## 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

• a special character: @$#!%*?& If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. ## SAT (Fall 2023) Course: sat (fall 2023) > unit 6. • Solving linear equations and linear inequalities | Lesson • Understanding linear relationships | Lesson ## Linear inequality word problems | Lesson • Graphing linear equations | Lesson • Systems of linear inequalities word problems | Lesson • Solving systems of linear equations | Lesson • Systems of linear equations word problems | Lesson ## What are linear inequality word problems, and how frequently do they appear on the test? • Understanding linear relationships • Solving linear equations and linear inequalities ## How do I write linear inequalities based on word problems? Using inequalities to solve problems, linear inequality word problems, what are some key phrases to look out for, let's look at some examples. • Your answer should be • an integer, like 6 6 6 6 • a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5 • a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4 • a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4 • an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75 • a multiple of pi, like 12 pi 12\ \text{pi} 1 2 pi 12, space, start text, p, i, end text or 2 / 3 pi 2/3\ \text{pi} 2 / 3 pi 2, slash, 3, space, start text, p, i, end text • (Choice A) 56 + 52 + 38 + x ≥ 50 56+52+38+x\geq50 5 6 + 5 2 + 3 8 + x ≥ 5 0 56, plus, 52, plus, 38, plus, x, is greater than or equal to, 50 A 56 + 52 + 38 + x ≥ 50 56+52+38+x\geq50 5 6 + 5 2 + 3 8 + x ≥ 5 0 56, plus, 52, plus, 38, plus, x, is greater than or equal to, 50 • (Choice B) 56 4 + 52 4 + 38 4 + x ≥ 50 \dfrac{56}{4}+\dfrac{52}{4}+\dfrac{38}{4}+x\geq50 4 5 6 ​ + 4 5 2 ​ + 4 3 8 ​ + x ≥ 5 0 start fraction, 56, divided by, 4, end fraction, plus, start fraction, 52, divided by, 4, end fraction, plus, start fraction, 38, divided by, 4, end fraction, plus, x, is greater than or equal to, 50 B 56 4 + 52 4 + 38 4 + x ≥ 50 \dfrac{56}{4}+\dfrac{52}{4}+\dfrac{38}{4}+x\geq50 4 5 6 ​ + 4 5 2 ​ + 4 3 8 ​ + x ≥ 5 0 start fraction, 56, divided by, 4, end fraction, plus, start fraction, 52, divided by, 4, end fraction, plus, start fraction, 38, divided by, 4, end fraction, plus, x, is greater than or equal to, 50 • (Choice C) 56 + 52 + 38 3 + x ≥ 50 \dfrac{56+52+38}{3}+x\geq 50 3 5 6 + 5 2 + 3 8 ​ + x ≥ 5 0 start fraction, 56, plus, 52, plus, 38, divided by, 3, end fraction, plus, x, is greater than or equal to, 50 C 56 + 52 + 38 3 + x ≥ 50 \dfrac{56+52+38}{3}+x\geq 50 3 5 6 + 5 2 + 3 8 ​ + x ≥ 5 0 start fraction, 56, plus, 52, plus, 38, divided by, 3, end fraction, plus, x, is greater than or equal to, 50 • (Choice D) 56 + 52 + 38 + x 4 ≥ 50 \dfrac{56+52+38+x}{4}\geq50 4 5 6 + 5 2 + 3 8 + x ​ ≥ 5 0 start fraction, 56, plus, 52, plus, 38, plus, x, divided by, 4, end fraction, is greater than or equal to, 50 D 56 + 52 + 38 + x 4 ≥ 50 \dfrac{56+52+38+x}{4}\geq50 4 5 6 + 5 2 + 3 8 + x ​ ≥ 5 0 start fraction, 56, plus, 52, plus, 38, plus, x, divided by, 4, end fraction, is greater than or equal to, 50 ## Things to remember Want to join the conversation. • Upvote Button opens signup modal • Downvote Button opens signup modal • Flag Button opens signup modal ## Solving Quadratic Inequalities ... and more ... A Quadratic Equation (in Standard Form) looks like: The above is an equation (=) but sometimes we need to solve inequalities like these: Solving inequalities is very like solving equations ... we do most of the same things. So this is what we do: • find the "=0" points • greater than zero (>0), or • less than zero (<0) • then pick a test value to find out which it is (>0 or <0) Here is an example: ## Example: x 2 − x − 6 < 0 x 2 − x − 6 has these simple factors (what luck!): (x+2)(x−3) < 0 Firstly , let us find where it is equal to zero: (x+2)(x−3) = 0 It is equal to zero when x = −2 or x = +3 because when x = −2, then (x+2) is zero or when x = +3, then (x−3) is zero So between −2 and +3, the function will either be • always greater than zero, or • always less than zero We don't know which ... yet! Let's pick a value in-between (say x=0) and test it: So between x=−2 and x=+3, the function is less than zero. And that is the region we want, so... x 2 − x − 6 < 0 in the interval (−2, 3) Note: x 2 − x − 6 > 0 in the interval (−∞,−2) and (3, +∞) Also try the Inequality Grapher . ## What If It Doesn't Go Through Zero? A "real world" example, a stuntman will jump off a 20 m building. a high-speed camera is ready to film him between 15 m and 10 m above the ground.. When should the camera film him? We can use this formula for distance and time: d = 20 − 5t 2 • d = distance above ground (m), and • t = time from jump (seconds) (Note: if you are curious about the formula, it is simplified from d = d 0 + v 0 t + ½a 0 t 2 , where d 0 =20 , v 0 =0 , and a 0 =−9.81 the acceleration due to gravity.) OK, let's go. ## First , let us sketch the question: The distance we want is from 10 m to 15 m : 10 < d < 15 And we know the formula for d : 10 < 20 − 5t 2 < 15 ## Now let's solve it! First, let's subtract 20 from both sides: −10 < −5t 2 <−5 Now multiply both sides by −(1/5). But because we are multiplying by a negative number, the inequalities will change direction ... read Solving Inequalities to see why. 2 > t 2 > 1 To be neat, the smaller number should be on the left, and the larger on the right. So let's swap them over (and make sure the inequalities still point correctly): 1 < t 2 < 2 Lastly, we can safely take square roots, since all values are greater then zero: √1 < t < √2 We can tell the film crew: "Film from 1.0 to 1.4 seconds after jumping" ## Higher Than Quadratic The same ideas can help us solve more complicated inequalities: ## Example: x 3 + 4 ≥ 3x 2 + x First, let's put it in standard form: x 3 − 3x 2 − x + 4 ≥ 0 This is a cubic equation (the highest exponent is a cube, i.e. x 3 ), and is hard to solve, so let us graph it instead: The zero points are approximately : And from the graph we can see the intervals where it is greater than (or equal to) zero: • From −1.1 to 1.3, and • From 2.9 on In interval notation we can write: Approximately: [−1.1, 1.3] U [2.9, +∞) • Pre-Algebra Topics • Algebra Topics • Algebra Calculator • Algebra Cheat Sheet • Algebra Practice Test • Algebra Readiness Test • Algebra Formulas • Want to Build Your Own Website? Sign In / Register ## Solving Word Problems in Algebra Inequality Word Problems How are you with solving word problems in Algebra? Are you ready to dive into the "real world" of inequalities? I know that solving word problems in Algebra is probably not your favorite, but there's no point in learning the skill if you don't apply it. I promise to make this as easy as possible. Pay close attention to the key words given below, as this will help you to write the inequality. Once the inequality is written, you can solve the inequality using the skills you learned in our past lessons. I've tried to provide you with examples that could pertain to your life and come in handy one day. Think about others ways you might use inequalities in real world problems. I'd love to hear about them if you do! Before we look at the examples let's go over some of the rules and key words for solving word problems in Algebra (or any math class). ## Word Problem Solving Strategies • Read through the entire problem. • Highlight the important information and key words that you need to solve the problem. • Identify your variables. • Write the equation or inequality. • Write your answer in a complete sentence. • Check or justify your answer. I know it always helps too, if you have key words that help you to write the equation or inequality. Here are a few key words that we associate with inequalities! Keep these handy as a reference. ## Inequality Key Words • at least - means greater than or equal to • no more than - means less than or equal to • more than - means greater than • less than - means less than Ok... let's put it into action and look at our examples. ## Example 1: Inequality Word Problems Keith has$500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws$25 each week for food, clothes, and movie tickets.

• Write an inequality that represents Keith's situation.
• How many weeks can Keith withdraw money from his account? Justify your answer.

Step 1: Highlight the important information in this problem.

Note:  At least is a key word that notes that this problem must be written as an inequality.

Step 2 : Identify your variable. What don't you know? The question verifies that you don't know how many weeks.

Let w = the number of weeks

500 - 25w > 200

I know you are saying, "How did you get that inequality?" I know the "at least" part is tricky. You would probably think that at least means less than.

But... he wants the amount in his account to be at least $200 which means$200 or greater. So, we must use the greater than or equal to symbol.

Step 4 : Solve the inequality. The number of weeks that Keith can withdraw money from his account is 12 weeks or less.

I'm going to prove that the largest number of weeks is 12 by substituting 12 into the inequality for w. You could also substitute any number less than 12. Since 200 is equal to 200, my answer is correct. Any more than 12 weeks and his account balance would be less than $200. Any number of weeks less than 12 and his account would stay above$200.

That wasn't too bad, was it? Let's take a look at another example.

## Example 2: More Inequality Word Problems

Yellow Cab Taxi charges a $1.75 flat rat e in addition to$0.65 per mile . Katie has no more than $10 to spend on a ride. • Write an inequality that represents Katie's situation. • How many miles can Katie travel without exceeding her budget? Justify your answer. Note: No more than are key words that note that this problem must be written as an inequality. Step 2 : Identify your variable. What don't you know? The question verifies that you don't know the number of miles Katie can travel. Let m = the number of miles Step 3: Write the inequality. 0.65m + 1.75 < 10 Are you thinking, "How did you write that inequality?" The "no more than" can also be tricky. "No more than" means that you can't have more than something, so that means you must have less than! Step 4: Solve the inequality. Since this is a real world problem and taxi's usually charge by the mile, we can say that Katie can travel 12 miles or less before reaching her limit of$10. Are you ready to try some on your own now? Yes... of course you are! Click here to move onto the word problem practice problems.

## Take a look at the questions that other students have submitted: Quite a complicated problem about perimeter and area of a rectangle

This is a toughie.... a compound inequalities word problem

• Inequalities
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• School Guide
• Class 11 Syllabus
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## Chapter 1: Sets

• Representation of a Set
• Types Of Sets
• Universal Sets
• Venn Diagram
• Operations on Sets
• Union of Sets

## Chapter 2: Relations & Functions

• Cartesian Product of Sets
• Relations and Functions
• Introduction to Domain and Range
• Piecewise Function
• Range of a Function

## Chapter 3: Trigonometric Functions

• Measuring Angles
• Trigonometric Functions
• Trigonometric Functions of Sum and Difference of Two Angles

## Chapter 4: Principle of Mathematical Induction

• Principle of Mathematical Induction

## Chapter 5: Complex Numbers and Quadratic Equations

• Complex Numbers
• Algebra of Real Functions
• Algebraic Operations on Complex Numbers | Class 11 Maths
• Polar Representation of Complex Numbers
• Absolute Value of a Complex Number
• Conjugate of Complex Numbers
• Imaginary Numbers
• Compound Inequalities
• Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation – Linear Inequalities | Class 11 Maths
• Graphical Solution of Linear Inequalities in Two Variables

## Solving Linear Inequalities Word Problems

Chapter 7: permutations and combinations.

• Fundamental Principle of Counting
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## Chapter 8: Binomial Theorem

• Binomial Theorem
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## Chapter 9: Sequences and Series

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## Chapter 10: Straight Lines

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## Chapter 11: Conic Sections

• Conic Sections
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## Chapter 12: Introduction to Three-dimensional Geometry

• Coordinate Axes and Coordinate planes in 3D space
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## Chapter 13: Limits and Derivatives

• Introduction to Limits
• Formal Definition of Limits
• Strategy in Finding Limits
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• Limits of Trigonometric Functions
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## Chapter 14: Mathematical Reasoning

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• Axiomatic Approach to Probability
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• CBSE Class 11 Maths Formulas
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We are well versed with equations in multiple variables. Linear Equations represent a point in a single dimension, a line in a two-dimensional, and a plane in a three-dimensional world. Solutions to linear inequalities represent a region of the Cartesian plane. It becomes essential for us to know how to translate real-life problems into linear inequalities.

## Linear Inequalities

Before defining the linear inequalities formally, let’s see them through a real-life situation and observe why their need arises in the first place. Let’s say Albert went to buy some novels for himself at the book fair. He has a total of Rs 200 with him. The book fair has a special sale policy which offers any book at Rs 70. Now he knows that he may not be able to spend the full amount on the books. Let’s say x is the number of books he bought. This situation can be represented mathematically by the following equation,

70x < 200

Since he can’t spend all the amount on books, and also the amount spent by him will always be less than Rs 200. The present situation can only be represented by the equation given above. Now let’s study the linear inequalities with a formal description,

Two real numbers or two algebraic expressions which are related by symbols such as ‘>’, ‘<‘, ‘≥’ and’≤’ form the inequalities. Linear inequalities are formed by linear equations which are connected with these symbols. These inequalities can be further classified into two parts:  Strict Inequalities: The inequalities with the symbols such as ‘>’ or ‘<‘. Slack Inequalities: The inequalities with the symbols such as ‘≥’ or ‘≤’.

Rules of Solving Linear Inequalities:

There are certain rules which we should keep in mind while solving linear inequalities.

• Equal numbers can be added or subtracted from both sides of the inequality without affecting its sign.
• Both sides of Inequality can be divided or multiplied by any positive number but when they are multiplied or divided by a negative number, the sign of the linear inequality is reversed.

Now with this brief introduction to linear inequalities, let’s see some word problems on this concept.

## Sample Problems

Question 1: Considering the problem given in the beginning. Albert went to buy some novels for himself at the book fair. He has a total of Rs 200 with him. The book fair has a special sale policy which offers any book at Rs 70. Now he knows that he may not be able to spend the full amount on the books. Let’s say x is the number of books he bought. Represent this situation mathematically and graphically.

Solution:

We know that Albert cannot buy books for all the money he has. So, let’s say the number of books he buys is “x”. Then,  70x < 200 ⇒ x <  To plot the graph of this inequality, put x = 0.  0 <   Thus, x = 0 satisfies the inequality. So, the graph for the following inequality will look like,

Question 2: Consider the performance of the strikers of the football club Real Madrid in the last 3 matches. Ronaldo and Benzema together scored less than 9 goals in the last three matches. It is also known that Ronaldo scored three more goals than Benzema. What can be the possible number of goals Ronaldo scored?

Let’s say the number of goals scored by Benzema and Ronaldo are y and x respectively.  x = y + 3 …..(1) x + y < 9  …..(2)  Substituting the value of x from equation (1) in equation (2).  y + 3 + y < 9  ⇒2y  < 6  ⇒y < 3 Possible values of y: 0,1,2  Possible values of x: 3,4,5

Question 3: A classroom can fit at least 9 tables with an area of a one-meter square. We know that the perimeter of the classroom is 12m. Find the bounds on the length and breadth of the classroom.

It can fit 9 tables, that means the area of the classroom is atleast 9m 2 . Let’s say the length of the classroom is x and breadth is y meters.  2(x + y) = 12 {Perimeter of the classroom} ⇒ x + y = 6  Area of the rectangle is given by,  xy > 9  ⇒x(6 – x) > 9  ⇒6x – x 2 > 9  ⇒ 0 > x 2 – 6x + 9  ⇒ 0 > (x – 3) 2 ⇒ 0 > x – 3  ⇒ x < 3  Thus, length of the classroom must be less than 3 m.  So, then the breadth of the classroom will be greater than 3 m.

Question 4: Formulate the linear inequality for the following situation and plot its graph.

Let’s say Aman and Akhil went to a stationery shop. Aman bought 3 notebooks and Akhil bought 4 books. Let’s say cost of each notebook was “x” and each book was “y”. The total expenditure was less than Rs 500.

Cost of each notebook was “x” and for each book, it was “y”. Then the inequality can be described as, 3x + 4y < 500 Putting (x,y) → (0,0)  3(0) + 4(0) < 500 Origin satisfies the inequality. Thus, the graph of its solutions will look like, x.

Question 5: Formulate the linear inequality for the following situation and plot its graph.

A music store sells its guitars at five times their cost price. Find the shopkeeper’s minimum cost price if his profit is more than Rs 3000.

Let’s say the selling price of the guitar is y, and the cost price is x. y – x > 3000 ….(1) It is also given that,  y = 5x ….(2)  Substituting the value of y from equation (2) to equation (1).  5x – x > 3000  ⇒ 4x > 3000  ⇒  x >  ⇒ x> 750  Thus, the cost price must be greater than Rs 750.

Question 6: The length of the rectangle is 4 times its breadth. The perimeter of the rectangle is less than 20. Formulate a linear inequality in two variables for the given situation, plot its graph and calculate the bounds for both length and breadth.

Let’s say the length is “x” and breadth is “y”.  Perimeter = 2(x + y) < 20 ….(1) ⇒ x + y < 10  Given : x = 4y  Substituting the value of x in equation (1).  x + y < 10 ⇒ 5y < 10  ⇒ y < 2 So, x < 8 and y < 2.

Question 7: Rahul and Rinkesh play in the same football team. In the previous game, Rahul scored 2 more goals than Rinkesh, but together they scored less than 8 goals. Solve the linear inequality and plot this on a graph.

The equations obtained from the given information in the question, Suppose Rahul scored x Number of goals and Rinkesh scored y number of goals, Equations obtained will be, x = y+2 ⇢ (1) x+y< 8 ⇢ (2) Solving both the equations, y+2 + y < 8 2y < 6 y< 3 Putting this value in equation (2), x< 5

Question 8: In a class of 100 students, there are more girls than boys, Can it be concluded that how many girls would be there?

Let’s suppose that B is denoted for boys and G is denoted for girls. Now, since Girls present in the class are more than boys, it can be written in equation form as, G > B  The total number of students present in class is 100 (given), It can be written as, G+ B= 100 B = 100- G Substitute G> B in the equation formed, G> 100 – G 2G > 100 G> 50 Hence, it is fixed that the number of girls has to be more than 50 in class, it can be 60, 65, etc. Basically any number greater than 50 and less than 100.

Question 9: In the previous question, is it possible for the number of girls to be exactly 50 or exactly 100? If No, then why?

No, It is not possible for the Number of girls to be exactly 50 since while solving, it was obtained that, G> 50  In any case if G= 50 is a possibility, from equation G+ B= 100, B = 50 will be obtained. This simply means that the number of boys is equal to the number of girls which contradicts to what is given in the question. No, it is not possible for G to be exactly 100 as well, as this proves that there are 0 boys in the class.

Question 10: Solve the linear inequality and plot the graph for the same,

7x+ 8y < 30

The linear inequality is given as, 7x+ 8y< 30  At x= 0, y= 30/8= 3.75 At y= 0, x= 30/7= 4.28 These values are the intercepts. The graph for the above shall look like,  Putting x= y/2, that is, y= 2x in the linear inequality, 7x + 16x < 30 x = 1.304 y = 2.609

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## How to Solve Systems of Linear Inequalities?

Linear inequalities are expressions in which two linear expressions are compared using the inequality symbols. In this step-by-step guide, you will learn about solving systems of linear inequalities. The solution to a system of a linear inequality is the region where the graphs of all linear inequalities in the system overlap.

## Related Topics

• How to Solve Linear Inequalities
• How to Solve Quadratic Inequalities

## A step-by-step guide to solving systems of linear inequalities

The system of linear inequalities is a set of equations of linear inequality containing the same variables. Several methods of solving systems of linear equations translate to the system of linear inequalities. However, solving a system of linear inequalities is somewhat different from linear equations because the signs of inequality prevent us from solving by the substitution or elimination method. Perhaps the best way to solve systems of linear inequalities is by graphing the inequalities.

To solve a system of inequalities, graph each linear inequality in the system on the same $$x-y$$ axis by following the steps below:

• Solve the inequality for $$y$$.
• Treat the inequality as a linear equation and graph the line as either a solid line or a dashed line depending on the inequality sign. If the inequality sign does not contain an equals sign $$(< or >)$$ then draw the line as a dashed line. If the inequality sign does have an equals sign $$(≤ or ≥)$$ then draw the line as a solid line.
• Shade the region that satisfies the inequality.
• Repeat steps $$1 – 3$$ for each inequality.
• The solution set will be the overlapped region of all the inequalities.

## Solving Systems of Linear Inequalities – Example 1:

Solve the following system of inequalities.

$$\begin{cases}x\:-\:5y\ge \:6\\ \:3x\:+\:2y>1\end{cases}$$

First, isolate the variable $$y$$ to the left in each inequality:

$$x -5y≥ 6$$

$$x≥6 + 5y$$

$$5y≤ x- 6$$

$$y≤0.2 x -1.2$$

$$3x+ 2y> 1$$

$$2y>1-3x$$

$$y> 0.5-1.5x$$

Now, graph $$y≤ 0.2x-1.2$$ and $$y > 0.5 -1.5x$$ using a solid line and a broken one, respectively.

The solution of the system of inequality is the darker shaded area which is the overlap of the two individual solution regions.

## Exercises for Solving Systems of Linear Inequalities

Determine the solution to the following system of inequalities..

• $$\color{blue}{\begin{cases}5x-2y\le 10 \\ \:3x+2y>\:6\end{cases}}$$
• $$\color{blue}{\begin{cases}-2x-y<\:-1 \\ \:4x+\:2y\:\le -6\end{cases}}$$

by: Effortless Math Team about 1 year ago (category: Articles )

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## This is how we solve economic inequality

By Faiza Shaheen

09 July 2014 If you’re reading this, chances are you’re already fed up hearing how critical tackling economic inequality is to a sustainable future . You know high levels of disparity are destabilising our economy, fuelling social problems and undermining democracy – and you are probably one of the 80% of Brits who want the government to stop talking about economic inequality and do something.

But what? Today I’m speaking alongside Lisa Nandy MP, at the Westminster launch of a new NEF report setting out the five pressure points as a country we must target if we are serious about a future for the 99% .

## A national target

I’ve blogged before about the most insidious aspect of economic inequality: its ability to hijack the democratic process, as the wealthy elite move seamlessly from Oxbridge into positions of power from where they have the clout to out-lobby, out-publicise and out-finance anything that challenges the status quo.

Setting a binding national target for reducing economic inequality, much like the recent target for reducing child poverty, would be a vital first step in defusing this effect. It would solidify government commitment to act, serve as a barometer of success and, most importantly, provide an important means for the public to hold them to account.  Download the report to read more about how an economic inequality reduction target could work and the indicators we could use to measure it.

## The inequality hit list

No one policy can single-handedly beat inequality: the roots of the problem extend into the very structure of our economy. The solution will instead require ambitious, concerted action on several fronts. Our proposals focus on five major policy areas that, targeted together, could help reverse the vicious cycle.

## 1.  Make high-quality childcare available to all

We all know how critical the first five years of a person’s life are to social and cognitive development, yet the UK still has an eye-wateringly expensive childcare system that puts high quality care out of reach to those on low incomes.

Overhauling the system so that good childcare is affordable to all would help address unequal starting points and lay the foundations for a more equal society. NEF proposes state support to cap the costs of care at 15% family income, and a vast improvement in the pay, working conditions, training and status of childcare workers. More about the feasibility and cost of this in NEF’s recent report, The value of childcare .

Better, more affordable childcare would also have the bonus effect of giving mums and dads more choice over how to juggle their children, working lives and other important commitments – good for economy and well-being alike.

## 2.  Tackle polarised pay

The economy may be growing overall, but the share of wealth going into employee pay packets (as opposed to shareholder profits) is shrinking. Average real wages have been falling continuously for decades, while executive pay rockets skyward. In-work poverty has got so bad that the largest group of people claiming benefits are from families with at least one working adult.

Clearly this is not the route to a healthy, more equal economy. NEF proposes a department of labour tasked with rebuilding the link between the UKs overall economic prosperity and wages. There are plenty of places they could start: raising the minimum wage; requiring companies to publish the difference between the highest and lowest salaries they pay out; introducing pay ratios; and restoring the bargaining power of workers through embedding collective voice in the workplace. The opportunity exists for the public sector to lead the way, as spelled out in our recent report – Raising the benchmark .

## 3.  Create good jobs around the country

Our jobs market is not only geographically skewed towards London and the South East – it is hollow in the middle, as positions are increasingly divided between low-paid jobs in care, retail and hospitality and highly-paid jobs in sectors such finance, law and IT.

NEF has previously called on the government to extend the mandate of the planned British Investment Bank to not only boost lending to small and medium businesses, but ensure these businesses are capable of delivering well-paid, rewarding and environmentally viable jobs around the country.

## 4.  Transform jobs into careers with better training

It is often implied that inequality is the result of the unwillingness of those at the bottom to work hard and climb the ladder. But as young people – graduates and non-graduates alike – are increasingly sucked into dead-end jobs with scant opportunity for progression, the reality is that, for many, this ladder does not exist.

We need a major investment drive in training and skills development, at all levels of industry from junior to management (which the UK scores famously poorly on). This could involve promoting pooled training investment by sector and channelling state support towards apprenticeships that lead to progression.

## 5.  Fairer taxes

When you take account of direct and indirect taxes, those on low incomes in the UK are being hit too hard, while billions of pounds each year are being lost through tax avoidance and evasion at the top. Progressive tax reforms, such as a Land Value Tax, would help address inequality at root and redistribute economic power. Shifting the burden of taxes onto environmentally unfriendly activities would kill two birds with stone by relieving struggling families and speeding up the transition to a low-carbon economy.

Policy makers can no longer claim that there is little they can do to address growing income and wealth divides.  Our report , produced in partnership with Friedrich-Ebert-Stiftung London, provides a menu of policies that the government could apply — some of which would cost very little. Political will is now the only barrier to action.

## Image credit: cityofstrangers via Flickr

Topics Health & social care Inequality Work & pay

## Stop airport expansions

Despite the urgent need to cut carbon emissions, seven airport expansions are underway in the UK. NEF supports communities with expert analysis to challenge these expansions. But more groups are coming to NEF for help than we have resources to support.

If you’d like to support communities to stop airport expansions, donate to our Big Give #GreenMatchFund appeal before noon on Friday 29 April.

If you value great public services, protecting the planet and reducing inequality, please support NEF today.

## Make a one-off donation

Make a monthly donation. ## Investing in universal early years education pays for itself

Extending high-quality provision to those on low incomes brings biggest benefits

Jeevun Sandher , Thomas Stephens

18 July 2023 ## Migrant agricultural workers face absolute poverty while supermarkets profit

New report reveals exploitation and sets out vital reforms

Christian Jaccarini

12 July 2023 ## New Economics Podcast: Why antiracism means anticapitalism

Ayeisha Thomas-Smith is joined by Arun Kundnani

Ayeisha Thomas-Smith

19 June 2023 ## The gap between universal credit and the cost of living is growing

Universal credit is falling £890 a month short of essentials, despite inflation-linked uprating

24 May 2023

Subscribe to Here’s the Deal, our politics newsletter for analysis you won’t find anywhere else. ## Frances Tiafoe shares rollercoaster journey to becoming one of tennis’s top players Amna Nawaz Amna Nawaz Anne Azzi Davenport Anne Azzi Davenport

• Copy URL https://www.pbs.org/newshour/show/frances-tiafoe-shares-rollercoaster-journey-to-becoming-one-of-tenniss-top-players

Some say tennis is a metaphor for life, involving anticipation, problem-solving and incredibly hard work. For 25-year-old Frances Tiafoe, now one of the top 10 players in the world, those were lessons learned early, both on and off the court. Amna Nawaz caught up with Tiafoe at the U.S. Open in New York for our arts and culture series, CANVAS.

Amna Nawaz:

So, some say tennis is a metaphor for life, involving anticipation, problem-solving and incredibly hard work.

For 25-year-old Frances Tiafoe, now one of the top 10 players in the world, those were lessons learned early both on and off the court.

I caught up with Tiafoe at the U.S. Open in Flushing Meadows, New York, as part of our arts and culture series, Canvas.

… is no strange feat. Frances Tiafoe, native son, what's good?

Frances Tiafoe's trademark smile is both charming and disarming. But, in battle, a fierce competitor roars to life, propelling Tiafoe to the world's top 10 and a place in tennis history.

Frances Tiafoe Jr., Professional Tennis Player:

I'm sure a lot of people have high expectations for me, but I have high expectations for myself.

Those expectations soared after last year's U.S. Open.

Even Rafa says, too good.

In the round of 16, Tiafoe took down the number two-seated Rafael Nadal.

Frances Tiafoe says, it's my time. And that's it.

Then, in the quarterfinals, beat Andrey Rublev in straight sets, before losing to Carlos Alcaraz in the semifinals, making Tiafoe the first American man to make it to the U.S. Open semifinals since Andy Roddick in 2006 and the first Black American man since Arthur Ashe in 1972.

And, this spring, Big Foe, as he's known, became only the third Black American man in history to break into the top 10, following Ashe and James Blake. At this year's Open, the world is expecting big things from Big Foe.

Frances Tiafoe Jr.:

I want to approach it like another year somewhere where I just want to do well. I don't want to make it this big ordeal, because, ultimately, I know, when I'm at my best and having fun out there, I can do some special things.

Everyone we talked to describes you as always happy, happy-go-lucky, easygoing, laid back.

But there is this fierce competitor in you that comes out when you play. I wonder how — how do you balance those two?

Yes, I think Frances Tiafoe off the court is a totally different beast, a guy who just likes to enjoy and have fun.

But I know what I'm out there competing for. I'm competing for my family, friends, obviously, myself. I want to achieve great things, the whole DMV area. So, got a lot of people I want to continue to make happy, continue to make proud. And, yes, so that's probably what helps the balance.

Tiafoe knows where he comes from, his 6'2" frame often draped in hometown jerseys. The Maryland native is proudly DMV-made from the District/Maryland/Virginia area around Washington, D.C.

He went pro at 16, competing in Grand Slams since he was 17. He won his first in Delray Beach, Florida, in 2018, made the quarterfinals at the 2019 Australian Open and again that same year in Miami. But Tiafoe's story, he says, began long before he ever picked up a racket, when his parents, Frances Tiafoe Sr. and Alphina Kamara, fled civil war in Sierra Leone in the 1990s, raising Frances and his twin brother, Franklin, in the U.S.

His father worked in construction, helping to build the U.S. Tennis Association's Junior Tennis Champion Center, or JTCC, in College Park, Maryland, then becoming its custodian. He slept in a spare office. The Tiafoe boys did too a few days every week for over a decade.

It was there Frances Tiafoe first picked up a tennis racket, started training, and launched a career that has already inspired a new generation of fans, fans who gathered in the center his dad helped to build to cheer on Tiafoe during last year's U.S. Open, fans who now see themselves in Tiafoe.

You are only the third American Black man to make it into the top 10. Why haven't there been more?

How many people have gotten the op, though, right?

I mean, I think opportunity is everything. I'm a product of it. And how do we make the game more accessible? Obviously, USTA is doing a great job, the NJTLs, and making it more accessible across the country.

But then you also need to have the right coach, a certain amount of passion for the game. How do you have these kids wanting to continue the game at a high level, courts and what have you? So, there's a lot — a lot goes into it, but, hopefully, there's a lot more than three, right, when I'm done.

You guys gave me the best opportunity in the world to play the game of tennis.

Success, Tiafoe says, is measured by how many you bless. On a recent return to the JTCC, Tiafoe announced a $250,000 fund, seed money to boost tennis education and access in 270 communities across the country, as his mother and father looked on. Alphina Kamara, Mother of Frances Tiafoe Jr.: It's a wonderful thing for the children, because when they see somebody that grew up in the same facility that they are trying to come to, or whichever facility they are going to, and they see Frances has become what he has become, by the grace of God. Frances Tiafoe Sr., Father of Frances Tiafoe Jr.: Well, they will learn a lot from this today. They want to be like Frances or better than Frances, so they can not even be like him or more than him. Billie Jean King, Former U.S. Tennis Champion: He's great, and the crowd loves him. He's got Hollywood in him. He's got that. He loves the crowd too, right? Billie Jean King: He loves the crowd. Tennis legend Billie Jean King says Tiafoe is just what tennis needs. I think the players have to understand we're there for the audience, not the audience is there for us. And great performers know that,when they walk out there, whether there's one person in the audience or it's full, you want them to go home and say, God, that was great. Tennis great Mary Joe Fernandez, the youngest player ever to win at the U.S. Open, says Tiafoe poised for even bigger things. Mary Joe Fernandez, Tennis Champion: I love Frances. I don't know anybody that doesn't love Frances. He brings so much entertainment to the game. You really see his enjoyment of the game. And I feel like Frances can compete, he can focus, but he can also entertain, and that's very difficult. There is a mental fortitude that's unique to tennis, right? You're out there alone. You're competing at the highest level. The pressure is high. There's a lot of conversation around mental health now in sports more generally, but really in tennis too. And I just wonder how you think about that. You don't seem to struggle with it. But, again, a lot of that stuff, people don't show to the rest of the world. Yes, I think, obviously, it's a tough sport, man. It's a tough sport. You win, it's on you. You lose, it's on you, right? Traveling the world, a lot of time away from family, a lot of time away from home, dealing with expectations. When you don't get them, you got all these hate messages, people going on you and stuff like that. Do you read those messages? So, that's the funny thing. I always tell my girl, I'm not really into that, reading those messages or whatever. For me, I think it's kind of a mind-set thing. I'm not envious of anybody else or whatever. What's meant for me is meant for me. But, ultimately, these people who are sending hate messages, they're going to follow my life anyways, right. You love the crowd. You feed off the crowd. People wrote about it last time too. You would hold your arm up and gesture to them and put your hand to your ear and call for them to be louder, which is not normal in tennis culture, right? What do you love about that? What's it feel like when you're out there? I mean, a guy like me would never think he would be able to play and 23,000 people and pack the whole arena up, right, and have them whistling and yelling their name for hours and hours while I'm competing at the highest level. So I'm just loving that moment. And, again, this is about having fun, and I play better that way. He may be out on that court alone, but Tiafoe says he is held up by a cast of dozens. Who's your first call, who's your first text when you need that little moment? Yes, I definitely call both my parents, twin brother, obviously, my girlfriend who I have been with for years. She helps me so much. But, yes, I have cousin — I have a big squad, cousins, friends, I mean, a lot of people I like to stay in touch with. So… You roll deep. I roll deep, yes. You ready, right? You got the wristband and everything. With his family and a legion of fans behind him, Tiafoe is laser-focused on the road ahead and enjoying every step along the way. And our next piece on trailblazers at the U.S. Open features Billie Jean King and her fight for pay equity 50 years ago. That's coming up next week. And there's more online, including a lightning round with Frances Tiafoe. That is on our Instagram. ## Listen to this Segment Watch the Full Episode Amna Nawaz serves as co-anchor of PBS NewsHour. Anne Azzi Davenport is the Senior Producer of CANVAS at PBS NewsHour. Support Provided By: Learn more ## More Ways to Watch • PBS iPhone App • PBS iPad App  #### IMAGES 1. MathCamp321: Algebra 2 2. PPT 3. Solve each Inequality then Graph the Solution 4. Solve Absolute Value Inequality: 6|x + 3| greater than 42 5. Module 1 Inequalities Problem #3 6. Solving Equations/Inequalities #### VIDEO 1. 7.11A solving inequalities 2. Solving Inequalities..... Step by Step 3. Inequalities 2 Part 1 Solving Inequalities 4. 3 methods presented for this inequality question. How about yours? 5. How To Solve Problem Based On Inequality? 6. Inequality Reasoning Tricks #### COMMENTS 1. Solving Inequalities Simplify: x > 10 Solved! How to Solve Solving inequalities is very like solving equations, we do most of the same things ... ... but we must also pay attention to the direction of the inequality. Direction: Which way the arrow "points" Some things can change the direction! < becomes > > becomes < ≤ becomes ≥ ≥ becomes ≤ Safe Things To Do 2. Solving equations & inequalities Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions. Linear equations with variables on both sides Learn Why we do the same thing to both sides: Variable on both sides Intro to equations with variables on both sides 3. 3.6: Solve Applications with Linear Inequalities Many real-life situations require us to solve inequalities. In fact, inequality applications are so common that we often do not even realize we are doing algebra. For example, how many gallons of gas can be put in the car for$20? Is the rent on an apartment affordable? Is there enough time before class to go get lunch, eat it, and return?

4. Using Inequalities To Solve Problems

This video looks at different word problems involving inequalities. This first video introduces how to solve the problems. Khan Academy exercise to practice:...

5. Inequality Calculator

Step 1: 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. Step 2: Click the blue arrow to submit. Choose "Simplify" from the topic selector and click to see the result in our Algebra Calculator! Examples Simplify

6. Solving and graphing linear inequalities (video)

The equation y>5 is a linear inequality equation. y=0x + 5. So whatever we put in for x, we get x*0 which always = 0. So for whatever x we use, y always equals 5. The same thing is true for y>5. y > 0x + 5. And again, no matter what x we use, y is always greater than 5.

7. Linear inequality word problems

Test prep > SAT (Fall 2023) > About the SAT Math Test > Heart of Algebra: lessons by skill Linear inequality word problems | Lesson Google Classroom What are linear inequality word problems, and how frequently do they appear on the test? Linear inequalities are very common in everyday life.

8. 1.1: Solving Linear Equations and Inequalities

Solving Basic Linear Equations. An equation 129 is a statement indicating that two algebraic expressions are equal. A linear equation with one variable 130, $$x$$, is an equation that can be written in the standard form $$ax + b = 0$$ where $$a$$ and $$b$$ are real numbers and $$a ≠ 0$$.For example $$3 x - 12 = 0$$ A solution 131 to a linear equation is any value that can replace the ...

Now let's solve it! First, let's subtract 20 from both sides: −10 < −5t 2 <−5. Now multiply both sides by −(1/5). But because we are multiplying by a negative number, the inequalities will change direction ... read Solving Inequalities to see why. 2 > t 2 > 1. To be neat, the smaller number should be on the left, and the larger on the ...

If the inequality involves "less than," then determine the $$x$$-values where the function is below the $$x$$-axis. If the inequality involves "greater than," then determine the $$x$$-values where the function is above the $$x$$-axis. We can streamline the process of solving quadratic inequalities by making use of a sign chart.

11. Solving word problems involving inequalities

0:00 / 4:26 Solving word problems involving inequalities MooMooMath and Science 353K subscribers Subscribe 979 Share 111K views 6 years ago Math Help Learn to solve word problems that...

12. Algebra

Chapter 2 : Solving Equations and Inequalities. Here are a set of assignment problems for the Solving Equations and Inequalities chapter of the Algebra notes. Please note that these problems do not have any solutions available. These are intended mostly for instructors who might want a set of problems to assign for turning in.

13. How to Solve Word Problems Involving One-step Inequalities?

A word problem involving one-step inequalities is a type of mathematical problem that describes a situation using words and requires solving an inequality that can be solved with just one step. In other words, the problem can be solved by performing one mathematical operation, such as adding, subtracting, multiplying, or dividing.

14. Solving Word Problems in Algebra

Solution Step 1: Highlight the important information in this problem. Note: At least is a key word that notes that this problem must be written as an inequality.

15. SOLVING PROBLEMS INVOLVING LINEAR INEQUALITIES IN TWO ...

WOW MATH 523K subscribers Subscribe 1.2K 81K views 2 years ago GRADE 8 || SECOND QUARTER ‼️SECOND QUARTER‼️ 🟡 GRADE 8: SOLVING PROBLEMS INVOLVING SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES

16. Solving Word Problems Involving Inequalities

Thanks to all of you who support me on Patreon. You da real mvps! \$1 per month helps!! :) https://www.patreon.com/patrickjmt !! Solving Word Problems Invo...

17. Solving Linear Inequalities Word Problems

Rules of Solving Linear Inequalities: There are certain rules which we should keep in mind while solving linear inequalities. Equal numbers can be added or subtracted from both sides of the inequality without affecting its sign.

18. HOW TO SOLVE PROBLEMS INVOLVING INEQUALITIES

0:00 / 9:10 HOW TO SOLVE PROBLEMS INVOLVING INEQUALITIES SIR ARIEL MATH 9.25K subscribers 5.4K views 1 year ago Grade 7 2nd Quarter How to solve Problems Involving Inequalities...

19. Word Problems Involving Linear Inequalities in two Variables

Grade 8 Math Word Problems Involving Linear Inequalities in two Variables Math A Matic 562 subscribers Subscribe 6 265 views 1 year ago This video is focused on how to solve word problems...

20. How to Solve Systems of Linear Inequalities?

To solve a system of inequalities, graph each linear inequality in the system on the same $$x-y$$ axis by following the steps below: Solve the inequality for $$y$$. Treat the inequality as a linear equation and graph the line as either a solid line or a dashed line depending on the inequality sign.

21. Why can't we solve the problem of inequality?

Answer (1 of 30): Firstly, it's not a problem and there for it does not need to be solved. The problem is that people have an emotional reaction to something that is not an actual problem. And emotional reactions to non-problems usually then creates problems where none actually existed originally...

22. This is how we solve economic inequality

4. Transform jobs into careers with better training. It is often implied that inequality is the result of the unwillingness of those at the bottom to work hard and climb the ladder. But as young people - graduates and non-graduates alike - are increasingly sucked into dead-end jobs with scant opportunity for progression, the reality is that ...

23. Frances Tiafoe shares rollercoaster journey to becoming one of ...

Some say tennis is a metaphor for life, involving anticipation, problem-solving and incredibly hard work. For 25-year-old Frances Tiafoe, now one of the top 10 players in the world, those were ...