• The final solution is the intersection of the solutions of the independent inequalities if we have "and" between them.
• The final solution is the union of the solutions of the independent inequalities if we have "or" between them.

Example: Solve the compound inequality 2x + 3 < -5 and x + 6 < 3.

By first inequality: 2x + 3 < -5 2x < -8 x < -4

By second inequality, x + 6 < 3 x < -3

Since we have "and" between them, we have to find the intersection of the sets x < -4 and x < -3. A number line may be helpful in this case. Then the final solution is:

x < -3 (or) (-∞, -3).

Solving Quadratic Inequalities

A quadratic inequality involves a quadratic expression in it. Here is the process of solving quadratic inequalities . The process is explained with an example where we are going to solve the inequality x 2 - 4x - 5 ≥ 0.

• Step 1: Write the inequality as equation. x 2 - 4x - 5 = 0
• Step 2: Solve the equation. Here we can use any process of solving quadratic equations . Then (x - 5) (x + 1) = 0 x = 5; x = -1.

• Step 5: The inequalities with "true" from the above table are solutions. Therefore, the solutions of the quadratic inequality x 2 - 4x - 5 ≥ 0 is (-∞, -1] U [5, ∞).

We can use the same process for solving cubic inequalities, biquadratic inequalities, etc.

Solving Absolute Value Inequalities

An absolute value inequality includes an algebraic expression inside the absolute value sign. Here is the process of solving absolute value inequalities where the process is explained with an example of solving an absolute value inequality |x + 3| ≤ 2. If you want to learn different methods of solving absolute value inequalities, click here .

• Step 1: Consider the absolute value inequality as equation. |x + 3| = 2
• Step 2: Solve the equation. x + 3 = ±2 x + 3 = 2; x + 3 = -2 x = -1; x = -5

• Step 5: The intervals that satisfied the inequality are the solution intervals. Therefore, the solution of the absolute value inequality |x + 3| ≤ 2 is [-5, -1].

Solving Rational Inequalities

Rational inequalities are inequalities that involve rational expressions (fractions with variables). To solve the rational inequalities (inequalities with fractions), we just use the same procedure as other inequalities but we have to take care of the excluded points . For example, while solving the rational inequality (x + 2) / (x - 2) < 3, we should note that the rational expression (x + 2) / (x - 2) is NOT defined at x = 2 (set the denominator x - 2 = 0 ⇒x = 2). Let us solve this inequality step by step.

• Step 1: Consider the inequality as the equation. (x + 2) / (x - 2) = 3
• Step 2: Solve the equation. x + 2 = 3(x - 2) x + 2 = 3x - 6 2x = 8 x = 4

• Step 5: The intervals that have come up with "true" in Step 4 are the solutions. Therefore, the solution of the rational inequality (x + 2) / (x - 2) < 3 is (-∞, 2) U (4, ∞).

Important Notes on Inequalities:

Here are the notes about inequalities:

• If we have strictly less than or strictly greater than symbol, then we never get any closed interval in the solution.
• We always get open intervals at ∞ or -∞ symbols because they are NOT numbers to include.
• Write open intervals always at excluded values when solving rational inequalities.
• Excluded values should be taken care of only in case of rational inequalities.

☛  Related Topics:

• Linear Inequalities Calculator
• Triangle Inequality Theorem Calculator
• Rational Inequalities Calculator

Inequalities Examples

Example 1: Using the techniques of solving inequalities, solve: -19 < 3x + 2 ≤ 17 and write the answer in the interval notation.

Given that -19 < 3x + 2 ≤ 17.

This is a compound inequality.

Subtracting 2 from all sides,

-21 < 3x ≤ 15

Dividing all sides by 3,

-7 < x ≤ 5

Answer: The solution is (-7, 5].

Example 2: While solving inequalities, explain why each of the following statements is incorrect. Also, correct them. a) 2x < 5 ⇒ x > 5/2 b) x > 3 ⇒ x ∈ [3, ∞) c) -x > -7 ⇒ x > 7.

a) 2x < 5. Here, when we divide both sides by 2, which is a positive number, the sign does not change. So the correct inequality is x < 5/2.

b) x > 3. It does not include an equal to symbol. So 3 should NOT be included in the interval. So the correct interval is (3, ∞).

c) -x > -7. When we divide both sides by -1, a negative number, the sign should change. So the correct inequality is x < 7.

Answer: The corrected ones are a) x < 5/2; b) x ∈ (3, ∞); c) x < 7.

Example 3: Solve the inequality x 2 - 7x + 10 < 0.

First, solve the equation x 2 - 7x + 10 = 0.

(x - 2) (x - 5) = 0.

x = 2, x = 5.

If we represent these numbers on the number line, we get the following intervals: (-∞, 2), (2, 5), and (5, ∞).

Let us take some random numbers from each interval to test the given quadratic inequality.

Therefore, the only interval that satisfies the inequality is (2, 5).

Answer: The solution is (2, 5).

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Practice Questions on Inequalities

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FAQs on Inequalities

What are inequalities in math.

When two or more algebraic expressions are compared using the symbols <, > ≤, or ≥, then they form an inequality. They   are the mathematical expressions in which both sides are not equal.

How Do you Solve Inequalities On A Number Line?

To plot an inequality in math, such as x>3, on a number line,

• Step 1: Draw a circle over the number (e.g., 3).
• Step 2: Check if the sign includes equal to (≥ or ≤) or not. If equal to sign is there along with > or <, then fill in the circle otherwise leave the circle unfilled.
• Step 3: On the number line, extend the line from 3(after encircling it) to show it is greater than or equal to 3.

How to Calculate Inequalities in Math?

To  calculate inequalities :

• just make it an equation
• mark the zeros  on the number line to get intervals
• test the intervals by taking any one number from it against the inequality.

Explain the Process of Solving Inequalities Graphically.

Solving inequalities graphically is possible when we have a system of two inequalities in two variables. In this case, we consider both inequalities as two linear equations and graph them. Then we get two lines. Shade the upper/lower portion of each of the lines that satisfies the inequality. The common portion of both shaded regions is the solution region.

What is the Difference Between Equations and Inequalities?

Here are the differences between equations and inequalities.

What Happens When you Square An Inequality?

A square of a number is always greater than or equal to zero p 2  ≥ 0. Example: (4) 2 = 16, (−4) 2 = 16, (0) 2 = 0

What are the Steps to Calculate Inequalities with Fractions?

Calculating inequalities with fractions is just like solving any other inequality. One easy way of solving such inequalities is to multiply every term on both sides by the LCD of all denominators so that all fractions become integers . For example, to solve (1/2) x + 1 > (3/4) x + 2, multiply both sides by 4. Then we get 2x + 4 > 3x + 8 ⇒ -x > 4 ⇒ x < -4.

What are the Steps for Solving Inequalities with Variables on Both Sides?

When an inequality has a variable on both sides, we have to try to isolate the variable. But in this process, flip the inequality sign whenever we are dividing or multiplying both sides by a negative number. Here is an example. 3x - 7 < 5x - 11 ⇒ -2x < -4 ⇒ x > 2.

How Do you Find the Range of Inequality?

You can find the range of values of x, by solving the inequality by considering it as a normal linear equation.

What Are the 5 Inequality Symbols?

The 5 inequality symbols are less than (<), greater than (>), less than or equal (≤), greater than or equal (≥), and the not equal symbol (≠).

How Do you Tell If It's An Inequality?

Equations and inequalities are mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are supposed to be equal and shown by the symbol =. Whereas in inequality, the two expressions are not necessarily equal and are indicated by the symbols: >, <, ≤ or ≥.

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• -x+3\gt 2x+1
• (x+5)(x-5)\gt 0
• 2x^2-x\gt 0
• (x+3)^2\le 10x+6
• \left|3+2x\right|\le 7
• \frac{\left|3x+2\right|}{\left|x-1\right|}>2
• What are the 4 inequalities?
• There are four types of inequalities: greater than, less than, greater than or equal to, and less than or equal to.
• What is a inequality in math?
• In math, inequality represents the relative size or order of two values.
• How do you solve inequalities?
• To solve inequalities, isolate the variable on one side of the inequality, If you multiply or divide both sides by a negative number, flip the direction of the inequality.
• What are the 2 rules of inequalities?
• The two rules of inequalities are: If the same quantity is added to or subtracted from both sides of an inequality, the inequality remains true. If both sides of an inequality are multiplied or divided by the same positive quantity, the inequality remains true. If we multiply or divide both sides of an inequality by the same negative number, we must flip the direction of the inequality to maintain its truth.

inequalities-calculator

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Equations and Inequalities Involving Signed Numbers

In chapter 2 we established rules for solving equations using the numbers of arithmetic. Now that we have learned the operations on signed numbers, we will use those same rules to solve equations that involve negative numbers. We will also study techniques for solving and graphing inequalities having one unknown.

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

Upon completing this section you should be able to solve equations involving signed numbers.

Example 1 Solve for x and check: x + 5 = 3

Using the same procedures learned in chapter 2, we subtract 5 from each side of the equation obtaining

Example 2 Solve for x and check: - 3x = 12

Dividing each side by -3, we obtain

LITERAL EQUATIONS

• Identify a literal equation.
• Apply previously learned rules to solve literal equations.

An equation having more than one letter is sometimes called a literal equation . It is occasionally necessary to solve such an equation for one of the letters in terms of the others. The step-by-step procedure discussed and used in chapter 2 is still valid after any grouping symbols are removed.

Example 1 Solve for c: 3(x + c) - 4y = 2x - 5c

First remove parentheses.

At this point we note that since we are solving for c, we want to obtain c on one side and all other terms on the other side of the equation. Thus we obtain

Sometimes the form of an answer can be changed. In this example we could multiply both numerator and denominator of the answer by (- l) (this does not change the value of the answer) and obtain

The advantage of this last expression over the first is that there are not so many negative signs in the answer.

The most commonly used literal expressions are formulas from geometry, physics, business, electronics, and so forth.

Notice in this example that r was left on the right side and thus the computation was simpler. We can rewrite the answer another way if we wish.

GRAPHING INEQUALITIES

• Use the inequality symbol to represent the relative positions of two numbers on the number line.
• Graph inequalities on the number line.

The symbols are inequality symbols or order relations and are used to show the relative sizes of the values of two numbers. We usually read the symbol as "greater than." For instance, a > b is read as "a is greater than b." Notice that we have stated that we usually read a < b as a is less than b. But this is only because we read from left to right. In other words, "a is less than b" is the same as saying "b is greater than a." Actually then, we have one symbol that is written two ways only for convenience of reading. One way to remember the meaning of the symbol is that the pointed end is toward the lesser of the two numbers.

In simpler words this definition states that a is less than b if we must add something to a to get b. Of course, the "something" must be positive.

If you think of the number line, you know that adding a positive number is equivalent to moving to the right on the number line. This gives rise to the following alternative definition, which may be easier to visualize.

Example 1 3 < 6, because 3 is to the left of 6 on the number line.

Example 2 - 4 < 0, because -4 is to the left of 0 on the number line.

Example 3 4 > - 2, because 4 is to the right of -2 on the number line.

Example 4 - 6 < - 2, because -6 is to the left of -2 on the number line.

The mathematical statement x < 3, read as "x is less than 3," indicates that the variable x can be any number less than (or to the left of) 3. Remember, we are considering the real numbers and not just integers, so do not think of the values of x for x < 3 as only 2, 1,0, - 1, and so on.

As a matter of fact, to name the number x that is the largest number less than 3 is an impossible task. It can be indicated on the number line, however. To do this we need a symbol to represent the meaning of a statement such as x < 3.

The symbols ( and ) used on the number line indicate that the endpoint is not included in the set.

Example 5 Graph x < 3 on the number line.

Note that the graph has an arrow indicating that the line continues without end to the left.

Example 6 Graph x > 4 on the number line.

Example 7 Graph x > -5 on the number line.

Example 8 Make a number line graph showing that x > - 1 and x < 5. (The word "and" means that both conditions must apply.)

Example 9 Graph - 3 < x < 3.

Example 10 x >; 4 indicates the number 4 and all real numbers to the right of 4 on the number line.

The symbols [ and ] used on the number line indicate that the endpoint is included in the set.

Example 13 Write an algebraic statement represented by the following graph.

Example 14 Write an algebraic statement for the following graph.

Example 15 Write an algebraic statement for the following graph.

SOLVING INEQUALITIES

Upon completing this section you should be able to solve inequalities involving one unknown.

The solutions for inequalities generally involve the same basic rules as equations. There is one exception, which we will soon discover. The first rule, however, is similar to that used in solving equations.

If the same quantity is added to each side of an inequality , the results are unequal in the same order.

Example 1 If 5 < 8, then 5 + 2 < 8 + 2.

Example 2 If 7 < 10, then 7 - 3 < 10 - 3.

We can use this rule to solve certain inequalities.

Example 3 Solve for x: x + 6 < 10

If we add -6 to each side, we obtain

Graphing this solution on the number line, we have

We will now use the addition rule to illustrate an important concept concerning multiplication or division of inequalities.

Suppose x > a.

Now add - x to both sides by the addition rule.

Now add -a to both sides.

The last statement, - a > -x, can be rewritten as - x < -a. Therefore we can say, "If x > a, then - x < -a. This translates into the following rule:

If an inequality is multiplied or divided by a negative number, the results will be unequal in the opposite order.

Example 5 Solve for x and graph the solution: -2x>6

To obtain x on the left side we must divide each term by - 2. Notice that since we are dividing by a negative number, we must change the direction of the inequality.

Take special note of this fact. Each time you divide or multiply by a negative number, you must change the direction of the inequality symbol. This is the only difference between solving equations and solving inequalities.

Once we have removed parentheses and have only individual terms in an expression, the procedure for finding a solution is almost like that in chapter 2.

Let us now review the step-by-step method from chapter 2 and note the difference when solving inequalities.

First Eliminate fractions by multiplying all terms by the least common denominator of all fractions. (No change when we are multiplying by a positive number.) Second Simplify by combining like terms on each side of the inequality. (No change) Third Add or subtract quantities to obtain the unknown on one side and the numbers on the other. (No change) Fourth Divide each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will be reversed. (This is the important difference between equations and inequalities.)

• A literal equation is an equation involving more than one letter.
• The symbols are inequality symbols or order relations .
• a a is to the left of b on the real number line.
• To solve a literal equation for one letter in terms of the others follow the same steps as in chapter 2.
• To solve an inequality use the following steps: Step 1 Eliminate fractions by multiplying all terms by the least common denominator of all fractions. Step 2 Simplify by combining like terms on each side of the inequality. Step 3 Add or subtract quantities to obtain the unknown on one side and the numbers on the other. Step 4 Divide each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will be reversed. Step 5 Check your answer.

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Examples of Solving Harder Linear Inequalities

Intro & Formatting Worked Examples Harder Examples & Word Prob's

Once you'd learned how to solve one-variable linear equations, you were then given word problems. To solve these problems, you'd have to figure out a linear equation that modelled the situation, and then you'd have to solve that equation to find the answer to the word problem.

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Solving Inequalities

So, now that you know how to solve linear inequalities — you guessed it! — they give you word problems.

• The velocity of an object fired directly upward is given by V = 80 − 32 t , where the time t is measured in seconds. When will the velocity be between 32 and 64 feet per second (inclusive)?

This question is asking when the velocity, V , will be between two given values. So I'll take the expression for the velocity,, and put it between the two values they've given me. They've specified that the interval of velocities is inclusive, which means that the interval endpoints are included. Mathematically, this means that the inequality for this model will be an "or equal to" inequality. Because the solution is a bracket (that is, the solution is within an interval), I'll need to set up a three-part (that is, a compound) inequality.

I will set up the compound inequality, and then solve for the range of times t :

32 ≤ 80 − 32 t ≤ 64

32 − 80 ≤ 80 − 80 − 32 t ≤ 64 − 80

−48 ≤ −32 t ≤ −16

−48 / −32 ≥ −32 t / −32 ≥ −16 / −32

1.5 ≥ t ≥ 0.5

Note that, since I had to divide through by a negative, I had to flip the inequality signs.

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Note also that you might (as I do) find the above answer to be more easily understood if it's written the other way around, with "less than" inequalities.

And, because this is a (sort of) real world problem, my working should show the fractions, but my answer should probably be converted to decimal form, because it's more natural to say "one and a half seconds" than it is to say "three-halves seconds". So I convert the last line above to the following:

0.5 ≤ t ≤ 1.5

Looking back at the original question, it did not ask for the value of the variable " t ", but asked for the times when the velocity was between certain values. So the actual answer is:

The velocity will be between 32 and 64 feet per second between 0.5 seconds after launch and 1.5 seconds after launch.

Okay; my answer above was *extremely* verbose and "complete"; you don't likely need to be so extreme. You can probably safely get away with saying something simpler like, "between 0.5 seconds and 1.5 seconds". Just make sure that you do indeed include the approprioate units (in this case, "seconds").

Always remember when doing word problems, that, once you've found the value for the variable, you need to go back and re-read the problem to make sure that you're answering the actual question. The inequality 0.5 ≤  t  ≤ 1.5 did not answer the actual question regarding time. I had to interpret the inequality and express the values in terms of the original question.

• Solve 5 x + 7 < 3( x  + 1) .

First I'll multiply through on the right-hand side, and then solve as usual:

5 x + 7 < 3( x + 1)

5 x + 7 < 3 x + 3

2 x + 7 < 3

2 x < −4

x < −2

In solving this inequality, I divided through by a positive 2 to get the final answer; as a result (that is, because I did *not* divide through by a minus), I didn't have to flip the inequality sign.

• You want to invest $30,000 . Part of this will be invested in a stable 5% -simple-interest account. The remainder will be "invested" in your father's business, and he says that he'll pay you back with 7% interest. Your father knows that you're making these investments in order to pay your child's college tuition with the interest income. What is the least you can "invest" with your father, and still (assuming he really pays you back) get at least $1900 in interest?

First, I have to set up equations for this. The interest formula for simple interest is I = Prt , where I is the interest, P is the beginning principal, r is the interest rate expressed as a decimal, and t is the time in years.

Since no time-frame is specified for this problem, I'll assume that t  = 1 ; that is, I'll assume (hope) that he's promising to pay me at the end of one year. I'll let x be the amount that I'm going to "invest" with my father. Then the rest of my money, being however much is left after whatever I give to him, will be represented by "the total, less what I've already given him", so 30000 −  x will be left to invest in the safe account.

Then the interest on the business investment, assuming that I get paid back, will be:

I = ( x )(0.07)(1) = 0.07 x

The interest on the safe investment will be:

(30 000 − x )(0.05)(1) = 1500 − 0.05 x

The total interest is the sum of what is earned on each of the two separate investments, so my expression for the total interest is:

0.07 x + (1500 − 0.05 x ) = 0.02 x + 1500

I need to get at least $1900 ; that is, the sum of the two investments' interest must be greater than, or at least equal to, $1,900 . This allows me to create my inequality:

0.02 x + 1500 ≥ 1900

0.02 x ≥ 400

x ≥ 20 000

That is, I will need to "invest" at least $20,000 with my father in order to get $1,900 in interest income. Since I want to give him as little money as possible, I will give him the minimum amount:

I will invest $20,000 at 7% .

• An alloy needs to contain between 46% copper and 50% copper. Find the least and greatest amounts of a 60% copper alloy that should be mixed with a 40% copper alloy in order to end up with thirty pounds of an alloy containing an allowable percentage of copper.

This is similar to a mixture word problem , except that this will involve inequality symbols rather than "equals" signs. I'll set it up the same way, though, starting with picking a variable for the unknown that I'm seeking. I will use x to stand for the pounds of 60% copper alloy that I need to use. Then 30 −  x will be the number of pounds, out of total of thirty pounds needed, that will come from the 40% alloy.

Of course, I'll remember to convert the percentages to decimal form for doing the algebra.

How did I get those values in the bottom right-hand box? I multiplied the total number of pounds in the mixture ( 30 ) by the minimum and maximum percentages ( 46% and 50% , respectively). That is, I multiplied across the bottom row, just as I did in the " 60% alloy" row and the " 40% alloy" row, to get the right-hand column's value.

The total amount of copper in the mixture will be the sum of the copper contributed by each of the two alloys that are being put into the mixture. So I'll add the expressions for the amount of copper from each of the alloys, and place the expression for the total amount of copper in the mixture as being between the minimum and the maximum allowable amounts of copper:

13.8 ≤ 0.6 x + (12 − 0.4 x ) ≤ 15

13.8 ≤ 0.2 x + 12 ≤ 15

1.8 ≤ 0.2 x ≤ 3

9 ≤ x ≤ 15

Checking back to my set-up, I see that I chose my variable to stand for the number of pounds that I need to use of the 60% copper alloy. And they'd only asked me for this amount, so I can ignore the other alloy in my answer.

I will need to use between 9 and 15 pounds of the 60% alloy.

Per yoozh, I'm verbose in my answer. You can answer simply as " between 9 and 15 pounds ".

• Solve 3( x − 2) + 4 ≥ 2(2 x − 3)

First I'll multiply through and simplify; then I'll solve:

3( x − 2) + 4 ≥ 2(2 x − 3)

3 x − 6 + 4 ≥ 4 x − 6

3 x − 2 ≥ 4 x − 6

−2 ≥ x − 6            (*)

Why did I move the 3 x over to the right-hand side (to get to the line marked with a star), instead of moving the 4 x to the left-hand side? Because by moving the smaller term, I was able to avoid having a negative coefficient on the variable, and therefore I was able to avoid having to remember to flip the inequality when I divided through by that negative coefficient. I find it simpler to work this way; I make fewer errors. But it's just a matter of taste; you do what works for you.

Why did I switch the inequality in the last line and put the variable on the left? Because I'm more comfortable with inequalities when the answers are formatted this way. Again, it's only a matter of taste. The form of the answer in the previous line, 4 ≥ x , is perfectly acceptable.

As long as you remember to flip the inequality sign when you multiply or divide through by a negative, you shouldn't have any trouble with solving linear inequalities.

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10.2.1: Solving One-Step Inequalities

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Learning Objectives

• Represent inequalities on a number line.
• Use the addition property of inequality to isolate variables and solve algebraic inequalities, and express their solutions graphically.
• Use the multiplication property of inequality to isolate variables and solve algebraic inequalities, and express their solutions graphically.

Introduction

Sometimes there is a range of possible values to describe a situation. When you see a sign that says “Speed Limit 25,” you know that it doesn’t mean that you have to drive exactly at a speed of 25 miles per hour (mph). This sign means that you are not supposed to go faster than 25 mph, but there are many legal speeds you could drive, such as 22 mph, 24.5 mph or 19 mph. In a situation like this, which has more than one acceptable value, inequalities are used to represent the situation rather than equations .

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using an inequality sign. In an inequality, one expression of the inequality can be greater or less than the other expression. Special symbols are used in these statements. The box below shows the symbol, meaning, and an example for each inequality sign.

Inequality Signs

\(\ x \neq y \quad x \text{ is} {\bf\text { not equal }}\text{to } y\).

Example : The number of days in a week is not equal to 9.

\(\ x>y \quad x {\bf\text { is greater than }} y . \text { Example: } 6>3\)

Example : The number of days in a month is greater than the number of days in a week.

\(\ x<y \quad x {\bf\text { is less than }} y\)

Example : The number of days in a week is less than the number of days in a year.

\(\ x \geq y \quad x {\bf\text { is greater than or equal to }} y\)

Example : 31 is greater than or equal to the number of days in a month.

\(\ x \leq y \quad x {\bf\text { is less than or equal to }} y\)

Example : The speed of a car driving legally in a 25 mph zone is less than or equal to 25 mph.

The important thing about inequalities is that there can be multiple solutions. For example, the inequality “31 ≥ the number of days in a month” is a true statement for every month of the year—no month has more than 31 days. It holds true for January, which has 31 days (\(\ 31 \geq 31\)); September, which has 30 days (≥ 30); and February, which has either 28 or 29 days depending upon the year (\(\ 31 \geq 28 \text { and } 31 \geq 29\)).

The inequality \(\ x>y\) can also be written as \(\ y<x\). The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.

Representing Inequalities on a Number Line

Inequalities can be graphed on a number line. Below are three examples of inequalities and their graphs.

Each of these graphs begins with a circle—either an open or closed (shaded) circle. This point is often called the end point of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to (≥) or less than or equal to (≤) . The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution.

The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of \(\ x \geq-3\) shown above, the end point is -3, represented with a closed circle since the inequality is greater than or equal to -3. The blue line is drawn to the right on the number line because the values in this area are greater than -3. The arrow at the end indicates that the solutions continue infinitely.

Solving Inequalities Using Addition & Subtraction Properties

You can solve most inequalities using the same methods as those for solving equations. Inverse operations can be used to solve inequalities. This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the blue box below.

Addition and Subtraction Properties of Inequality

\(\ \text { If } a>b, \text { then } a+c>b+c\)

\(\ \text { If } a>b, \text { then } a-c>b-c\)

Because inequalities have multiple possible solutions, representing the solutions graphically provides a helpful visual of the situation. The example below shows the steps to solve and graph an inequality.

Solve for \(\ x\).

\(\ x+3<5\)

\(\ x<2\)

The graph of the inequality \(\ x<2\) is shown below.

Just as you can check the solution to an equation, you can check a solution to an inequality. First, you check the end point by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions. This can also help you check that your graph is correct.

The example below shows how you could check that \(\ x<2\) is the solution to \(\ x+3<5\).

Check that \(\ x<2\) is the solution to \(\ x+3<5\).

\(\ x<2\) is the solution to \(\ x+3<5\)

The following examples show additional inequality problems. The graph of the solution to the inequality is also shown. Remember to check the solution. This is a good habit to build!

Advanced Example

\(\ \frac{15}{2}+x>-\frac{37}{4}\)

\(\ x>-\frac{67}{4}\)

\(\ x-10 \leq-12\)

\(\ x \leq-2\)

The graph of this solution in shown below. Notice that a closed circle is used because the inequality is “less than or equal to” (≤). The blue arrow is drawn to the left of the point -2 because these are the values that are less than -2.

Check that \(\ x \leq-2\) is the solution to \(\ x-10 \leq-12\).

\(\ x \leq-2\) is the solution to \(\ x-10 \leq 12\).

Solve for \(\ a\).

\(\ a-17>-17\)

\(\ a>0\)

The graph of this solution in shown below. Notice that an open circle is used because the inequality is “greater than” (>). The arrow is drawn to the right of 0 because these are the values that are greater than 0.

Check that \(\ a>0\) is the solution to

\(\ a-17>-17\).

\(\ a>0\) is the solution to \(\ a-17>-17\).

Advanced Question

Solve for \(\ x\): \(\ 0.5 x \leq 7-0.5 x\).

• \(\ x \leq 0\)
• \(\ x>35\)
• \(\ x \leq 7\)
• \(\ x \geq 5\)
• Incorrect. To find the value of \(\ x\), try adding \(\ 0.5 x\) to both sides. The correct answer is \(\ x \leq 7\).
• Correct. When you add \(\ 0.5 x\) to both sides it creates \(\ 1 x\), so \(\ x \leq 7\).

Solving Inequalities Involving Multiplication

Solving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let’s look at what happens to the inequality when you multiply or divide each side by the same number.

When you multiply by a negative number, “reverse” the inequality sign.

Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed in order to keep a true statement.

These rules are summarized in the box below.

Multiplication and Division Properties of Inequality

\(\ \begin{aligned} \text { If } a>b \text { , then } a c>b c \text { , if } c>0\\ \text { If } a>b \text { , then } a c<b c \text { , if } c<0\\ \\ \text { If } a>b \text { , then } \frac{a}{c}>\frac{b}{c} \text { , if } c>0\\ \text { If } a>b \text { , then } \frac{a}{c}<\frac{b}{c} \text { , if } c<0 \end{aligned}\)

Keep in mind that you only change the sign when you are multiplying and dividing by a negative number. If you add or subtract a negative number, the inequality stays the same.

\(\ -\frac{1}{3}>-12 x\)

\(\ x>\frac{1}{36}\)

\(\ 3 x>12\)

\(\ x>4\)

The graph of this solution is shown below.

There was no need to make any changes to the inequality sign because both sides of the inequality were divided by positive 3. In the next example, there is division by a negative number, so there is an additional step in the solution!

\(\ -2 x>6\)

\(\ x<-3\)

Because both sides of the inequality were divided by a negative number, -2, the inequality symbol was switched from > to <. The graph of this solution is shown below.

Solve for \(\ y\): \(\ -10 y \geq 150\)

• \(\ y=-15\)
• \(\ y \geq-15\)
• \(\ y \leq-15\)
• \(\ y \geq 15\)
• Incorrect. While -15 is a solution to the inequality, it is not the only solution. The solution must include an inequality sign. The correct answer is \(\ y \leq-15\).
• Incorrect. This solution does not satisfy the inequality. For example \(\ y=0\), which is a value greater than -15, results in an untrue statement. 0 is not greater than When dividing by a negative number, you must change the inequality symbol. The correct answer is \(\ y \leq-15\).
• Correct. Dividing both sides by -10 leaves \(\ y\) isolated on the left side of the inequality and -15 on the right. Since you divided by a negative number, the ≥ must be switched to ≤.
• Incorrect. Divide by -10, not 10, to isolate the variable. The correct answer is \(\ y \leq-15\).

Solve for \(\ a\): \(\ -\frac{a}{5}<\frac{35}{8}\)

• \(\ a>-\frac{175}{8}\)
• \(\ a<-\frac{175}{8}\)
• \(\ a>-\frac{7}{8}\)
• \(\ a<-\frac{7}{8}\)
• Correct. By multiplying both sides by -5 and flipping the inequality sign from < to >, you found that \(\ a>-\frac{175}{8}\).
• Incorrect. You correctly multiplied by -5, but remember that the inequality sign flips when you multiply by a negative number. The correct response is: \(\ a>-\frac{175}{8}\).
• Incorrect. It looks like you divided both sides by -5. While you remembered to flip the inequality sign correctly, division is not the correct operation here. The correct response is: \(\ a>-\frac{175}{8}\).
• Incorrect. It looks like you divided both sides by -5. Division is not the correct operation here, and remember to flip the inequality sign when you multiply or divide by a negative number. The correct response is: \(\ a>-\frac{175}{8}\).

Solving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. Since inequalities can have multiple solutions, it is customary to represent the solution to an inequality graphically as well as algebraically. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality.

Inequality Calculator

Enter the inequality below which you want to simplify.

The inequality calculator simplifies the given inequality. You will get the final answer in inequality form and interval notation.

Click the blue arrow to submit. Choose "Simplify" from the topic selector and click to see the result in our Algebra Calculator!

Popular Problems

Solve for x 3 - 2 ( 1 - x ) ≤ 2 Solve for x 5 + 5 ( x + 4 ) ≤ 2 0 Solve for x 4 - 3 ( 1 - x ) ≤ 3 Solve for x - x - x + 7 ( x - 2 ) Solve for x 8 x ≤ - 3 2

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Seventh grade math

IXL offers hundreds of seventh grade math skills to explore and learn! Not sure where to start? Go to your personalized Recommendations wall to find a skill that looks interesting, or select a skill plan that aligns to your textbook, state standards, or standardized test.

A. Integers

• 1 Understanding integers
• 2 Integers on number lines
• 3 Graph integers on horizontal and vertical number lines
• 4 Understanding absolute value
• 5 Absolute value and opposite integers
• 6 Quantities that combine to zero: word problems
• 7 Compare and order integers
• 8 Integer inequalities with absolute values

B. Operations with integers

• 1 Add integers using counters
• 2 Add integers using number lines
• 3 Integer addition rules
• 4 Add integers
• 5 Add three or more integers
• 6 Subtract integers using counters
• 7 Subtract integers using number lines
• 8 Integer subtraction rules
• 9 Subtract integers
• 10 Integer addition and subtraction rules
• 11 Add and subtract integers using counters
• 12 Add and subtract integers
• 13 Complete addition and subtraction equations with integers
• 14 Add and subtract integers: word problems
• 15 Understand multiplying by a negative integer using a number line
• 16 Integer multiplication rules
• 17 Multiply integers
• 18 Integer division rules
• 19 Equal quotients of integers
• 20 Divide integers
• 21 Integer multiplication and division rules
• 22 Multiply and divide integers
• 23 Complete multiplication and division equations with integers
• 24 Add, subtract, multiply, and divide integers
• 25 Evaluate numerical expressions involving integers

C. Decimals

• 1 Decimal numbers review
• 2 Compare and order decimals
• 3 Round decimals

D. Operations with decimals

• 1 Add and subtract decimals
• 2 Add and subtract decimals: word problems
• 3 Multiply decimals
• 4 Multiply decimals and whole numbers: word problems
• 5 Divide decimals
• 6 Divide decimals by whole numbers: word problems
• 7 Estimate sums, differences, and products of decimals
• 8 Add, subtract, multiply, or divide two decimals
• 9 Add, subtract, multiply, and divide decimals: word problems
• 10 Multi-step inequalities with decimals
• 11 Maps with decimal distances
• 12 Evaluate numerical expressions involving decimals

E. Number theory

• 1 Prime factorization
• 2 Greatest common factor
• 3 Least common multiple
• 4 GCF and LCM: word problems

F. Fractions

• 1 Understanding fractions: word problems
• 2 Equivalent fractions
• 3 Write fractions in lowest terms
• 4 Fractions: word problems with graphs and tables
• 5 Least common denominator
• 6 Compare and order fractions
• 7 Compare fractions: word problems
• 8 Convert between mixed numbers and improper fractions
• 9 Compare mixed numbers and improper fractions
• 10 Round mixed numbers

G. Operations with fractions

• 1 Add and subtract fractions
• 2 Add and subtract fractions: word problems
• 3 Add and subtract mixed numbers
• 4 Add and subtract mixed numbers: word problems
• 5 Inequalities with addition and subtraction of fractions and mixed numbers
• 6 Estimate sums and differences of mixed numbers
• 7 Multiply two fractions using models
• 8 Multiply fractions
• 9 Multiply mixed numbers
• 10 Multiply fractions and mixed numbers: word problems
• 11 Multiplicative inverses
• 12 Divide fractions
• 13 Divide mixed numbers
• 14 Divide fractions and mixed numbers: word problems
• 15 Estimate products and quotients of fractions and mixed numbers
• 16 Add, subtract, multiply, and divide fractions and mixed numbers: word problems
• 17 Maps with fractional distances
• 18 Evaluate numerical expressions involving fractions

H. Rational numbers

• 1 Convert fractions or mixed numbers to decimals
• 2 Convert decimals to fractions or mixed numbers
• 3 Convert between decimals and fractions or mixed numbers
• 4 Identify rational numbers
• 5 Classify rational numbers using a diagram
• 6 Classify rational numbers
• 7 Absolute value of rational numbers
• 8 Compare rational numbers
• 9 Put rational numbers in order
• 10 Classify real numbers

I. Operations with rational numbers

• 1 Add and subtract positive and negative decimals
• 2 Add and subtract positive and negative fractions
• 3 Add and subtract rational numbers
• 4 Rational numbers: find the sign
• 5 Apply addition and subtraction rules
• 6 Identify quotients of rational numbers: word problems
• 7 Multiply and divide positive and negative decimals
• 8 Multiply and divide positive and negative fractions
• 9 Multiply and divide rational numbers
• 10 Apply multiplication and division rules
• 11 Multi-step word problems with positive rational numbers

J. Exponents

• 1 Understanding exponents
• 2 Evaluate powers
• 3 Solve equations with variable exponents
• 4 Powers with negative bases
• 5 Powers with decimal and fractional bases
• 6 Powers of ten
• 7 Evaluate numerical expressions involving exponents
• 8 Scientific notation
• 9 Compare numbers written in scientific notation

K. Square roots

• 1 Square roots of perfect squares
• 2 Estimate square roots

L. Ratios, rates, and proportions

• 1 Understanding ratios
• 2 Identify equivalent ratios
• 3 Write an equivalent ratio
• 4 Equivalent ratios: word problems
• 5 Unit rates
• 6 Calculate unit rates with fractions
• 7 Compare ratios: word problems
• 8 Compare rates: word problems
• 9 Do the ratios form a proportion?
• 10 Do the ratios form a proportion: word problems
• 11 Solve proportions
• 12 Solve proportions: word problems
• 13 Estimate population size using proportions

M. Coordinate plane

• 1 Coordinate plane review
• 2 Quadrants and axes
• 3 Follow directions on a coordinate plane
• 4 Distance between two points

N. Proportional relationships

• 1 Find the constant of proportionality from a table
• 2 Write equations for proportional relationships from tables
• 3 Identify proportional relationships by graphing
• 4 Find the constant of proportionality from a graph
• 5 Write equations for proportional relationships from graphs
• 6 Identify proportional relationships from graphs and equations
• 7 Identify proportional relationships from tables
• 8 Complete a table and graph a proportional relationship
• 9 Graph a proportional relationship: word problems
• 10 Interpret graphs of proportional relationships
• 11 Write and solve equations for proportional relationships

O. Percents

• 1 What percentage is illustrated?
• 2 Convert between percents, fractions, and decimals
• 3 Compare percents to fractions and decimals
• 4 Estimate percents of numbers
• 5 Solve percent problems using strip models
• 6 Percents of numbers and money amounts
• 7 Percents of numbers: word problems
• 8 Solve percent equations
• 9 Solve percent equations: word problems
• 10 Percent of change
• 11 Percent of change: word problems
• 12 Percent of change: find the original amount word problems
• 13 Percent error: word problems

P. Consumer math

• 1 Add, subtract, multiply, and divide money amounts: word problems
• 2 Price lists
• 3 Unit prices
• 4 Unit prices with unit conversions
• 5 Unit prices: find the total price
• 6 Percent of a number: tax, discount, and more
• 7 Which is the better coupon?
• 8 Find the percent: tax, discount, and more
• 9 Sale prices: find the original price
• 10 Multi-step problems with percents
• 11 Estimate tips
• 12 Simple interest
• 13 Compound interest

Q. Units of measurement

• 1 Compare and convert customary units
• 2 Mixed customary units
• 3 Compare and convert metric units
• 4 Convert between customary and metric systems
• 5 Celsius and Fahrenheit temperatures

R. Expressions

• 1 Write variable expressions: one operation
• 2 Write variable expressions: two or three operations
• 3 Write variable expressions: word problems
• 4 Evaluate linear expressions
• 5 Evaluate multi-variable expressions
• 6 Evaluate absolute value expressions
• 7 Evaluate nonlinear expressions
• 8 Identify terms and coefficients
• 9 Sort factors of variable expressions

S. Equivalent expressions

• 1 Properties of addition and multiplication
• 2 Simplify expressions by combining like terms: with algebra tiles
• 3 Simplify expressions by combining like terms
• 4 Multiply using the distributive property: area models
• 5 Multiply using the distributive property
• 6 Write equivalent expressions using properties
• 7 Add and subtract linear expressions
• 8 Add and subtract like terms: with exponents
• 9 Factor linear expressions: area models
• 10 Factors of linear expressions
• 11 Identify equivalent linear expressions using algebra tiles
• 12 Identify equivalent linear expressions I
• 13 Identify equivalent linear expressions II
• 14 Identify equivalent linear expressions: word problems

T. One-variable equations

• 1 Which x satisfies an equation?
• 2 Write an equation from words
• 3 Solve equations using properties
• 4 Model and solve equations using algebra tiles
• 5 Write and solve equations that represent diagrams
• 6 Solve one-step equations
• 7 Solve two-step equations without parentheses
• 8 Solve two-step equations with parentheses
• 9 Solve two-step equations
• • New! Solve two-step equations with fractions
• 10 Choose two-step equations: word problems
• 11 Solve two-step equations: word problems
• 12 Solve equations involving like terms
• 13 Solve equations: complete the solution

U. One-variable inequalities

• 1 Solutions to inequalities
• 2 Graph inequalities on number lines
• 3 Write inequalities from number lines
• 4 Solve one-step inequalities
• 5 Graph solutions to one-step inequalities
• 6 One-step inequalities: word problems
• 7 Solve two-step inequalities
• 8 Graph solutions to two-step inequalities

V. Sequences

• 1 Identify arithmetic and geometric sequences
• 2 Arithmetic sequences
• 3 Geometric sequences
• 4 Sequences: mixed review
• 5 Sequences: word problems
• 6 Evaluate variable expressions for sequences
• 7 Write variable expressions for arithmetic sequences
• 1 Find the slope from a graph
• 2 Find the slope from two points
• 3 Find a missing coordinate using slope
• 4 Graph a line using slope
• 5 Constant rate of change
• 6 Rate of change: tables
• 7 Rate of change: graphs

X. Two-variable equations

• 1 Identify independent and dependent variables
• 2 Find a value using two-variable equations
• 3 Evaluate two-variable equations: word problems
• 4 Complete a table for a two-variable relationship
• 5 Write a two-variable equation
• 6 Identify the graph of an equation
• 7 Graph a two-variable equation
• 8 Interpret a graph: word problems
• 9 Write an equation from a graph using a table
• 10 Write a linear equation from a graph

Y. Lines and angles

• 1 Name, measure, and classify angles
• 2 Lines, line segments, and rays
• 3 Parallel, perpendicular, and intersecting lines
• 4 Identify complementary, supplementary, vertical, and adjacent angles
• 5 Find measures of complementary, supplementary, vertical, and adjacent angles
• 6 Write and solve equations using angle relationships
• 7 Identify alternate interior and alternate exterior angles
• 8 Transversals of parallel lines: name angle pairs
• 9 Transversals of parallel lines: find angle measures
• 10 Find lengths and measures of bisected line segments and angles

Z. Two-dimensional figures

• 1 Identify and classify polygons
• 2 Classify triangles
• 3 Triangle inequality
• 4 Identify trapezoids
• 5 Classify quadrilaterals I
• 6 Classify quadrilaterals II
• 7 Graph triangles and quadrilaterals
• 8 Find missing angles in triangles
• 9 Find missing angles in triangles using ratios
• 10 Find missing angles in quadrilaterals I
• 11 Find missing angles in quadrilaterals II
• 12 Interior angles of polygons
• 13 Parts of a circle
• 14 Central angles of circles

AA. Three-dimensional figures

• 1 Bases of three-dimensional figures
• 2 Nets of three-dimensional figures
• 3 Front, side, and top view
• 4 Cross sections of three-dimensional figures

BB. Perimeter and area

• 1 Perimeter
• 2 Area of rectangles and parallelograms
• 3 Area of triangles and trapezoids
• 4 Area and perimeter: word problems
• 5 Circumference of circles
• 6 Area of circles
• 7 Circles: word problems
• 8 Area of semicircles and quarter circles
• 9 Semicircles: calculate area, perimeter, radius, and diameter
• 10 Quarter circles: calculate area, perimeter, and radius
• 11 Area of compound figures made of rectangles
• 12 Area of compound figures with triangles
• 13 Area of compound figures with triangles, semicircles, and quarter circles
• 14 Area between two shapes

CC. Surface area and volume

• 1 Surface area of cubes and prisms
• 2 Surface area of pyramids
• • New! Lateral area of prisms and pyramids
• 3 Surface area of cylinders
• 4 Volume of cubes and prisms
• 5 Volume of cubes and rectangular prisms: word problems
• 6 Volume of prisms: advanced
• 7 Volume of pyramids
• 8 Volume of cylinders

DD. Scale drawings

• 1 Scale drawings of polygons
• 2 Scale drawings: word problems
• 3 Scale drawings: scale factor word problems
• 4 Perimeter and area: changes in scale

EE. Transformations

• 1 Identify reflections, rotations, and translations
• 2 Translations: graph the image
• 3 Translations: find the coordinates
• 4 Reflections over the x- and y-axes: graph the image
• 5 Reflections over the x- and y-axes: find the coordinates
• 6 Reflections: graph the image
• 7 Reflections: find the coordinates
• 8 Rotations: graph the image
• 9 Rotations: find the coordinates

FF. Congruence and similarity

• 1 Similar and congruent figures
• 2 Congruence statements and corresponding parts
• 3 Side lengths and angle measures of congruent figures
• 4 Side lengths and angle measures of similar figures
• 5 Similar figures and indirect measurement

GG. Data and graphs

• 1 Interpret line plots
• 2 Create line plots
• 3 Create and interpret line plots with fractions
• 4 Interpret stem-and-leaf plots
• 5 Create stem-and-leaf plots
• 6 Interpret bar graphs
• 7 Create bar graphs
• 8 Interpret histograms
• 9 Create histograms
• 10 Create frequency charts
• 11 Interpret circle graphs
• 12 Circle graphs and central angles
• 13 Box plots

HH. Statistics

• 1 Calculate mean, median, mode, and range
• 2 Interpret charts and graphs to find mean, median, mode, and range
• 3 Mean, median, mode, and range: find the missing number
• 4 Changes in mean, median, mode, and range
• 5 Calculate mean absolute deviation
• 6 Calculate quartiles and interquartile range
• 7 Identify an outlier
• 8 Identify representative, random, and biased samples
• • New! Make inferences from multiple samples
• 9 Compare populations using measures of center and spread

II. Probability

• 1 Probability of simple events
• 2 Probability of simple events and opposite events
• 3 Probability of mutually exclusive events and overlapping events
• 4 Experimental probability
• 5 Make predictions using experimental probability
• 6 Use collected data to find probabilities and make predictions
• 7 Make predictions using theoretical probability
• 8 Compound events: find the number of outcomes
• 9 Compound events: find the number of sums
• 10 Find the number of outcomes: word problems
• 11 Probability of compound events
• 12 Which simulation represents the situation?
• 13 Identify independent and dependent events
• 14 Probability of independent and dependent events

Seventh grade lessons

These lessons help you brush up on important math topics and prepare you to dive into skill practice!

• Absolute value
• Additive inverses
• Adding and subtracting integers
• Multiplying and dividing integers
• Adding decimals
• Subtracting decimals
• Multiplying decimals by powers of 10
• Multiplying decimals by whole numbers
• Multiplying decimals
• Dividing decimals by powers of 10
• Dividing decimals by whole numbers
• Dividing decimals
• Rational numbers
• Comparing rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers

Exponents and powers of 10

• Powers of 10

Properties and mixed operations

• The distributive property
• Order of operations

Percents, ratios, and rates

• Unit rates with fractions
• Simple interest
• Percent change

Proportions and proportional relationships

• Proportions
• Proportional relationships
• Constant of proportionality
• Expressions and equations
• Writing algebraic expressions
• Evaluating expressions
• Equivalent expressions
• Simplifying expressions
• Expanding expressions
• Factoring expressions
• Solving equations
• Multi-step equations
• Independent and dependent variables
• Inequalities on a number line
• Solving inequalities
• Area of triangles
• Area of parallelograms
• Area of rhombuses
• Area of trapezoids
• Parts of a circle
• Area of circles
• Circumference of circles
• Complementary angles
• Supplementary angles
• Adjacent angles
• Vertical angles
• What is surface area?
• Surface area formulas
• Volume formulas
• Volume of rectangular prisms
• Volume of prisms
• Volume of cylinders
• Volume of pyramids
• Cross sections
• Triangle inequality theorem

Statistics and probability

• Mean, median, mode, and range
• Mean absolute deviation
• Box and whisker plots
• Statistical questions
• Probability
• Compound probability

Summer and Fall classes are open for enrollment. Schedule today !

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South Africa's future depends on an 'unholy alliance' solving its real-world problems before time runs out

Analysis South Africa's future depends on an 'unholy alliance' solving its real-world problems before time runs out

From Constantia Nek, a pass across the spine of rugged mountain that runs all the way to the Cape of Good Hope, day trippers are afforded a stunning view across centuries-old vineyards all the way to Cape Town's city centre.

A quick jaunt down the other side past Michelin-star restaurants and gated housing estates leads to Hout Bay, a thriving fishing port on the city's Atlantic coast.

There, butted up against architect-designed trophy houses, sits Imizamu Yethu, an 18-hectare "informal settlement" of mostly corrugated iron shacks that is home to almost 35,000 residents.

The contrast couldn't be more incongruous. But it is a stark illustration of the unachieved ambitions and dashed hopes that accompanied the end of apartheid 30 years ago and stands as a monument to the failure of the African National Congress (ANC) to deliver meaningful change after three decades of uninterrupted rule.

It's not an isolated case.

Right across the country, townships have sprung up and existing ones have expanded as African workers have flooded into urban and regional centres, graphically hammering home the message that while forced racial segregation may have ended, wealth still is largely delineated along racial lines.

The rich, most of whom are white, live very, very well. The rest struggle to eat regularly.

A fortnight ago, the ANC's iron grip on power was smashed, an idea once considered unthinkable. For decades, it was considered that democracy played out through the party framework rather than through the electoral process.

But the country's crumbling infrastructure, endless revelations of corruption at the highest levels and an inability to deliver basic services saw the ANC garner just 42 per cent of the national vote.

After a fortnight of post-poll haggling, the ANC late last week announced a coalition with the Democratic Alliance (DA), which it has long accused of pandering to the interests of white South Africans, and two minor parties.

Under the deal, Cyril Ramaphosa will remain as president with the DA nominating the deputy.

In reality, the ANC had little choice. Financial markets were rattled by the prospect of the alternative; an alliance with the two major left-wing parties which advocate the resumption of mines and land without compensation.

The DA, supported by white and mixed-race South Africans, has long controlled the Western Cape province that includes Cape Town and has a far better track record on service delivery than its rivals in other provinces.

For many voters, it was the outcome for which they were hoping; a coalition government held to greater accountability.

Economy on the slide

There aren't too many statistics that put South Africa in a competitive global position.

But crime is one. Cape Town slots in at 10th place when it comes to murders, with Mexican cities occupying seven of the top nine.

Then there's unemployment. At 40 per cent, and with a social security net that offers minimal support, many South Africans have little option but to resort to crime simply to put food on the table.

Most suburban houses now hide behind high walls with barbed wire and electric fencing, festooned with placards denoting which armed response company has been contracted for protection.

While many would point to these developments as a sign of a deteriorating society, in reality, there has been little change in the past 30 years.

The main point of difference is that crime is now more visible. During the apartheid era, Africans were prohibited from moving beyond designated areas, controlled by a violent regime determined to brutally suppress any form of opposition.

More heartening was that the recent elections were unhindered by any major incidents, apart from long lines at polling stations, and campaigning largely was peaceful. Even the results largely were accepted until last week when former president Jacob Zuma claimed the elections were rigged.

Uncertainty over the results, however, temporarily dented international confidence in the currency which has been on the slide for decades.

On my first visit almost 40 years ago, I was stunned to discover the rand had slumped from parity with the Aussie and was delivering two rand to the dollar. Last week, it was yielding just shy of 13 rand.

Former finance minister and once senior ANC official Trevor Manuel, now the chair of investment group Old Mutual, argues global investors have dumped about 1 trillion rand worth of South African shares in the past decade.

"This money is being redirected to competing markets that appear to be on a more sound governance and regulatory footing," he wrote in this year's annual report.

A vocal critic of the government in recent years, Manuel has frequently lashed out about rising corruption that has seen vast amounts of taxpayer funds squandered.

When friends fall out

Former president Jacob Zuma loomed large over the most recent polls.

The octogenarian, who regularly has pleaded ill health during the many court proceedings against him, assumed a leading role in the relatively new MK Party.

Having been previously jailed for contempt, his status as a future politician is under a cloud. And he continues to fight multiple corruption allegations including taking bribes from French group Thales in 2000 over a defence contract.

Even his surprise elevation to lead the MK Party has been subject to legal action from the former leader who claims Zuma's daughter penned his "resignation letter".

Despite that, the MK Party did surprisingly well, garnering 14.6 per cent of the vote, and it has joined forces with the Economic Freedom Fighters as the official opposition with almost 30 per cent of seats in the National Assembly.

As breakaways from the ANC, they are largely responsible for the collapse in support for the party that has dominated South African politics for the past 30 years. But both are now wedded to a policy of forced resumption of assets, including mines and farms.

Adding to the disquiet, Zuma's party last week launched proceedings in the country's highest court, alleging vote rigging, a claim the court quickly dismissed.

Over the weekend, the former president lashed out against the ANC liaison with the DA, labelling it a "white-led", "unholy alliance".

Zuma and Ramaphosa are archenemies. The former president managed to strike a blow against Ramaphosa four years ago after it was revealed that two thieves had broken into his farmhouse and stolen $US580,000 ($870,000) in cash that had been stuffed into a sofa .

Exactly why the money was there has never been explained but Ramaphosa came close to losing his position over the scandal.

New battles, old territory

On Sunday, Zuma raised the spectre of the age-old fight against white colonialism and apartheid in his attack on the ANC and his successor Ramaphosa.

The new Patriotic Front would operate both within and outside parliament, his spokesman said, reading from a prepared statement, where he placed his nemesis alongside the apartheid-era Afrikaner leaders.

It may be a clever domestic political ploy but threats to seize property will do little to instil confidence in the country's ability to compete globally. Further deterioration of the currency will only exacerbate the extreme cost-of-living pressures facing South African households and businesses.

It also ignores history. The DA's forebears once were labelled left wing for their efforts to overthrow apartheid.

Ultimately, however, dissatisfaction and unrest within the electorate have been fuelled by the abject failure to provide even basic services, much of which occurred during Zuma's decade-long presidency.

Delivery of new housing, education and jobs fell well short of promises. And the country's crumbling infrastructure has seen crippling power shortages during the past year as aging, coal-fired generators have been shut but not replaced.

Oddly, the "load shedding", which was really blackouts, stopped shortly before the election was announced.

For a country with so much promise and natural bounty, so culturally rich and diverse, time is running short to find solutions to real-world problems.

Otherwise, South Africa may well become another textbook case on what happens when inequality is allowed to fester.

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What is the 3-body problem, and is it really unsolvable?

The three-body problem is a physics conundrum that has boggled scientists since Isaac Newton's day. But what is it, why is it so hard to solve and is the sci-fi series of the same name really possible?

A rocket launch. Our nearest stellar neighbor. A Netflix show. All of these things have something in common: They must contend with the "three-body problem." But exactly what is this thorny physics conundrum?

The three-body problem describes a system containing three bodies that exert gravitational forces on one another. While it may sound simple, it's a notoriously tricky problem and "the first real worry of Newton," Billy Quarles , a planetary dynamicist at Valdosta State University in Georgia, told Live Science.

In a system of only two bodies, like a planet and a star, calculating how they'll move around each other is fairly straightforward: Most of the time, those two objects will orbit roughly in a circle around their center of mass, and they'll come back to where they started each time. But add a third body, like another star, and things get a lot more complicated. The third body attracts the two orbiting each other, pulling them out of their predictable paths .

The motion of the three bodies depends on their starting state — their positions, velocities and masses. If even one of those variables changes, the resulting motion could be completely different. 

"I think of it as if you're walking on a mountain ridge," Shane Ross , an applied mathematician at Virginia Tech, told Live Science. "With one small change, you could either fall to the right or you could fall to the left. Those are two very close initial positions, and they could lead to very different states."  

There aren't enough constraints on the motions of the bodies to solve the three-body problem with equations, Ross said. 

Related: Cosmic 'superbubbles' might be throwing entire galaxies into chaos, theoretical study hints

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But some solutions to the three-body problem have been found. For example, if the starting conditions are just right, three bodies of equal mass could chase one another in a figure-eight pattern. Such tidy solutions are the exception, however, when it comes to real systems in space.

Certain conditions can make the three-body problem easier to parse. Consider Tatooine , Luke Skywalker's fictional home world from "Star Wars" — a single planet orbiting two suns. Those two stars and the planet make up a three-body system. But if the planet is far enough away and orbiting both stars together, it's possible to simplify the problem. 

"When it's the Tatooine case, as long as you're far enough away from the central binary, then you think of this object as just being a really fat star," Quarles said. The planet doesn't exert much force on the stars because it's so much less massive, so the system becomes similar to the more easily solvable two-body problem. So far, scientists have found more than a dozen Tatooine-like exoplanets , Quarles told Live Science.

But often, the orbits of the three bodies never truly stabilize, and the three-body problem gets "solved" with a bang. The gravitational forces could cause two of the three bodies to collide, or they could fling one of the bodies out of the system forever — a possible source of "rogue planets" that don't orbit any star , Quarles said. In fact, three-body chaos may be so common in space that scientists estimate there may be 20 times as many rogue planets as there are stars in our galaxy.

When all else fails, scientists can use computers to approximate the motions of bodies in an individual three-body system. That makes it possible to predict the motion of a rocket launched into orbit around Earth, or to predict the fate of a planet in a system with multiple stars.

— 'Mathematically perfect' star system being investigated for potential alien technology

— How common are Tatooine worlds?

— Mathematicians find 12,000 new solutions to 'unsolvable' 3-body problem

With all this tumult, you might wonder if anything could survive on a planet like the one featured in Netflix's "3 Body Problem," which — spoiler alert — is trapped in a chaotic orbit around three stars in the Alpha Centauri system , our solar system 's nearest neighbor. 

"I don't think in that type of situation, that's a stable environment for life to evolve," Ross said. That's one aspect of the show that remains firmly in the realm of science fiction.

Skyler Ware is a freelance science journalist covering chemistry, biology, paleontology and Earth science. She was a 2023 AAAS Mass Media Science and Engineering Fellow at Science News. Her work has also appeared in Science News Explores, ZME Science and Chembites, among others. Skyler has a Ph.D. in chemistry from Caltech.

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The trump rallying cry that’s also a math problem.

His supporters can’t quite agree on whether to call him the 45th president or the 47th president.

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By Jess Bidgood

Donald Trump and his supporters can’t quite seem to agree: Should he be labeled the 45th president, the 47th president or both?

As he takes the stage at rallies, he is sometimes introduced with both titles, making it almost sound as if he were two different people.

Last Friday, he was treated to a birthday celebration in West Palm Beach, Fla., by a group of supporters called Club 47 USA — which used to be called Club 45 USA, but changed its name. The group’s website, however, is still club45usa.com .

And the day before, Republican senators regaled him with a birthday cake containing two sets of numbered candles — a 45 and a 47. (According to a video posted on social media by one of his campaign accounts, only the 45 appeared to be lit when Trump received the cake.)

Trump was, of course, the country’s 45th president, and now might become its 47th — a number he has plastered all over his campaign’s infrastructure, including the name of his joint fund-raising committee, a URL for his fund-raising website and his grass-roots organizing program.

There may well be a strategy at play here. Trump has not been elected the 47th president, and his embrace of the figure came well before it was even clear that he would be his party’s nominee — making it an attempt to burnish the air of inevitability he often tries to project.

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How to Solve Your Company’s Toughest Problems

A conversation with Harvard Business School professor Frances Frei on how to solve any problem in five clear steps.

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You’ve likely heard the phrase, “Move fast and break things.” But Harvard Business School professor Frances Frei says speed and experimentation are not enough on their own. Instead, she argues that you should move fast and fix things. (That’s also the topic and title of the book she coauthored with Anne Morriss .)

In this episode, Frei explains how you can solve any problem in five clear steps. First, she says, start by identifying the real problem holding you back. Then move on to building trust and relationships, followed by a narrative for your solution — before you begin implementing it.

Key episode topics include: leadership, strategy execution, managing people, collaboration and teams, trustworthiness, organizational culture.

HBR On Leadership curates the best case studies and conversations with the world’s top business and management experts, to help you unlock the best in those around you. New episodes every week.

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HANNAH BATES: Welcome to HBR on Leadership , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock the best in those around you.

Maybe you’ve heard the phrase, “move fast and break things.” It refers to a certain approach for rapid innovation that was popularized in Silicon Valley and invoked by many tech firms. But Harvard Business School professor Frances Frei says that speed and experimentation are not enough on their own. Instead, Frei argues that you should “move fast and fix things.” That’s the topic and title of the book she co-authored with Anne Morriss.

In this episode, Harvard Business Review’s editorial audience director Nicole Smith sits down with Frei to discuss how you can solve any problem in five quick steps. You’ll learn how to start by uncovering your true problem. Then, move on to build trust, relationships, and a narrative for your solution before you dive in on the actual work of implementing your fix.

This conversation was originally part of HBR’s “Future of Business” virtual conference in November 2023. Here it is.

FRANCES FREI: So, I would love to talk to you about how to move fast and fix things. And I’ll tell you the reason that Anne and I wrote this book – and it’s really a quest we’ve been on – is that Mark Zuckerberg, in his IPO for Facebook, famously said, “we’re going to move fast and break things.” And the problem with that is that it gave the world a false trade-off. It convinced so many of us that you could either move fast and break things or you could take care of people, one or the other. And we have found that there is a third, much better way. And that is, we can move even faster if we fix things along the way. And so, that’s what I’d love to talk to you about right now. And the way that we think about this is that if you want to move fast and fix things, we have to do it on a foundation of trust. And so, the first thing to do is to experience high trust. And we’re going to talk about how to build trust. But the way we see the world can be described in this grid. And in the presence of trust, we can move really fast. That’s how we move fast and fix things. We call it accelerating excellence. It’s only when we’re in the presence of low trust that we move fast and break things, or what we call being reckless disruption. And as I said, so many organizations are afraid of reckless disruption that they actually end up in this state of responsible stewardship, which is really just going slowly. And so, we wrote the book to get those that are in responsible stewardship to realize that we could go across the way to accelerating excellence. And we didn’t have to go down to reckless disruption. So, the way that we think about this, and it’s the way we wrote the book, is that there’s a five-step plan to do it. We organized the book for days of the week. We think that the metabolic rate of organizations can be improved significantly and that many, many hard problems can be solved in just one week. So, we wrote the book in the structure of a week. Step one is we have to find our real problem, that if we’re… for far too many of us, we’re addressing the symptom and not the cause. At any problem, there’s going to be trust broken at the bottom of it. And we’re going to solve for trust. We’re then going to learn how to get more perspectives to make our plans even better. Learn how to tell a narrative that works. And then, and only then, on Friday, do we get to go as fast as we can. And what typically happens in the move fast and break things is that we move Friday too forward in the week. So, our goal is to put ourselves in a position to move fast. And you have to wait till Friday to do that. So, what do I mean by finding the real problem? Most of us, a problem gets presented as a symptom. So, I’ll give you a recent example that got presented to me and Anne. We got called by a company. And they said, we’re having a gender problem. Will you come in and help us? And we’ve been able to help many organizations solve gender problems. So, we go in there. And we just wanted to make sure that they really did have a gender problem. The symptoms were super clear. There were no women at the top of the organization. Not very many women were coming into the organization. And great women were leaving the organization. So, they had… it looked like a gender problem. But it took, I don’t know, an hour. It took 60 minutes, certainly not even all of Monday, to uncover that their actual problem was not a gender problem. Their actual problem was a communication problem. And if we did all of the things that we know exist in our gender tool kit on how to fix gender, that would have all been wasted effort. But instead, what we found out is that the founders of this organization, and they were two cofounders, and they were very similar to each other, and they’d worked together and known each other for decades. They had a really uncomfortably and aggressively direct communication style. That communication style repelled all women and most men. So yes, the symptoms were gender. But oh, my goodness, the cause was that the two founders were succumbing to a problem many of us succumb to, which is, we were treating others as we like to be treated. They loved to be treated with aggressively direct communication. But nobody else loved it. And when we simply confronted them with that and taught them that instead of treating others as you want to be treated, now it’s a puzzle. Find out how they want to be treated, and treat them that way. Gets fixed. And all of a sudden, women and lots of other men are flowing to the organization. So, Monday… and we take a whole day for this. Let’s make sure we’re solving the real problem. And symptoms are rarely the cause. So, we just want to do some due diligence, some due diligence there. Once we know we’re solving for the real problem, there’s going to be trust broken down somewhere in the… amidst the problem. Well, very fortunately, we now understand trust super well. If I’m going to earn your trust, you will have an involuntary reaction of trusting me if you experience my authenticity, logic, and empathy all at the same time. When these three things are present, you will trust me. But if any one of these three is missing, you will not trust me. And here’s the catch. If trust is broken, and we know it’s only ever broken for one of these three reasons, we need to know which of the three, because the prescriptions to solve a broken authenticity pillar versus logic pillar versus empathy pillar, they’re entirely different from one another. So, you can think about rebuilding trust. It’s just a matching game. Know which one is at stake. And then bring in the curated prescription for that. There is a myth about trust that it takes a lifetime to build and a moment to destroy. And then you can never rebuild it. None of those things are true, that we can actually build trust very quickly when we understand the architecture of it. We can rebuild it quickly and just as strong as it was before. So, this notion that trust is a Faberge egg, it’s catchy and not true. Trust is being rebuilt all the time. But we want to do it with a deep understanding of the stable architecture. So, Tuesday takes all day. We solve for trust. On Wednesday, we call Wednesday making new friends. And what we mean by that is whichever collection of people you bring to the table who are the people that maybe are on your senior team or the people that you bring to the table to solve problems. And here, I’ve represented a table. And there’s eight check marks for eight seats. I encourage you to bring four extra chairs to that table. If you have eight seats, bring four extra chairs. Point to the extra chairs and ask yourself, who’s not here? Who has a stake in our problem who’s not represented at the table? I was recently in a conversation with our senior colleagues at the Harvard Business School. And we were talking about how to do junior faculty development. And we came up with what we thought were great ideas. And then we looked around and we were like, Oh, my goodness, there’s no junior faculty here. How on Earth do we know if these are good ideas? So, we got the empty seats. We invited people in. And sure enough, the junior faculty helped improve our plans dramatically. The equivalent of that always happens. So, on Wednesday, we want to make new friends. So, one is inviting them into the room. But then the second part is, how do you make sure that their voices are heard? And what we need to do is that when someone comes to the room, they’re going to be awfully tempted to say things that they think we want to hear. They’re going to be awfully tempted to conform to what we’re already saying. So, what we need to do is learn how to be inclusive of their unique voices. And the way we do that is by going through this four-step progressive process, which is, first, we have to make sure they feel safe and that they feel… they’re going to feel physically and emotionally safe, I’m sure, but that they feel psychologically safe. And that’s a shout-out to Amy Edmondson and all of her beautiful work there. But we have to make sure that we feel safe. Once we feel safe, then it’s our job to make sure that the new voices feel welcome. You can think of that as table stakes. Then when we’re doing is we’re really trying to move people up the inclusion dial. And here, this is when it really starts to make a big difference. And now what we want to do is make sure that they feel celebrated for their unique contribution. And so, what we’re doing is moving them up the inclusion dial. Now, here’s why that’s kind of hard. Most of us tend to celebrate sameness. And here, I’m asking you to celebrate uniqueness. And what I mean by celebrating sameness is that for the most part, like, when I watch my students in class, if one student says something, and then another student was going to say that, after class, they go and seek out the first person. And they’re like, you’re awesome. You said what I was going to say. They didn’t realize this. They’re celebrating sameness. They’re encouraging sameness. So, what I do is I advise my students to not share that verbal treat, that what we playfully refer to as a Scooby snack. Don’t share that Scooby snack for when somebody says something you were going to say. Share it for when somebody says something you could never have said on your own, and that it comes from their lived experience and learned experience, and how they metabolize successes and failures, and their ambition, if they’re lucky enough to have neurodiversity, their worldview, all of that. It’s a beautiful cocktail. Wait till they say something that comes uniquely from all of that. Celebrate that. When we celebrate uniqueness, that’s when we get the blossoming of the perspectives. And what we want to do to make somebody really feel included is we celebrate them when they are in our presence. But if you really want somebody to feel included, and we bring folks into the room for this, make sure that you champion them when they’re in the absence. So, let’s not just ask the junior faculty to come along. Or if it’s a senior team, and it’s mostly men, and the board of directors is coming in, and we’re like, oh, goodness. Let’s make sure we can show some women too. So, we bring some women along. We celebrate them in our presence. Let’s make sure that we champion them in our absence as well, which is celebrate their uniqueness in our presence and champion them in rooms that they’re not yet allowed into in their absence. So that’s Wednesday. Let’s make new friends. Let’s include their voices. Let’s champion those new voices in their absence. Thursday, we tell a good story. And stories have three parts to it: past, present, and future. It is really important – if you’re going to change something, if you’re going to fix something, it is critical to honor the past. People that were here before us, if they don’t feel like we see the past, we see them, we’re honoring the past, I promise you, they’re going to hold us back. And they’re going to be like The Godfather movie and keep pulling us back. So, we have to honor the past with clear eyes, both the good part of the past and the bad part of the past. Then we have to answer the question, why should we change now? Like, why shouldn’t we change maybe next week, maybe the week after, maybe the month after, maybe next year? So, it’s really important that we give a clear and compelling change mandate that answers the question, why now? Why not in a little while? I find that if you’re a retailer, and you have the metaphor of Walmart just opened up next door, clear, compelling. We have to… that should be our metaphor. How can we be, with as crisp of a language, clear and compelling about why now? And then we’ve honored the past. We have a clear and compelling change mandate. You want people to follow us in the improved future, we have to have a super rigorous and a super optimistic way forward. We have seen so many people be optimistic without rigor. Nobody’s going to follow. And similarly, rigor without optimism, also, nobody’s going to follow. So, it’s our job to keep refining and refining and refining until we can be both rigorous and optimistic. Now, how do we know when our plan is working? Well, here are the four parts of storytelling that we know. Our job is to understand this plan so deeply that we can describe it simply. When we describe it, we want to make sure if I describe it to you, and you describe it to the next person, that the next person understands it as if I described it to them. So, our job is to understand so deeply that we can describe simply that it’s understood in our absence. And the ultimate test is it’s understood when they go home and share it with their family. They have the same understanding we want. We find this to be the four-stage litmus test to make sure we have been effective in our communication. And when people understand it this well, then they can act on it in our absence. And that’s when we’re now in the position to go as fast as we can. And when all of that infrastructure is in place, well, then we can go super fast. And there are all kinds of clever ways that we can do that. So, I look forward to opening this up and having a conversation with you.

NICOLE SMITH: That was excellent. Professor, we got several questions. I want to just dive right into it. Tessa asked, what tools, practices, and skills do you use to uncover the underlying superficial problems? It sounded like you talked a lot about questions and asking questions.

FRANCES FREI: Yeah, it’s right. So, the Toyota production system would famously refer to the five whys. And they had… and that was root cause analysis, which we all know. But essentially, what they found is that it’s about five… why does this exist? Well, why does that exist? Well, why does that exist? Like, if you ask why five times, they found that that’s how you got to the root cause. We find, in practice, the answer is closer to three. It’s rarely one. So, it would be, the symptom and the cause are usually a few layers. And you want to keep asking why. So, that’s the first thing I would say, is that we want to have… make sure that you’re doing root cause analysis. But the second thing on a specific tool, the tool that we like the most, we call the indignities list. And what you do is that… and the way we found out the symptom is we went to women in this company, because that’s what… they said they were having a gender problem. And we asked the women, is there anything that’s going on at work that just… it feels like it’s just nicking your dignity? And it occurs for… is it happening to you, or you observe it happening to other women? So, you go in search of the indignities list. Every time we do this, you’ll get a list of issues. Often, they will sound trivial. When you start to get convergence on those indignities, we then ask you to convert those indignities to the dignity list. And in this case, it was the communication style. And you know what the awesome thing about that was? It was free.

NICOLE SMITH: Wow.

FRANCES FREI: You can’t beat free.

NICOLE SMITH: Monique asks, can you speak more about how to amplify others’ ideas and perspectives, especially when they’re from underrepresented stakeholders?

FRANCES FREI: Oh, I love that question. Thank you very much. And so, I’m going to go to… here is my favorite visual on the amplification part, which is the team I’ve drawn in the middle, it’s a three-person team. And each circle represents a person on the team. And I’m showing that there’s three circles in the middle, that those folks are very similar to one another. And then on either side, we have a team where there’s difference among us. And this is where the underrepresented might come in. If we’re not careful, when we have underrepresented voices, we’re only going to be seeking from them the parts that overlap with us. So, this is when we’ve invited them to the table, but we’re not inclusive of their voices. What we want to do is make sure that everybody feels comfortable bringing all of their richness to the table, not just the part that overlaps. And so, what we find we need to do is be very solicitous about… and same with questions. From your perspective, how does this sound to you? What else are we missing? What I’m trying to do is get you off the scent of saying what you think I want to say or even asking you to say what I want to say because it makes me feel better. But I want to be inclusive of all of the gorgeous uniqueness. And this, of course, ties to diversity, equity, and inclusion, which I know has gotten a rocky go of things in the press. But what I’ll tell you is, if I got to rewrite diversity, equity, and inclusion, I would have written it as inclusion, equity, and diversity, because I have seen teams bring… I have seen organizations bring in diverse and underrepresented talent and not get the benefit from it.

NICOLE SMITH: Yeah.

FRANCES FREI: So, diversity may or may not beget inclusion. But I have never, ever seen an organization that was inclusive that didn’t beget gorgeous diversity.

NICOLE SMITH: Right.

FRANCES FREI: So, be inclusive first.

NICOLE SMITH: I appreciate you saying that, not just sitting at the table, but actually including and giving lift to people’s voices. I also want to talk about this friends thing you keep talking about, making new friends. First of all, how do I identify who’s a friend?

FRANCES FREI: Yeah. So, in this case, I want the friend to be someone who is as different from you as possible. So, the new friends. Like, who’s worthy of friendship? Not someone who you’re already attracted to, not somebody who you’re already hanging out with. So, here’s the thing about humans. We really like people who are really like us. It doesn’t make us bad people. But it just makes us human. And so, what I want you to do is seek difference. Find people from different perspectives. And that will be demographic difference, different lived experience, different learned experience. And so, if we’re senior faculty, let’s invite in junior faculty. If we’re all women, let’s invite in a man. If we’re all engineers, let’s make sure we’re bringing in the perspective of marketing. So, what I would say is my guiding principle is seek difference. Those are your potential new friends.

NICOLE SMITH: OK, so Steve wants to hone in on Friday, right? And Steve asks, can you paint a quick sketch of what’s going fast after this being slower – a slower, more thoughtful process?

FRANCES FREI: I sure can. Thank you, Steve. And so, here’s how I would think about Friday. We need ruthless prioritization. And what I mean by that is that for the most part, organizations have… that we work equally on everything. We think everything is equally important. But what we know is that organizations that win, they have ruthless prioritization. And they know, this is what I’m designed to be great at. And this is what I’m designed to be bad at. Not bad for sport, bad in the service of great. And if an organization can’t discern between these two, they’re going to end up with exhausted mediocrity. And so, what we have to do for our employees and the rest of the organization is, here’s what we’re going to optimize on. That’s half the story. And here’s what we’re not. So, I’ll give you an example of this. And the example is from Steve Jobs. And if those of you that are a bit techie, and you remember 20 years ago, when Steve Jobs walked out on that Worldwide Developer Conference stage with a manila envelope, and it had a MacBook Air in it. And he slid out that MacBook Air. And the crowd and the world went crazy, because it was the lightest-weight laptop in the world. Well, he very, very openly said, we are best in class at weight because we are worst in class at physical features. We could have been best in class at physical features. But then we would have been worst in class at weight. Or we could have chosen to be average at both. But then we would have had to rename our company. And then he made fun of another company that I won’t say here. So, we will end up… if we aren’t deliberate, we’re going to end up with exhausted mediocrity, constantly getting better at the things we’re bad at, which, without realizing it, means we’re getting worse at the things we’re good at. So, the most important thing we can do on Friday is to articulate, this is what we want to be disproportionately good at. And thus, this is what we want to be disproportionately bad at. And there’s a whole other series of things. But that’s the most important one.

NICOLE SMITH: Mm-hmm. Speaking of Steve Jobs, we have a question where they ask, do you think that the culture in Silicon Valley is changing from break things to fix things, particularly as it pertains to not only their own companies, but broader societal problems?

FRANCES FREI: Yeah, so I – not in all of Silicon Valley. So, I think we can famously see, it’s not clear to me that Twitter is moving fast and fixing things. But what I will say is that, look at Uber today. And I had the pleasure of going and working with Uber back in 2017, when they were going to move fast and break things. They are moving fast and fixing things now, and going at a catapulting speed. Or ServiceNow didn’t ever even go through move fast and break things. It’s just moving fast and fixing things. Stripe is doing the same thing. Airbnb is now moving fast and fixing things. So, what I would say is that Silicon Valley can now choose to move fast and fix things, whereas, in the past, I think they only thought they had the choice of going slow or moving fast and breaking things. Today, we have the choice. And more and more companies are making that choice.

NICOLE SMITH: Mm-hmm. And so, Bill asked, which one of these steps do you find the most commonly in need of… that companies need the most help with? So, you laid out Monday through Friday. Is there something that sticks out often?

FRANCES FREI: Well, I’ll tell you that if companies are really pressed for time, they skip Thursday. And that’s to their peril, because if we skip Thursday, that means we have to be present. And we’re a bottleneck for everything. That means people need us to translate why this is important. So, I would say that Thursday is the one that’s most often skipped. And I encourage you not to. And then I would say that Tuesday is the one that’s most often misunderstood because of all of the myths I mentioned that we have about trust. And we just think, oh, if trust is broken, we have to work around it, as opposed to going right through it and rebuilding trust.

NICOLE SMITH: So, Thursday, that’s the storytelling, honoring the past, describing it simply, right? So why do we struggle to describe things simply?

FRANCES FREI: Oh, I don’t know what your inbox looks like on your email. But you tell me how many long emails you have.

NICOLE SMITH: I refuse to deal with my inbox. I’ll deal with it later.

FRANCES FREI: So, Mark Twain was right. I apologize for sending you a long letter. I didn’t have the time to send you a short letter. It’s the metaphor for all of this, that when we understand something in a complicated way, we want to benefit people from the entirety of our knowledge. And we just throw up all of it on people, as opposed to realizing the beautiful curation and skill that’s required to go from understanding it deeply to understanding it elegantly in its simplicity. So, I think it takes time. It’s also… it takes skill. Like, this is… there are professional communicators for a reason. They’re really good at it. But if you’re on your second draft of something, you have no chance of describing it simply. So, I would say, unless you’re on your 10th draft, you’re probably describing it in too complicated of a way.

NICOLE SMITH: Yeah. So, can I ask you a little bit more of a personal question, Professor?

FRANCES FREI: Yeah, anything.

NICOLE SMITH: So, Abby asks, how do you apply the essential steps to moving fast and fixing things in your own consulting role? So, Uber and all the places that you go.

FRANCES FREI: Yeah. Yeah, so I’ll tell you, when we’ve been successful, it’s when organizations come to us, and they say, here’s our problem. Will you help us? When we’ve been unsuccessful is when we go to the organizations, and we’re like, we think you’re having a problem. So, pull works. Push doesn’t. So, the only thing we can’t provide is the desire to change. And so, I would say personally, make sure there’s an opening. And then you can be super helpful in fixing a problem. And I also would say that all of this applies to yourself. I mean, that ruthless prioritization – so many of us are trying to be good at as many things as possible – at work, at home, daughter, sister, cousin, parent, friend – as opposed to, I’m going to kill it at work, kill it at home. And I am not going to be good… not now. I’m not going to be as good at all of these other things. So, you can either choose exhausted mediocrity, or you can have the nobility of excellence. These things are choices. So, I think all of this applies to ourselves.

NICOLE SMITH: So, let’s go back to Tuesday, where you drew that triangle with logic, and empathy, and authenticity. So, Hung asks, between logic and empathy, which one would you say an individual should develop first? And Hung really describes just having a left foot and right foot and not knowing which one to go forward.

FRANCES FREI: Yeah. So, here’s what I would say, Hung, is, ask yourself… I bet you’re trusted most of the time, which means people are experiencing your authenticity, logic, and empathy most of the time. But ask yourself, the last time, or the most recent times you had a skeptic, you had someone who was doubting you, who they were wobbling on your trust, ask yourself, what is it that they doubted about you? And if it’s that they doubted your logic, double click there. If they doubted your empathy, double click there. And that is, each of us has what we call a wobble. Each one of us has a pattern where the distribution of these is higher for one or the other. That’s the sequence I would go in. There’s not some generic sequence that is better. All three of these pillars are equally important. But I bet, for each one of us, one tends to be more shaky than the other. And that’s what I would go after. Now, I will just tell you the distribution in the world. The vast majority of us have empathy wobbles, then logic wobbles, then authenticity wobbles. But that doesn’t help any of us specifically. It just tells us we have lots of company.

NICOLE SMITH: OK. So, we got a lot more questions and a little time. I want to get as many as I can in, but…

FRANCES FREI: OK, I’ll go super quick. Yeah.

NICOLE SMITH: No, take your time. But I just want to let you know, you’re pretty popular in this conversation. Rock star, as Allison said. Tara asks, how can company leadership make sure that their messaging is actually heard and understood? I feel like you touched on this a bit with simplicity.

FRANCES FREI: Yeah. Yeah, and I think that the way to do it is, talk to people about your message that didn’t hear it directly from you. And see how well they understood. That tells you whether or not it’s reaching. So, don’t ask the people that were in the room. Ask the people that were spoken to by other people in the room. That will tell you how well it’s there. And if it took you a long time to describe it, I promise you, it’s not going to be heard.

NICOLE SMITH: Mm. Oh, wow. Yeah, thinking about it, probably need to shorten my own stories a little bit here. So, Karen asks you, how do you handle employees who are not willing to accept others’ points of view and be open minded? I mean, you described this uniqueness and diversity. But there are people who are holdouts that don’t see the advantage of that.

FRANCES FREI: So, I often find those folks are an education away, because if I can let you know that if I get to benefit from everyone’s point of view, and you only get to benefit from some people’s point of view, I will competitively thump you. So, let’s say you don’t have the moral imperative wanting to do it. Well, the performance imperative… we have found that organizations that are inclusive get a 200% to 500% boost on employee engagement and team performance with no new people, no new technology, simply the act of being inclusive. So, the person who doesn’t want to be inclusive, I’m going to ask them, can they afford… can their career afford performing so suboptimally?

NICOLE SMITH: Mm. And so, we have a question. The person didn’t leave their name, so I don’t have a name. But how much time do you spend on each stage? Some folks like to spend more time on stages than others. Does the team not move forward until everyone’s satisfied with the current step? What do you do when you hit a roadblock on each stage, and not everyone is in agreement?

FRANCES FREI: Yeah. Well, I don’t like consensus, so I’ll just… I’ll say there. And so, what I try to do is work on momentum, which is that I want to make sure that everybody’s voices have been heard. But then you have to leave the decision to someone else. So, we want to do is make sure everybody’s voices are heard, and they had a chance to do it. But we don’t hold out until the very last person. We move forward. And then we can retrace and see if the momentum can bring people forward. So, not consensus. I would consider it not consensus, and we have to make sure that everybody gets to air out what their problems are.

NICOLE SMITH: OK. Well, Christopher asks our last question. How does transparency fit into this model, specifically this trust, authenticity, logic model? Does it have a place?

FRANCES FREI: Yeah. It sure does. And I find that the most important part for transparency is on the logic side. So, if you’re going to say… if you’re going to inspect whether or not I have good rigor, and I have a good plan, I could say, oh, just have faith. I did all of this hard work. Or I could give you a glimpse inside so that you can see the inner workings. Now, I often call it a window of transparency, because there’s actually a cost of full transparency that I’m not always willing to take. But a window of transparency, I think we always need. So, to me, the transparency part is, let’s be transparent about our logic so people can see it for themselves, and they don’t have to do it in too much of a faith-based way.

NICOLE SMITH: Professor, that was all dynamic. And thank you for the illustrations. You made it simple with the illustrations.

FRANCES FREI: Yeah, all right. Awesome. Thanks so much.

NICOLE SMITH: Thank you for your time.

FRANCES FREI: OK.

HANNAH BATES: That was Harvard Business School professor Frances Frei in conversation with HBR’s editorial audience director Nicole Smith at the “Future of Business” virtual conference in November 2023.

We’ll be back next Wednesday with another hand-picked conversation about leadership from Harvard Business Review. If you found this episode helpful, share it with your friends and colleagues, and follow our show on Apple Podcasts, Spotify, or wherever you get your podcasts. While you’re there, be sure to leave us a review.

When you’re ready for more podcasts, articles, case studies, books, and videos with the world’s top business and management experts, you’ll find it all at HBR.org.

This episode was produced by Anne Saini, and me, Hannah Bates. Ian Fox is our editor. Music by Coma Media. Special thanks to Dave Di Iulio, Terry Cole, and Maureen Hoch, Erica Truxler, Ramsey Khabbaz, Nicole Smith, Anne Bartholomew, and you – our listener. See you next week.

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The Music Industry Has a Plastic Waste Problem. Can This ‘Ugly’ Cup Solve It?

Produced by reuse company r.World, the cups are good for 300 uses and are already at more than 200 venues across the U.S. Their creator says this is just the beginning. 

By Katie Bain

Scan the ground after any given concert or music festival and one thing you’re almost certain to see is a scattering of empty plastic cups. According to a 2024 report by environmental advocacy agency Upstream, the live-event industry creates over 4 billion single-use cups that end up in landfills every year in North America alone.  

Depeche Mode’s Crypto.com Arena Shows to Eliminate Single-Use Cups in Sustainability…

It doesn’t have to be this way — and reuse company r.World wants to lead the change. The Minneapolis-based company provides reusable serveware — cups, food containers and more — for mass gatherings, with these products designed to mitigate the persistent single-use plastic waste problem in the live music industry and beyond.

Trending on Billboard

Founded in 2017, r.World provides reusable plastic cups and other serveware to more than 200 venues across the U.S., along with festivals like Long Beach’s 20,000 capacity Cali Vibes and San Francisco’s 30,000-capacity Portola. In late May, the company partnered with Los Angeles’ Crypto.com Arena, home of the NBA’s Lakers and Clippers, and Peacock Theater to launch a full-time reusable cup program in each venue.

But the mission extends far beyond concerts and sports, with r.World aiming to build the infrastructure for a national reuse economy that would extend to airlines, consumer packaged goods, restaurants and more, ultimately becoming “a one-stop national solution,” Martin says. “The music industry has essentially launched and is leading the reuse movement in the country, and it’s inspiring universities, corporate campuses, quick-service restaurants and others.”

At the center of this movement is the plastic cup itself. Good for 300 uses, r.Cups are made of thick plastic designed and manufactured to r.World specifications that Martin says “overhauled” the manufacturing process of a standard single use cup. Made in the United States to minimize carbon emissions from shipping, each cup is slapped with the words “please return our cup to an r.cup bin,” and when a cup reaches its maximum number of uses, it’s upcycled into other r.World products.

The sweeping project started 10 years ago, when Martin’s other company, the climate solutions-focused Effects Partners, was hired to analyze operations at Live Nation and create a sustainability strategy. While assembling a five-year plan for the live-events behemoth, Martin realized “the recycling and composting efforts at the venues were never going to work,” given that most everything ultimately just ended up in landfills. The realization made him “depressed for, like, six months,” until he considered the reuse programs he’d seen in European venues — and then developed r.World.

Through connections to U2 , Martin suggested the band try reuse on their 2017 tour. It was a success, and r.World was soon working with 13 acts, including the Rolling Stones , Dave Matthews Band , Bon Jovi , Radiohead and Maggie Rogers , all of whom gave Martin permission to go to venues on their behalf and request that the venue try reuse during their show.  

The first r.Cup cups were branded with band logos, until the team realized fans were just keeping them as souvenirs. In 2019, the model morphed into “an ugly cup” people were less inclined to take home.  

Cups are collected in yellow bins that sit alongside garbage cans and recycling containers at venues, then brought to an r.World-owned wash hub facility. These hubs are built in economically depressed areas of any given city to help spur the economy, and are where cups are washed and inspected, largely staffed by people living in halfway houses or who are getting back on their feet after getting out of prison. 

These local facilities are crucial because, as Martin says, “you can’t prioritize the environment if you’re shipping cups great distances across the country” due to the carbon emissions created by such transport. r.World plans to establish wash hubs and reuse solutions in the top 20-30 U.S. markets, having already launched in seven. The company expects to add another one or two cities in the coming months and is in conversation with officials from nearly every city they are targeting. “We know the demand and need is there,” says Martin. While the majority of r.World’s current business is cups, Martin cites “exploding” demand for food containers at venues, festivals, schools and corporate campuses.

Venue Sustainability Program GOAL Releases First Impact Report

r.Cup typically launches in a venue after a facility or concession manager reaches out to ask about reuse. (Martin notes that they have a 99% client retention rate, and the one venue that did let go of the program was having financial issues.) With an operational design developed via focus groups with national concessionaires like Levy Restaurants, Aramark and Sodexo US, r.World provides everything from cups and collection bins to signage, employee training materials and social media content to educate guests, offering “a complete turnkey solution so it’s a no brainer for the operators,” says Martin. Venues are also provided with environmental impact reporting that uses EPA guidelines to consider everything from the sourcing and shipping of cups to the temperature of the water used to clean them. (Martin says the company is “sort of obsessive” about these protocols, which he attributes to “being a numbers geek.)

Cost of implementation is based on the number of single use items required by a venue and varies by how much of their service is packaged drinks versus draft or fountain drinks. Martin says the biggest arenas that serve draft and fountain beverages go through 1.5 million-2.5 million single use cups per year. While upfront costs of r.World products are higher than single use, the cost over time is generally less given that venues must keep buying the reusable plastic cups that get thrown away after each event.

Some venues embed this added expense into the drink price, while others allow guests to opt out and get a single use cup for a slightly lower cost. (Over r.World’s millions of transactions, Martin has heard about “two or three” people opting out.) Drink servers are also into r.Cup, he says, “because they felt bad giving out all that single use waste, and cups are a conversation starter with guests.” Beyond the price differentials, Martin says the biggest hesitation venues and events have about adopting reusable cups is an “imagination gap,” along with other factors like existing vendor contracts, venue infrastructure and apathy and misinformation, such as thinking single-use aluminum or compostable cups are good for the environment.

To wit, reusable cups are alternatives to frequently-used compostable cups, which have a dicey record of being composted and behave as a regular single use plastic cup if they end up in a landfill. Aluminum cups and bottles also often end up in landfills given that recycling sorting at events can be spotty. A 2023 Upstream report states that “single-use aluminum cups are the worst option for the climate by far,” as they use 47% more energy over their life cycle and create 86% more carbon dioxide than other single-use plastic options. 

As sustainability initiatives become more common and more in-demand across the industry and culture at large, more than 150 national reuse companies have launched since the pandemic. In 2022, Live Nation invested in Turn Systems, a program that provides reusable cups, collection bins and mobile washing systems at venues and festivals. As such, r.World is partnered with Live Nation competitors including AEG, ASM and NIVA, and provides product washing for other reuse companies.

Beyond venues and events, r.World clients include the Coca-Cola Company, which is widely cited as one of the world’s leading single-use plastic waste producers. Coca-Cola has made a commitment to incorporating 25% reusable products by 2030 and is working with r.World to provide reuse services for Coca-Cola clients like music venues, movie theaters, the Olympics, the World Cup and wherever else Coca-Cola wants to implement reuse. r.World has also been selected by the EPA and the White House’s Council on Environmental Quality to help raise national awareness of reuse.

When It Comes To Making Real Eco-Conscious Change, Science Is An Artist’s Best Friend

Martin says that while an industry has developed to help solve the single use plastic problem, many waste management and consumer packaged goods companies would rather not see a large-scale shift to reuse happen. And despite the explosive growth in the sector, Martin says r.World’s biggest competitors are still single-use cups and serveware, whether plastic, compostable or aluminum.

This is where artists and fans can flex their power by requesting reuse programs in their riders and spending money at venues with reuse programs given, Martin says, that “businesses will give back what consumers are asking for.” 

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Pacmoon Almost Solved Crypto’s SocialFi Problem—Then Elon Musk Crushed It

Blast meme coin pacmoon appeared to solve a key problem that’s vexed crypto for years. the future was limitless—until a twitter update undid everything..

Decrypt’s Art, Fashion, and Entertainment Hub.

For years, countless crypto startups have tried—and failed—to get one innovation right: a sustainable, on-chain system for financially incentivizing online behavior (browsing social media, reading articles, watching videos) that rewards creators and consumers alike. 

The concept, dubbed SocialFi, speaks to the heart of what crypto once promised: an escape from the iron grip of billionaire tech elites who monetize your every online move, and a means to recapture that wealth for the many. 

Last month, a Blast meme coin community called Pacmoon (PAC) debuted what appeared to be a novel solution to crypto’s SocialFi problem. For a few weeks, the system, Pacmoon v2, soared: members of the PAC community began accumulating tokens with real monetary value in exchange for simple engagement on Twitter that, crucially, appeared to reward quality content and box out spammy token farmers and automated bots. 

The future, for Pacmoon and for crypto, looked bright. Then, last week, Elon Musk’s Twitter abruptly moved to make user “likes” data private—crippling Pacmoon’s nascent SocialFi scheme in an instant, and laying bare crypto’s continued dependence on the very tech billionaires it seeks to free us from.

Pacmoon was started earlier this year by a crypto trader and two friends who sought to create the first dominant meme coin on Blast —a new Ethereum layer-2 scaling network from the creators of Blur , the disruptive, incentivized NFT marketplace. 

From the beginning, “Bobby Big Yield,” Pacmoon’s pseudonymous founder, knew he needed to make PAC multifaceted if it was going to succeed. It couldn’t just be a meme coin, so many of which fade away with the next news cycle—it had to be a “community coin,” designed to operate as a social home base for Blast’s lucrative, finance-focused user base. 

If Bobby could find a way to incorporate SocialFi features into PAC, he figured, the token would never fade from relevance. It would serve as the backbone of a self-perpetuating ecosystem in which Blast users got rewarded to engage with each other. 

But the trader was under no illusions about SocialFi. He’d seen countless crypto projects amass activity with flashy promises of rewarding social media behavior, then collapse after forward momentum slowed. 

“Going viral once, getting a bunch of people to use your product, is not that hard for any decent crypto marketer and competent product team,” Bobby told Decrypt . “What's hard is to get people to do it twice.”

So Bobby and his colleagues devised a novel SocialFi solution, one they dubbed “social validation.” Borrowing inspiration from the structure of proof-of-stake networks like Ethereum— which rely on independent validators who have significant financial “stake” in the network to authenticate transactions—Pacmoon would tap the token’s biggest holders to curate community Twitter behavior and reward quality posts.

The logic behind empowering “social validators” was the same relied on by countless blockchain networks. Pacmoon validators (those holding 10,000 PAC or more; $2,300 worth at peak price) would naturally be invested in boosting the value of their own tokens, and thus the long-term success of the Pacmoon brand, by curating quality social media content—not the countless “BUY TOKEN NOW” posts that flood most Crypto Twitter timelines. 

If Pacmoon validators got sent PAC for liking Pacmoon-related tweets, and the authors of those tweets got rewarded in PAC too, Bobby figured the system would unlock a stream of genuinely engaging social media content that would bring the Pacmoon community closer together and keep PAC relevant in perpetuity.  

When Pacmoon v2—the version debuting “social validators”—launched in late May, Bobby’s thesis was largely proven right. Pacmoon holders loved the innovation, and immediately went to work creating elaborate artworks , raps , and animations to garner likes from the roughly 1,600 Pacmoon whales who signed up to become validators. 

Even Beeple , the world-renowned digital artist behind the most expensive NFT ever sold, jumped on the bandwagon. 

PACMOONIANS ENJOYING A GOLDEN SHOWER ✨ pic.twitter.com/DCFCtoeRPB — beeple (@beeple) June 3, 2024

Ovie Faruq (aka OSF), a crypto influencer with over 195,000 Twitter followers who signed up to become one of Pacmoon’s first social validators, told Decrypt shortly after the debut of v2 that he particularly liked how the system relied on Twitter likes—not reposts or any other form of active promotion that could dilute his brand or legitimacy. 

“For accounts with larger followings, you don’t want to be shilling,” Faruq said at the time. “Now I can still engage with it, but do it passively.”

He also added that the system encouraged him to organically engage with Pacmoon-related content as he would ordinarily—increasing the net quality of the Pacmoon ecosystem, and potentially offering a glimpse of a SocialFi model finally fit to gain long-term traction. 

“I think that's the first time I've seen anything like this,” Faruq said. “It’s a model that I think any brand new crypto project starting out could definitely use.”

Bobby, meanwhile, was blown away by Pacmoon v2’s success. Pacmoon holders were creating real social bonds; the system facilitating those bonds was spreading the Pacmoon brand across Twitter, for free; and all the while, PAC was pumping towards all-time highs.

“We’re creating an almost infinitely higher number of connections between members of our communities,” Bobby said at the end of May. “And this advertising… we can't pay for it.”

Less than two weeks later, it all fell apart.

Pivot to v3

On June 12, Elon Musk announced that Twitter had made all likes on the platform private, apparently in the name of user privacy .

Based on how Pacmoon collected data, the policy change immediately derailed Pacmoon v2. Unless Bobby and his team wanted to pay for exorbitantly expensive access to Twitter’s enterprise API platform—something they say they couldn’t afford—their “social validation” experiment was kaput. 

Important change: your likes are now private https://t.co/acUL8HqjUJ — Elon Musk (@elonmusk) June 12, 2024

As soon as they saw the post, Bobby and his fellow Pacmoon builders faced hard questions. Chief among them: Was their ultimate goal really cracking crypto’s SocialFi problem, even in the face of obstacles lobbed by one of the world’s richest men? Or were they just here to make a fun meme coin?

They chose the latter answer. Pacmoon v2 was halted; the team worked furiously to cook up a new path forward less dependent on SocialFi. The price of PAC immediately collapsed by over 50%. At writing, it’s sunk to roughly $.07 .

“It sucks to say ‘Well, we have to scale it back,’” Bobby recently said. “At least for now, it makes the most sense for us to focus on creating fun content and making sure everybody has a good experience, with the financial incentives more as just a fun cherry on top.”

Pacmoon v3 launched on Wednesday. Effectively, Pacmoon validators no longer validate. The job of curating community content has fallen back on the official Pacmoon Twitter account , which retweets 100-odd top posts of the day. Those chosen posters earn PAC, as do Pacmoon validators, so long as they like any one Pacmoon-related tweet on a daily basis. 

Pacmoon v3 is undeniably more centralized—and less democratic—than v2 was. 

Pacmoon V3 is now live! Create content and like tweets to earn $PAC How it works 👇 pic.twitter.com/jp2tGrd71k — PACMOON (@pacmoon_) June 19, 2024

Bobby insists that v2 wasn’t perfect to begin with. He says that the job of weeding out nefarious token farming among Pacmoon validators was becoming cumbersome and costly to his small team, and that playing the role of policeman was threatening Pacmoon’s feel-good “vibes.” He maintains the transition was for the best.

But he nonetheless believes that he stumbled across something crucial in innovating the concept of “social validators.” Perhaps a piece of a future SocialFi system that can really, finally work—maybe if harnessed by a large enough platform with enough resources to build it out, then let it soar.

A fight for another day, then, to be waged by someone with a lot more power than Bobby.

That’s the grim irony of crypto’s enduring SocialFi dilemma. Even if tech arises capable of propelling an incentivized social media engagement cycle forward in perpetuity, it currently must depend on hyper-centralized platforms like Twitter, which are popular enough to meet people where they are. 

Decentralized social media alternatives like Farcaster do exist. But they have a long, long way to go before they grow big enough to fuse with tools like Pacmoon v2 and unlock SocialFi’s futuristic potential. 

Just a few weeks ago, Bobby thought that future was here. Now, he’s feeling a lot less optimistic. 

“I'm kind of realizing, to be honest, decentralized social media is actually very, very hard to do right,” Bobby said this week. “I think it's virtually impossible to get it right now.”

Edited by Andrew Hayward

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Euro 2024: Three England problems Gareth Southgate must solve after dull Denmark draw

It's not the first time the Three Lions have had an uninspiring start to a tournament, and some of the issues facing the manager will remind fans of other failed attempts to win a second major title.

Sports correspondent @RobHarris

Friday 21 June 2024 13:24, UK

Completing their Euro 2024 group against Slovenia on Tuesday brings back memories for England of another tournament when they were booed by their own fans.

Slovenia were the only team beaten by England at the 2010 World Cup in South Africa in an era of underachievement, despite all the talent available.

Sound familiar?

Gareth Southgate endures what so many predecessors have struggled with - producing performances and results to match the high expectations.

The England manager saw it in 2018 - reaching the World Cup semi-finals but then wilting under the spotlight against Croatia.

He saw it in 2021 at Wembley - initially overpowering Italy in the Euros final only to lose on penalties.

And he saw it again on Thursday night in Frankfurt in the 1-1 draw with Denmark at these Euros, as England's control descended into capitulation.

More on England Football Team

Serbian FA charged after objects thrown at England's Euro 2024 match

England fans revel in 'good vibes' as tense 1-0 win against Serbia sees Three Lions top Euro 2024 group

Euro 2024: A new-look England and another moment for Jude Bellingham to shine

Related Topics:

• England football team
• Gareth Southgate
• Jude Bellingham

So where does it leave their campaign in Germany and what does Southgate need to address?

Remember, this isn't a unique situation. It is now 10 appearances at European Championships that Englishmen have failed to open the tournament with two wins.

• Plan ahead

Don't panic - England already seem to have enough points to qualify for the knockout phase.

Staying top of Group C avoids meeting a group winner.

But if England finish second, it could be hosts Germany standing in the way of the quarter-finals.

And if they finish third, they may not know their last-16 opponents until Wednesday, the day after their last group game against Slovenia.

• Midfield muddle

The lack of structure and cohesion in midfield is a problem.

Surely the Denmark game marks the end of seeing defender Trent Alexander-Arnold being deployed in midfield?

Perhaps Phil Foden is in need of a rest against Slovenia - ready to be unleashed later in Germany to recapture the form that saw him named the Premier League's Player of the Season and FWA Footballer of the Year while sweeping to a fourth title in a row with Manchester City.

Last night's laboured performance in the heat of Frankfurt - with the roof closed - was in desperate need of a spark of creativity.

Southgate left Jack Grealish and James Maddison at home after a season of struggles with fitness and form.

Why, though, is he yet to give any playing time at the Euros to Cole Palmer? The 22-year-old was Premier League Young Player of the Season with 22 goals and 11 assists as the shining light in an underwhelming Chelsea side.

Would Eberechi Eze benefit from more than the 22-minute cameo against Denmark in which he managed only 10 touches of the ball?

• Goal threat

After the Denmark draw, Harry Kane pointed to uncertainty in the side over pressing opponents.

Once the other team starts dropping deeper, England can't figure out how to exert more pressure on them.

Kane and Jude Bellingham did attempt to get ahead of the Danish midfielders in the second half - when it was already 1-1 - but they couldn't make an impact.

Read more: Southgate says 'huge amount of work' to do after Denmark draw Serbia threaten to pull out of Euro 2024 Eight England fans arrested at Euro 2024

Yet Bellingham, who scored the only goal in the opening win over Serbia, certainly knows how to be effective at club level.

Just look at his 23 goals and 10 assists in his first 42 games for Real Madrid last season.

And Kane was even more prolific at Bayern - contributing 44 goals and setting up another 12 in 45 games.

Related Topics