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GCSE Maths · EdexcelChapter 10 · Inequalities
Chapter 10 · Higher tier

GCSE Maths Inequalities (Higher): worked examples and questions with answers

By the end of this chapter you will be able to

  • Read and write the inequality symbols <, >, ≤ and ≥, and turn words such as “at least” into symbols
  • Show an inequality on a number line using open and closed circles
  • Solve linear inequalities, including when you must multiply or divide by a negative number
  • Solve double inequalities, and list the integers that satisfy an inequality
  • Form and solve inequalities from real-life situations
  • Higher tier: use set notation, solve quadratic inequalities and show regions on a graph

You should already know

  • Ordering positive and negative numbers (Chapter 1)
  • Solving linear equations (Chapter 9)
  • Substituting into expressions (Chapter 8)
  • Expanding brackets (Chapter 7)

Where this leads

  • Regions and graphs of straight lines (Chapter 12)
  • Quadratic graphs (Chapter 13)
  • Solving quadratic equations (Chapter 14)
  • Error intervals (Chapter 4)

Edexcel GCSE (9–1) Mathematics 1MA1, Higher tier. Content references: A22.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.
10.1

Inequality symbols

Edexcel 1MA1: A22
Grade 1-3

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Inequality symbolsComing soon

An equation says that two things are exactly equal, and uses the sign =. But many situations in real life are not exact. A fairground ride says “You must be at least 120 cm tall”. A lift says “Maximum 8 people”. A phone plan gives you “up to 5 GB” of data. None of these tells you one exact number. They tell you what is allowed: a whole range of numbers. In maths we write these ideas with inequality symbols.

There are four symbols. Each one compares two numbers or quantities.

SymbolRead asExample
<is less than3 < 7 (3 is smaller than 7)
>is greater than9 > 2 (9 is bigger than 2)
≤is less than or equal to5 ≤ 5 and 4 ≤ 5 are both true
≥is greater than or equal to8 ≥ 8 and 9 ≥ 8 are both true

How to remember which way round. Think of the symbol as a hungry mouth. The wide open end always faces the bigger number, and the small pointed end points at the smaller number. So 3 < 7 and 7 > 3 say exactly the same thing.

The line underneath. The extra line under ≤ and ≥ adds the words “or equal to”. So x ≥ 4 means “x is 4 or more”, and 4 itself is allowed. But x > 4 means “x is more than 4”, and 4 itself is not allowed.

Inequalities with a letter. When the inequality contains a letter, such as x > 3, it does not have one answer. It describes every number that makes it true: 3.1, 4, 7.5, 100 and so on. We call any such number a solution. To test a number, replace the letter with it and see whether the statement is true.

Negative numbers. On a number line, numbers get bigger as you go right and smaller as you go left. So −5 is smaller than −2, even though 5 looks bigger than 2. We write −5 < −2. A temperature of −5°C is colder than −2°C.

-6-5-4-3-2-10123456−5−23
Numbers get bigger to the right. So −5 < −2 and −2 < 3.

Turning words into symbols. These phrases appear in exam questions all the time. Learn which have the “or equal to” line.

WordsSymbol
more than, greater than, over, above, exceeds>
less than, fewer than, under, below<
at least, no less than, a minimum of≥
at most, no more than, a maximum of≤

Key ideas

  • < means “less than” and > means “greater than”. The open end of the symbol faces the bigger number.
  • ≤ and ≥ include the number itself (“or equal to”). < and > do not.
  • A solution of an inequality is any number that makes it true. There are usually many solutions.
  • “At least” means ≥ and “at most” means ≤.
  • With negative numbers, the number further to the left is smaller: −7 < −3.
Worked example 10.1.1
Write < or > in the gap: −6 ___ −2.

Key words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.

Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.

  1. Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.
    −6 and −2 on the same number line
  2. Decide which one is further to the right. The number further right is the bigger one. −2 is further right.
    −6 is to the left of −2
  3. Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.
    −6 < −2

Check. Read it back in words: “−6 is less than −2”. Both numbers are negative: the one nearer to zero is the bigger one. ✓

Watch out. With negative numbers, a number that looks “bigger” is actually smaller: −5 < −2 because −5 is further to the left.

Worked example 10.1.2
A ride says: “Riders must be at least 12 years old.” Write this as an inequality, using a for the age in years.

Key words. At least means this or more. At most means this or less.

Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.

  1. Find the key phrase. The phrase tells us which symbol to use.
    “at least”
  2. Decide if the number itself is allowed. “at least” allows the boundary number itself, so we need the “or equal to” line under the symbol.
    ≥
  3. Write it with the letter on the left. a stands for the age of the rider in years.
    a ≥ 12

Check. Read it back: “a is greater than or equal to 12” means the same as the sentence. ✓

Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).

Worked example 10.1.3
Here is an inequality: x ≤ 5. Which of these numbers are solutions?   −3,   5,   5.5,   4.9,   0

Key words. A solution is a number that makes the inequality true.

Method. Put each number in place of the letter and decide whether the inequality is true.

  1. Write the inequality and what it means. We test each number by replacing x with it.
    x ≤ 5, which says “x is less than or equal to 5”
  2. Test every number in turn. A number is a solution only if the inequality is true for it. Compare using a number line if you need to.
    −3 ≤ 5?   True  ✓ allowed
    5 ≤ 5?   True  ✓ allowed
    5.5 ≤ 5?   False  ✗ not allowed
    4.9 ≤ 5?   True  ✓ allowed
    0 ≤ 5?   True  ✓ allowed
  3. List the numbers that work. Keep only the ones marked true.
    −3, 5, 4.9, 0

Check. The boundary is 5. Numbers to the left of it work, and the boundary itself works, which agrees with the list. ✓

Watch out. Check the boundary number 5 carefully. With ≤ it is a solution.

Worked example 10.1.4
Is the statement −4 ≥ −4 true or false?

Key words. A statement is true if it is correct and false if it is wrong.

Method. Say the statement in words, compare the numbers on a number line, then decide.

  1. Read the statement in words. Say what the symbol means.
    −4 ≥ −4 reads “−4 is bigger than or equal to −4”
  2. Compare the two numbers. Use a number line if you need to: further right means bigger.
    −4 is equal to −4
  3. Decide whether the statement is true. For ≤ and ≥ the statement is true if the numbers are equal or the inequality holds.
    True

Check. Test the opposite claim: −4 < −4 would be false, which agrees. ✓

Watch out. ≤ and ≥ are true when the two numbers are equal as well. So 5 ≤ 5 is true.

Watch out

“At least 12” means 12 or more, so it is a ≥ 12, not a > 12. The word “at” is the clue that the number itself is allowed. Also be careful with negative numbers: −9 is smaller than −3.

Exam tip

If you are unsure which way a symbol points, put the two numbers on a number line. The bigger number is on the right, and the open end of the symbol faces it.

Exercise 10A  Reading and writing inequalities

Non-calculator
1
FWrite < or > in each gap.
(a)7  ___  3
(b)2  ___  9
(c)−4  ___  1
(d)−3  ___  −8
(e)0.5  ___  0.25
(f)−1.5  ___  −1.2
2
FWrite whether each statement is true or false.
(a)6 > 2
(b)−5 > −2
(c)4 ≤ 4
(d)9 ≥ 10
(e)−3 < −1
(f)0 ≥ −7
3
FWrite each statement as an inequality.
(a)A lift can carry at most 8 people. Use p for the number of people.
(b)You must be at least 14 years old to join. Use a for the age in years.
(c)The temperature, t °C, is less than 0°C.
(d)A parcel must weigh no more than 2 kg. Use w for the mass in kg.
(e)More than 500 people came to the match. Use n for the number of people.
(f)A pupil needs a minimum of 40 marks to pass. Use m for the marks.
4
FWrite each inequality in words, starting with the letter.
(a)x > 5
(b)n ≤ 3
(c)y ≥ 10
(d)t < 0.5
(e)a < −4
(f)k ≥ 0
5
FHere is an inequality: x > 3. Which of these numbers are solutions?   1,   3,   3.1,   5,   −4
6
FHere is an inequality: x ≤ −2. Which of these numbers are solutions?   −5,   −2,   −1,   0,   −2.5
7
PSWrite each rule as an inequality.
(a)A road sign says the maximum speed is 40 mph. Use s for the speed in mph.
(b)A ride needs the rider’s height, h cm, to be at least 120 cm.
(c)A café has fewer than 25 customers. Use c for the number of customers.
(d)A school needs over 90 volunteers. Use v for the number of volunteers.
8
PSThe speed limit on a road is 40 mph, so a driver’s speed, s mph, must satisfy s ≤ 40. Four cars are travelling at 38 mph, 40 mph, 43 mph and 39.5 mph. Which cars are obeying the limit?
9
PSAn airline allows bags of at most 23 kg. So a bag’s mass, w kg, must satisfy w ≤ 23. Which of these bags are allowed?   22.8 kg,   23 kg,   23.1 kg,   21.5 kg
10
FWrite down all the whole numbers n that satisfy each inequality.
(a)2 < n < 6
(b)−3 ≤ n < 1
(c)0 < n ≤ 4
(d)−5 < n < −1
11
RExplain why 5 ≤ 5 is a true statement, but 5 < 5 is false.
12
RAmir writes −9 > −3. Is he correct? Give a reason.
13
RBen writes “x is no more than 7” as x ≥ 7. Explain his mistake and write the correct inequality.
14
RWrite an inequality whose only whole-number solutions are 4, 5 and 6.
15
Rn is a positive whole number and n2 < 20. Write down all the possible values of n.
10.2

Inequalities on a number line

Edexcel 1MA1: A22
Grade 1-4

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Inequalities on a number lineComing soon

A number line is a way to see all the values an inequality allows. Imagine drawing a straight line with the numbers marked along it, then colouring every value that is a solution.

We need two pieces of information: where it starts or stops and which way it goes.

-3-2-101234567
x > 2: open circle at 2 (2 is not included), arrow pointing right.
−3−2−101
x ≤ −1: closed circle at −1 (included), arrow pointing left.

Two ends. Some inequalities have a smallest and a largest value, for example −2 ≤ x < 3. This is read “x is bigger than or equal to −2 and smaller than 3”. We draw a circle at each end (closed at −2, open at 3) and join them with a line.

−4−3−2−1012345
−2 ≤ x < 3: closed circle at −2, open circle at 3, line between.

Whole numbers only. If the question says the letter is a whole number (an integer), the solutions are separate points, not a continuous line. We mark them with dots.

−4−3−2−1012345
If x must be a whole number, the solutions of −2 ≤ x < 3 are −2, −1, 0, 1, 2.

Key ideas

  • Closed (filled) circle = the number is included (≤ or ≥).
  • Open circle = the number is not included (< or >).
  • The arrow points to the values that work: right for >, ≥; left for <, ≤.
  • For a range such as 1 < x ≤ 5, draw a circle at each end and join them with a line.
  • For whole-number solutions only, use separate dots.
Worked example 10.2.1
Write down the inequality shown on this number line.
12345

Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.

Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.

  1. Look at the circle. The circle is open, so the number is not included. We need < or >, with no “or equal to”.
    Circle at 3: open
  2. Look at the arrow. An arrow pointing left shows values smaller than the circle; an arrow pointing right shows larger values.
    Arrow points left
  3. Write the inequality. Put the letter in the middle for a range, or on the left for a single end.
    x < 3

Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓

Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.

Worked example 10.2.2
Show the inequality x ≤ −1 on a number line.

Key words. Closed circle (filled) = included. Open circle = not included.

Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.

  1. Find the boundary number(s). These are the numbers where the inequality starts or stops.
    −1
  2. Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.
    −1: closed
  3. Draw the line or arrow. Draw it over all the values that make the inequality true.
    a closed (filled) circle at −1 with an arrow pointing left
    −4−3−2−1012

Check. Pick a number on your line and put it into x ≤ −1: it makes the inequality true. ✓

Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.

Worked example 10.2.3
Write down the inequality shown on this number line.
−4−3−2−1012345

Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.

Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.

  1. Look at the two circles. A closed circle means the number is included (≤). An open circle means it is not (<).
    Left circle at −2: closed
    Right circle at 3: open
  2. Look at the line between them. Every number on the line is a solution, so the letter is between the two circles.
    −2 and 3 are the two ends
  3. Write the inequality. Put the letter in the middle for a range, or on the left for a single end.
    −2 ≤ x < 3

Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓

Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.

Worked example 10.2.4
Show the inequality 1 < x ≤ 5 on a number line.

Key words. Closed circle (filled) = included. Open circle = not included.

Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.

  1. Find the boundary number(s). These are the numbers where the inequality starts or stops.
    1, 5
  2. Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.
    1: open; 5: closed
  3. Draw the line or arrow. Draw it over all the values that make the inequality true.
    an open circle at 1, a closed (filled) circle at 5, and a line joining them
    −2−1012345678

Check. Pick a number on your line and put it into 1 < x ≤ 5: it makes the inequality true. ✓

Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.

Watch out

The most common mistake is the wrong circle. Open means “stop just before this number” (< or >). Closed means “include this number” (≤ or ≥). Also check the arrow: x < 3 points left towards the smaller numbers, even though the symbol itself looks like it opens to the right.

Exam tip

Check any number-line answer by picking one number on the line and one number off it. Put both into the inequality: the first should work and the second should not.

Exercise 10B  Number lines

Non-calculator
1
FWrite down the inequality shown on each number line.
(a)01234
(b)−3−2−101
(c)−5−4−3−2−1
(d)−2−1012
2
FWrite down the inequality shown on each number line. Use x for the letter.
(a)−4−3−2−1012345
(b)−1012345678
(c)−6−5−4−3−2−101
(d)−2−101234567
3
FShow each inequality on a number line.
(a)x > 3
(b)x ≤ −1
(c)x < 0
(d)x ≥ 2
(e)−1 < x ≤ 3
(f)2 ≤ x < 5
4
FWhich of these inequalities does this number line show?   x < 3,   x ≤ 3,   x > 3,   x ≥ 3
12345
5
FIs 3 a solution of each inequality?
(a)x > 3
(b)x ≥ 3
(c)x < 3
(d)x ≤ 3
6
FIs the number in brackets a solution of the inequality?
(a)x > −2   (−1)
(b)x ≤ 4   (4)
(c)x < 0.5   (1)
(d)x ≥ −3   (−5)
7
FShow each statement on a number line. Use x for the number.
(a)x is at least −2
(b)x is less than 4
(c)x is more than 0
(d)x is at most 6
8
PSA lift can carry at most 8 people. Let n be the number of people in the lift. Show the possible values of n on a number line and write an inequality using ≤ twice.
9
PSA ride is only for people whose height h cm is at least 110 cm but less than 195 cm. Write this as an inequality and show it on a number line.
10
RZoe says this number line shows x < 4.
23456
What mistake has she made? Write down the inequality that is actually shown.
11
RExplain the difference between x > 3 and x ≥ 3. Say what the number lines look like.
12
RFour whole numbers are shown on this number line.
−4−3−2−10123
Write down the four numbers, and an inequality for n using ≤ twice.
13
RThis number line shows two separate parts. Write down the inequalities.
−3−2−1012345
14
RThe number line for x > −1 and the number line for x ≤ 4 are drawn together.
−3−2−10123456
Write down the single inequality that gives the values satisfying both.
10.3

Solving linear inequalities

Edexcel 1MA1: A22
Grade 3-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Solving linear inequalitiesComing soon

You already know how to solve an equation such as 2x + 3 = 11. You do the same thing to both sides, like a pair of balanced scales, until the letter is on its own. Solving an inequality works in almost exactly the same way. The only difference is that the answer is a range of values, and there is one important new rule (see Section 10.4).

Why can we do the same to both sides? Start with a true statement: 3 < 7. Add 2 to both sides: 5 < 9, still true. Subtract 1 from both sides: 2 < 6, still true. Multiply both sides by 2: 6 < 14, still true. Divide both sides by 2: 1.5 < 3.5, still true. Adding, subtracting, and multiplying or dividing by a positive number keeps the inequality true and keeps the sign the same.

The method. Use inverse operations, in this order:

  1. Undo any adding or subtracting first (these are the furthest from the letter).
  2. Then undo any multiplying or dividing.
  3. Keep the inequality sign the same as you go (as long as you only multiply or divide by positive numbers).
  4. Write the answer as x < … or x > … (letter on the left).

Reading the answer. The solution x > 4 does not mean x equals 4. It means x can be any number bigger than 4. Always test your answer: choose a number that should be a solution and one that should not, and put each back into the original inequality.

Letter on the right. If the letter is on the right-hand side, for example 12 > 2x + 4, solve in the same way and you will get 4 < x. This is fine, but it is neater to turn it round: 4 < x says the same as x > 4 (the sign still points at the smaller number).

Key ideas

  • Adding or subtracting the same number on both sides keeps the sign the same.
  • Multiplying or dividing both sides by a positive number keeps the sign the same.
  • Undo +/− first, then ×/÷.
  • The solution is a range of values, so test one value inside and one outside.
  • Turn a result such as 4 < x round to x > 4 (the sign still points at the small number).
Worked example 10.3.1
Solve x + 7 > 12.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    x + 7 > 12
  2. Get the plain numbers on the other side. Subtract 7 from both sides. This is the inverse of adding 7.
    x > 5
  3. State the solution. This is the complete set of values that make the inequality true.
    x > 5

Check. Test a value inside the solution: x = 6 gives left side = 6 + 7 = 13, right side = 12. Is 13 > 12? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = 4 + 7 = 11, right side = 12. Is 11 > 12? No, it is false, so 4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Worked example 10.3.2
Solve 3x − 4 ≤ 11, and show the solution on a number line.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    3x − 4 ≤ 11
  2. Get the plain numbers on the other side. Add 4 to both sides. This is the inverse of subtracting 4.
    3x ≤ 15
  3. Divide both sides by 3. 3x means 3 lots of x. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
    x ≤ 15 ÷ 3 = 5
  4. State the solution. This is the complete set of values that make the inequality true.
    x ≤ 5
    On a number line: a closed (filled) circle at 5 with an arrow pointing left.
    2345678

Check. Test a value inside the solution: x = 4 gives left side = 3 × 4 − 4 = 8, right side = 11. Is 8 ≤ 11? Yes, it is true. ✓
Test a value outside it: x = 6 gives left side = 3 × 6 − 4 = 14, right side = 11. Is 14 ≤ 11? No, it is false, so 6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Worked example 10.3.3
Solve x/4 + 3 ≥ 5.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    x4 + 3 ≥ 5
  2. Multiply both sides by 4 to get rid of the fraction. Multiplying by the denominator cancels the division. 4 is a positive number, so the inequality sign does not change.
    x + 12 ≥ 20
  3. Get the plain numbers on the other side. Subtract 12 from both sides. This is the inverse of adding 12.
    x ≥ 8
  4. State the solution. This is the complete set of values that make the inequality true.
    x ≥ 8

Check. Test a value inside the solution: x = 9 gives left side = 94 + 3 = 5.25, right side = 5. Is 5.25 ≥ 5? Yes, it is true. ✓
Test a value outside it: x = 7 gives left side = 74 + 3 = 4.75, right side = 5. Is 4.75 ≥ 5? No, it is false, so 7 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Worked example 10.3.4
Solve 12 > 2x + 4.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    12 > 2x + 4
  2. Get the plain numbers on the other side. Subtract 4 from both sides.
    8 > 2x
  3. Divide both sides by 2. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
    8 ÷ 2 = 4
    4 > x
  4. Write it with x on the left. 4 > x says the same thing as x < 4: the sign still points to the smaller number.
    x < 4
  5. State the solution. This is the complete set of values that make the inequality true.
    x < 4

Check. Test a value inside the solution: x = 3 gives left side = 12, right side = 2 × 3 + 4 = 10. Is 12 > 10? Yes, it is true. ✓
Test a value outside it: x = 5 gives left side = 12, right side = 2 × 5 + 4 = 14. Is 12 > 14? No, it is false, so 5 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Watch out

Do not change the inequality sign into an equals sign half way through, and do not drop the “or equal to” line. If you start with ≤, you must finish with ≤. Also, only the letter on its own is the answer: 3x > 6 is not finished until you divide by 3.

Exam tip

Always finish with a quick test. Pick a whole number in your solution, put it into the original inequality, and check that it is true.

Exercise 10C  Solving one-step and two-step inequalities

Non-calculator
1
FSolve each inequality.
(a)x + 5 > 9
(b)x − 3 ≤ 4
(c)2x < 14
(d)x4 ≥ 3
(e)x + 7 ≤ 2
(f)5x > 35
2
FSolve each inequality.
(a)2x + 3 > 11
(b)3x − 4 ≤ 14
(c)5x + 1 < 26
(d)4x − 7 ≥ 9
(e)2x + 9 ≤ 3
(f)6x − 5 > −17
3
FSolve each inequality. Give your answer with x on the left.
(a)12 > 2x + 4
(b)20 ≤ 3x − 1
(c)7 < 5x − 8
(d)10 ≥ 4x + 2
(e)−4 < 2x + 6
(f)2 > 3x − 7
4
FSolve each inequality. (The letter is divided by a number.)
(a)x3 − 2 > 1
(b)x2 + 3 ≤ 5
(c)x5 + 1 ≥ 4
(d)2x3 < 8
5
RHana solves 2x − 5 < 9 and writes x < 2. Explain her mistake and give the correct solution.
6
RShow that x = 4 is a solution of 5x − 3 > 12, but x = 3 is not.
7
PSSolve 4x − 3 ≤ 17. Write down the largest whole number that satisfies the inequality.
8
PSSolve 3n + 5 > 20. Write down the smallest whole number that satisfies the inequality.
9
PSThe number n satisfies 5n − 4 < 31. What is the largest whole-number value of n?
10
PSSolve 2x + 7 ≥ 25 and show your solution on a number line.
11
RSolve 3x + 2 = 14 and 3x + 2 > 14. What is the difference between the two answers?
12
RIs it true that 2x < 10 and 10 > 2x have the same solutions? Show your working.
13
RMia and Leo both solve 3x + 4 > 19. Mia gets x > 5. Leo gets x ≥ 5. Who is right? Test x = 5 to decide.

Exercise 10D  Solving, number lines and whole-number solutions

Non-calculator
1
FSolve each inequality and show the solution on a number line.
(a)2x + 1 > 9
(b)3x − 2 ≤ 7
(c)4x + 5 < 1
(d)5x − 1 ≥ 14
2
FSolve each inequality, then write down the largest whole number that satisfies it.
(a)2x + 3 < 12
(b)5x − 1 ≤ 24
(c)3x + 2 < 9
(d)4x + 1 ≤ 20
3
FSolve each inequality, then write down the smallest whole number that satisfies it.
(a)3x + 1 > 13
(b)2x − 5 ≥ 4
(c)5x + 2 > 6
4
FSolve each inequality, then list the positive whole numbers that satisfy it.
(a)2x + 1 < 11
(b)3x ≤ 12
(c)5x − 3 < 12
(d)4x + 3 ≤ 20
5
PSSam thinks of a whole number, n. He multiplies it by 3 and adds 4. The answer is less than 30. What is the largest number Sam could be thinking of?
6
RDev solves 2x + 1 < 8 and gets x < 3.5. He says the largest whole number is 4. Is he right? Explain.
7
RExplain why the smallest whole number satisfying n > 1.3 is 2, and not 1.
8
FSolve each inequality.
(a)x + 32 ≥ 4
(b)2x − 13 < 5
(c)x + 45 > 3
9
FSolve each inequality.
(a)x + 32 ≥ x − 13
(b)2x − 13 < x + 42
(c)3x + 14 > x − 23
10.4

Negative numbers, brackets and letters on both sides

Edexcel 1MA1: A22
Grade 4-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Negative numbers, brackets and letters on both sidesComing soon

The one new rule. When you multiply or divide both sides by a negative number, the inequality sign must be reversed. Here is why. Start with a true statement, 2 < 5. Multiply both sides by −1: you get −2 and −5. On the number line −2 is to the right of −5, so −2 > −5. The order has flipped, so we must flip the sign: 2 < 5 becomes −2 > −5.

Multiplying by a negative number turns the number line round, like looking at it in a mirror, so bigger numbers become smaller and smaller numbers become bigger. So:

−3x > 12   becomes, after dividing by −3,   x < −4

Adding or subtracting a negative number never reverses the sign. It only happens when you multiply or divide.

A way to avoid it. You can avoid dividing by a negative number completely. In 7 − 2x ≥ 1, add 2x to both sides to get 7 ≥ 1 + 2x. Now the x term is positive. Solve that: 6 ≥ 2x, so 3 ≥ x, which is x ≤ 3. Both routes give the same answer, so use whichever you trust more, and use the other one to check.

Letter on both sides. Collect the x terms on one side by adding or subtracting the same x term from both sides. A good habit is to remove the smaller x term, so the one that is left is positive and you never need to divide by a negative.

Brackets. Expand brackets first (multiply every term inside by the number outside), then collect like terms. Fractions. Multiply both sides by the denominator to remove the fraction. The denominator is positive, so the sign stays the same.

Key ideas

  • Multiplying or dividing both sides by a negative number reverses the sign: < becomes >, ≤ becomes ≥.
  • Adding or subtracting never reverses the sign.
  • To avoid a negative coefficient, move the x term to the side where it is positive.
  • With x on both sides, remove the smaller x term first.
  • Expand brackets first. Multiply by the denominator to clear fractions.
Worked example 10.4.1
Solve −3x > 12.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    −3x > 12
  2. Divide both sides by −3, and reverse the inequality sign. −3x means −3 lots of x, so we divide by −3. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So > becomes <.
    x < 12 ÷ (−3) = −4
  3. State the solution. This is the complete set of values that make the inequality true.
    x < −4

Check. Test a value inside the solution: x = −5 gives left side = −3 × (−5) = 15, right side = 12. Is 15 > 12? Yes, it is true. ✓
Test a value outside it: x = −3 gives left side = −3 × (−3) = 9, right side = 12. Is 9 > 12? No, it is false, so −3 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −3x > 12 this gives −12 > 3x. Divide by 3 (positive, so no flip): −4 > x, which is the same as x < −4. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

Worked example 10.4.2
Solve 7 − 2x ≥ 1.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    −2x + 7 ≥ 1
  2. Get the plain numbers on the other side. Subtract 7 from both sides. This is the inverse of adding 7.
    −2x ≥ −6
  3. Divide both sides by −2, and reverse the inequality sign. −2x means −2 lots of x, so we divide by −2. This is a negative number, and dividing (or multiplying) by a negative flips the order of numbers: for example 2 < 5, but if we multiply both by −1 we get −2 > −5. So ≥ becomes ≤.
    x ≤ −6 ÷ (−2) = 3
  4. State the solution. This is the complete set of values that make the inequality true.
    x ≤ 3

Check. Test a value inside the solution: x = 2 gives left side = −2 × 2 + 7 = 3, right side = 1. Is 3 ≥ 1? Yes, it is true. ✓
Test a value outside it: x = 4 gives left side = −2 × 4 + 7 = −1, right side = 1. Is −1 ≥ 1? No, it is false, so 4 is correctly left out.
A different route that avoids dividing by a negative number: collect the x terms on the side where they are positive. From −2x + 7 ≥ 1 this gives 6 ≥ 2x. Divide by 2 (positive, so no flip): 3 ≥ x, which is the same as x ≤ 3. Same answer. ✓

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

Worked example 10.4.3
Solve 5x − 2 < 3x + 8.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    5x − 2 < 3x + 8
  2. Get the x terms on one side. There are x terms on both sides. We remove the smaller one, so that the x term left over is positive and we do not need to divide by a negative number later.
    Subtract 3x from both sides:
    2x − 2 < 8
  3. Get the plain numbers on the other side. Add 2 to both sides. This is the inverse of subtracting 2.
    2x < 10
  4. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
    x < 10 ÷ 2 = 5
  5. State the solution. This is the complete set of values that make the inequality true.
    x < 5

Check. Test a value inside the solution: x = 4 gives left side = 5 × 4 − 2 = 18, right side = 3 × 4 + 8 = 20. Is 18 < 20? Yes, it is true. ✓
Test a value outside it: x = 6 gives left side = 5 × 6 − 2 = 28, right side = 3 × 6 + 8 = 26. Is 28 < 26? No, it is false, so 6 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Worked example 10.4.4
Solve 5 − 3x < 2x + 20.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    −3x + 5 < 2x + 20
  2. Get the x terms on one side. There are x terms on both sides. We remove the smaller one, so that the x term left over is positive and we do not need to divide by a negative number later.
    Add 3x to both sides:
    5 < 5x + 20
  3. Get the plain numbers on the other side. Subtract 20 from both sides.
    −15 < 5x
  4. Divide both sides by 5. The inverse of multiplying by 5 is dividing by 5. 5 is positive, so the inequality sign stays the same.
    −15 ÷ 5 = −3
    −3 < x
  5. Write it with x on the left. −3 < x says the same thing as x > −3: the sign still points to the smaller number.
    x > −3
  6. State the solution. This is the complete set of values that make the inequality true.
    x > −3

Check. Test a value inside the solution: x = −2 gives left side = −3 × (−2) + 5 = 11, right side = 2 × (−2) + 20 = 16. Is 11 < 16? Yes, it is true. ✓
Test a value outside it: x = −4 gives left side = −3 × (−4) + 5 = 17, right side = 2 × (−4) + 20 = 12. Is 17 < 12? No, it is false, so −4 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Worked example 10.4.5
Solve 3(x + 2) > 2(x − 1).

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    3(x + 2) > 2(x − 1)
  2. Expand the bracket. Multiply every term inside the bracket by the number outside it.
    3 × x = 3x
    3 × 2 = 6
    2 × x = 2x
    2 × (−1) = −2
    3x + 6 > 2x − 2
  3. Get the x terms on one side. There are x terms on both sides. We remove the smaller one, so that the x term left over is positive and we do not need to divide by a negative number later.
    Subtract 2x from both sides:
    x + 6 > −2
  4. Get the plain numbers on the other side. Subtract 6 from both sides. This is the inverse of adding 6.
    x > −8
  5. State the solution. This is the complete set of values that make the inequality true.
    x > −8

Check. Test a value inside the solution: x = −7 gives left side = 3 × ((−7) + 2) = −15, right side = 2 × ((−7) − 1) = −16. Is −15 > −16? Yes, it is true. ✓
Test a value outside it: x = −9 gives left side = 3 × ((−9) + 2) = −21, right side = 2 × ((−9) − 1) = −20. Is −21 > −20? No, it is false, so −9 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Worked example 10.4.6
Solve (x + 3)/2 ≥ (x − 1)/3.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve it like an equation: do the same to both sides to get the letter on its own. The one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

  1. Write down the inequality. We want the letter on its own on one side. We do this by doing the same thing to both sides, exactly as when solving an equation.
    x + 32 ≥ x − 13
  2. Multiply both sides by 6 to get rid of the fraction. Multiplying by the denominator cancels the division. 6 is a positive number, so the inequality sign does not change.
    3(x + 3) ≥ 2(x − 1)
  3. Expand the bracket. Multiply every term inside the bracket by the number outside it.
    3 × x = 3x
    3 × 3 = 9
    2 × x = 2x
    2 × (−1) = −2
    3x + 9 ≥ 2x − 2
  4. Get the x terms on one side. There are x terms on both sides. We remove the smaller one, so that the x term left over is positive and we do not need to divide by a negative number later.
    Subtract 2x from both sides:
    x + 9 ≥ −2
  5. Get the plain numbers on the other side. Subtract 9 from both sides. This is the inverse of adding 9.
    x ≥ −11
  6. State the solution. This is the complete set of values that make the inequality true.
    x ≥ −11

Check. Test a value inside the solution: x = −10 gives left side = (−10) + 32 = −3.5, right side = (−10) − 13 = −113. Is −3.5 ≥ −113? Yes, it is true. ✓
Test a value outside it: x = −12 gives left side = (−12) + 32 = −4.5, right side = (−12) − 13 = −133. Is −4.5 ≥ −133? No, it is false, so −12 is correctly left out.

Watch out. The sign only reverses when you multiply or divide by a negative number. Otherwise leave it alone, and keep the “or equal to” part of ≤ and ≥ exactly as it was.

Watch out

Do not flip the sign when you add or subtract, and do not forget to flip it when you divide by a negative. A very common slip is to flip the sign because there is a negative number in the question (for example 5 − 3 > 2), when what matters is whether you multiply or divide by a negative. Always test the final answer with one number.

Exam tip

If you have to divide by a negative, write “÷ (−3), flip the sign” beside your working as a reminder. Or avoid the problem by moving the x term across so that it is positive.

Exercise 10E  Negative numbers, brackets and letters on both sides

Non-calculator
1
FSolve each inequality.
(a)−2x < 8
(b)−5x ≥ 20
(c)−x + 3 > 1
(d)−2x + 7 ≤ 1
(e)−3x + 10 > 4
(f)−4x + 1 ≥ 13
2
FSolve each inequality.
(a)5x + 2 < 3x + 10
(b)4x − 1 ≥ 2x + 9
(c)7x + 3 > 2x + 13
(d)3x + 5 ≤ 5x + 1
(e)2x − 3 < 6x + 5
(f)8x + 1 ≥ 3x − 9
3
FSolve each inequality. Expand the brackets first.
(a)3(x + 2) > 15
(b)2(x − 3) ≤ 10
(c)4(2x + 1) < 36
(d)5x ≥ 2(x + 6)
(e)2(3x − 1) > 4(x + 2)
(f)5(x − 2) ≥ 3(x + 4)
4
FSolve each inequality.
(a)x + 12 > 3
(b)2x − 13 ≤ 5
(c)x + 45 < −1
(d)x2 − 3 ≥ 1
5
RPriya solves −3x < 12 and writes x < −4. Explain her mistake and write the correct solution.
6
RJack solves 2 − x > 5 and gets x > −3. Is he correct? Test a value to decide.
7
RWithout solving first, use x = 0 to decide whether the solution of 4 − 2x < 10 is x > −3 or x < −3. Then solve it to check.
8
PSSolve 5 − 2x ≥ 17. Write down the largest whole number that satisfies it.
9
PSFind the largest whole number n such that 3n − 4 > 5n − 20.
10
PSSolve 2(3x − 1) > 4(x + 2). Then say whether x = 5 is a solution.
11
RSolve 3(2 − x) > 6 − 5x.
12
FSolve each inequality.
(a)9x < −4x + 3
(b)−4x + 9 < 2x + 3
(c)7x − 2 ≥ −3x + 8
13
FSolve each inequality.
(a)x + 32 ≥ x − 13
(b)2x − 13 < x + 42
(c)3x + 14 > x − 23
10.5

Double inequalities and whole-number solutions

Edexcel 1MA1: A22
Grade 4-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Double inequalities and whole-number solutionsComing soon

A double inequality has two inequality signs, and a value in the middle that is squeezed between two limits. For example, 3 < n ≤ 8 means “n is bigger than 3 and at most 8”. The numbers that work are 4, 5, 6, 7 and 8 (if n is a whole number), but also 3.5, 7.9 and so on if it can be any number.

Integers. An integer is a whole number: positive, negative or zero (… −2, −1, 0, 1, 2 …). Many questions ask you to “write down all the integers that satisfy” an inequality. The best method is: draw a number line, mark the two limits (open or closed circles), and list the whole numbers between them. Watch the circles: an open circle leaves the end number out, a closed circle keeps it in.

Solving a double inequality. When the middle is not just a letter, such as 3 < 2x + 1 ≤ 11, we do the same operation to all three parts.

  1. Subtract 1 from all three parts: 2 < 2x ≤ 10.
  2. Divide all three parts by 2: 1 < x ≤ 5.

Notice that the signs stay as they were. The answer says x is bigger than 1 and at most 5.

Two separate inequalities. Sometimes you are given two inequalities, such as x + 3 > 5 and 2x < 14. Solve each one, then find the values that satisfy both, where the two number lines overlap.

A negative in the middle (Higher). If the middle has a negative x term, such as 7 < 3 − 2x ≤ 11, dividing by −2 reverses both signs, and you then write the answer with the smaller number first.

−5−4−3−2−101234
−3 < n ≤ 2 for integers: −2, −1, 0, 1, 2.

Key ideas

  • A double inequality has a letter or expression squeezed between two limits.
  • Do every step to all three parts.
  • Integers are whole numbers (including negatives and zero). Use a number line to list them.
  • Open end = not included; closed end = included.
  • For two separate inequalities, keep only the values that satisfy both.
  • Higher: dividing all three parts by a negative reverses both signs.
Worked example 10.5.1
n is an integer. Write down all the values of n such that −3 < n ≤ 2.

Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.

Method. Draw the range on a number line, then list only the whole numbers inside it.

  1. Read the inequality. n is bigger than −3 and smaller than or equal to 2. The word integer means a whole number (positive, negative or zero).
    −3 < n ≤ 2
  2. Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
    −4−3−2−10123
  3. Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above −3. For the upper end: included, so up to 2.
    Smallest = −2, largest = 2
  4. List the integers from smallest to largest. Count up in ones.
    −2, −1, 0, 1, 2

Check. Test the end integers: −2 and 2 satisfy −3 < n ≤ 2, but −3 and 3 do not. ✓

Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.

Worked example 10.5.2
Solve 3 < 2x + 1 ≤ 11.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.

  1. Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.
    3 < 2x + 1 ≤ 11
  2. Subtract 1 from all three parts. This removes the plain number from the middle.
    2 < 2x ≤ 10
  3. Divide all three parts by 2. 2 is positive, so both signs stay the same.
    2 ÷ 2 = 1 and 10 ÷ 2 = 5
    1 < x ≤ 5
  4. State the solution. This is the range of values the letter can take.
    1 < x ≤ 5

Check. Test x = 4: the middle becomes 2 × 4 + 1 = 9, and 3 < 9 ≤ 11 is true. ✓
Test the end value x = 1: the middle becomes 3, which equals the left end 3, so the end is not included.

Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.

Worked example 10.5.3
x is an integer. Solve −4 ≤ 3x − 1 < 11 and list the possible values of x.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.

  1. Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.
    −4 ≤ 3x − 1 < 11
  2. Add 1 to all three parts. This removes the plain number from the middle.
    −3 ≤ 3x < 12
  3. Divide all three parts by 3. 3 is positive, so both signs stay the same.
    −3 ÷ 3 = −1 and 12 ÷ 3 = 4
    −1 ≤ x < 4
  4. State the solution. This is the range of values the letter can take.
    −1 ≤ x < 4
  5. List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.
    −2−1012345
    −1, 0, 1, 2, 3

Check. Test x = 1: the middle becomes 3 × 1 − 1 = 2, and −4 ≤ 2 < 11 is true. ✓
Test the end value x = −1: the middle becomes −4, which equals the left end −4, so the end is included.

Watch out. Do every step to all three parts, not just the middle. Keep the signs (< or ≤) exactly as they were unless you divide by a negative.

Worked example 10.5.4
x is an integer with x + 3 > 5 and 2x < 14. List the possible values of x.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Solve each inequality on its own, then keep only the values that satisfy both (where the number lines overlap).

  1. Solve the first inequality. Treat each inequality separately, using the same method as for one.
    x + 3 > 5
    gives x > 2
  2. Solve the second inequality. Do the same for the other one.
    2x < 14
    gives x < 7
  3. Show both on one number line. The values that satisfy both inequalities are where the two lines overlap.
    0123456789
  4. Write the values that satisfy both. Take the overlap of the two ranges.
    2 < x < 7
  5. List the integers. Write down the whole numbers in the overlap.
    3, 4, 5, 6

Check. Pick a value in the overlap and check it in both original inequalities. ✓

Watch out. The answer must satisfy both inequalities, not just one of them.

Worked example 10.5.5
Solve 7 < 3 − 2x ≤ 11, and list the integer solutions.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Do the same operation to all three parts until the letter is alone in the middle. Dividing by a negative reverses both signs.

  1. Write the double inequality. The middle expression must be bigger than the left number and smaller than the right number. We solve both at once by doing the same thing to all three parts.
    7 < −2x + 3 ≤ 11
  2. Subtract 3 from all three parts. This removes the plain number from the middle.
    4 < −2x ≤ 8
  3. Divide all three parts by −2 and reverse both signs. −2 is negative, so dividing flips the order. Both signs reverse.
    −2 > x ≥ −4
  4. Write it with the smaller number on the left. A double inequality is easiest to read with the smallest number first. Turning it round also turns each sign round.
    −4 ≤ x < −2
  5. State the solution. This is the range of values the letter can take.
    −4 ≤ x < −2
  6. List the integers (whole numbers) x can be. Use the number line to see which whole numbers fit inside the range.
    −5−4−3−2−1
    −4, −3

Check. Test x = −3: the middle becomes −2 × (−3) + 3 = 9, and 7 < 9 ≤ 11 is true. ✓
Test the end value x = −4: the middle becomes 11, which equals the right end 11, so the end is included.

Watch out. When you multiply or divide both sides by a negative number, reverse the inequality sign (< becomes >). Adding or subtracting a number never changes the sign.

Watch out

Do not forget that an open circle leaves the end number out. For −3 < n ≤ 2 the whole numbers are −2, −1, 0, 1, 2. Many students wrongly include −3 or leave out 2. And when solving, do every step to all three parts, not just the middle.

Exam tip

To list integers, draw a quick number line and put a dot on every whole number between the limits. Then double-check the two end numbers against the circles.

Exercise 10F  Double inequalities and integer solutions

Non-calculator
1
Fn is an integer. Write down all the values of n that satisfy each inequality.
(a)−3 ≤ n < 2
(b)1 < n ≤ 6
(c)−5 < n < −1
(d)0 ≤ n ≤ 4
(e)−2.5 < n ≤ 1.5
(f)−1 < n < 1
2
FSolve each double inequality.
(a)4 < 2x ≤ 12
(b)−3 ≤ x + 1 < 5
(c)6 < 3x ≤ 15
(d)1 ≤ 2x − 5 < 9
(e)0 < 5x + 10 ≤ 40
(f)−7 < 2x − 1 ≤ 3
3
Fx is an integer. Solve each inequality and list the possible values of x.
(a)3 < 2x + 1 ≤ 11
(b)−4 ≤ 3x − 1 < 11
(c)2 ≤ 4x − 2 < 18
(d)−6 < 2x ≤ 6
4
Fn is an integer. How many values of n satisfy −2 < n ≤ 3?
5
FThe number line shows the integer values of n that work.
−5−4−3−2−101234
Write down the integers, and an inequality for n.
6
PSx is an integer, 2x − 1 > 5 and x < 7. List all the possible values of x.
7
PSx is an integer, 3x − 2 ≥ 4 and 2x + 1 < 11. List all the possible values of x.
8
PSA rectangle has width 4 cm and length (x + 2) cm. Its area is more than 24 cm2 and at most 40 cm2. This means 24 < 4(x + 2) ≤ 40, which is 24 < 4x + 8 ≤ 40. If x is a whole number, write down the possible values of x.
9
PSA taxi costs £3 plus £2 for every whole mile, so a journey of m miles costs (2m + 3) pounds. Ellie pays at least £9 and at most £21. Write down all the possible values of m.
10
RHow many integers satisfy −7 < 2x + 1 ≤ 9?
11
RAli solves 2 < 3x − 1 < 11 and gets 13 < x < 103. Explain his mistake and give the correct answer.
12
RExplain why the only integer n satisfying −1 < n < 1 is 0.
13
Rn is an integer with n2 < 30 and 2n + 3 ≥ −5. List all the possible values of n.
14
FSolve each double inequality.
(a)7 < −2x + 3 ≤ 11
(b)−4 ≤ −3x + 5 < 8
(c)2 < −x + 1 ≤ 6
(d)−9 ≤ −4x + 3 < 0
15
PSx is an integer. Solve −3 ≤ 1 − 2x < 9 and list the possible values of x.
10.6

Forming and solving inequalities from context

Edexcel 1MA1: A22
Grade 4-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Forming and solving inequalities from contextComing soon

Many exam questions describe a real situation in words. You must turn the words into an inequality, solve it, and then explain what your answer means in the situation. Follow these steps.

  1. Choose a letter for the unknown and say what it stands for.
  2. Write an expression for each amount in the problem (for example, 7 tickets at £n each costs 7n pounds).
  3. Choose the sign from the key phrase: “at most” is ≤, “at least” is ≥, “less than” is <, “more than” is >.
  4. Solve the inequality.
  5. Interpret the answer: if you are counting things, the answer must be a whole number, and you must round the correct way. Then check it in the original problem.

Rounding the correct way. If the solution is n ≤ 6.29 (tickets), you cannot buy 6.29 tickets. The largest whole number that works is 6. If the solution is n ≥ 18.67 (weeks), the smallest whole number that works is 19, because 18 weeks is not enough. Always round so that the condition is still true, not to the nearest number.

Totals and means. A total is built by adding the pieces. A mean is the total divided by the number of values, so “the mean of four marks is at least 65” becomes (total of marks) ÷ 4 ≥ 65.

Key ideas

  • Choose a letter and say what it means.
  • Key phrases: at most = ≤, at least = ≥, less than = <, more than = >.
  • Solve, then interpret in context.
  • For counting problems, the answer must be a whole number: round down for an upper limit and up for a lower limit.
  • Always check the answer by substituting back into the story.
Worked example 10.6.1
Sam has £50 to spend. Cinema tickets cost £7 each, and he also buys a drink for £6. What is the greatest number of tickets he can buy?

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.

  1. Choose a letter. Let n be the number of tickets Sam buys.
  2. Write an inequality. The tickets cost 7n pounds, and the drink costs £6. The total must be at most £50, so we use ≤.
    7n + 6 ≤ 50
  3. Get the plain numbers on the other side. Subtract 6 from both sides. This is the inverse of adding 6.
    7n ≤ 44
  4. Divide both sides by 7. 7n means 7 lots of n. The inverse of multiplying by 7 is dividing by 7. 7 is positive, so the inequality sign stays the same.
    n ≤ 44 ÷ 7 = 447
  5. Answer the question in words. The inequality says the number is less than or equal to 447. Because the number of tickets must be a whole number, we need the largest whole number that fits.
    6 tickets

Check. Test a value inside the solution: n = 6 gives left side = 7 × 6 + 6 = 48, right side = 50. Is 48 ≤ 50? Yes, it is true. ✓
Test a value outside it: n = 7 gives left side = 7 × 7 + 6 = 55, right side = 50. Is 55 ≤ 50? No, it is false, so 7 is correctly left out.

Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.

Worked example 10.6.2
A rectangle has length (x + 4) cm and width 3 cm. Its perimeter is less than 30 cm. Find the possible values of x.

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.

  1. Choose a letter. Let x cm be the unknown length.
  2. Write an expression for the perimeter. A rectangle has two lengths and two widths, so the perimeter is 2(x + 4) + 2 × 3. Expand: 2x + 8 + 6.
    Perimeter = 2x + 14
  3. Get the plain numbers on the other side. Subtract 14 from both sides. This is the inverse of adding 14.
    2x < 16
  4. Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.
    x < 16 ÷ 2 = 8
  5. Answer the question in words. The length is (x + 4) cm, so x is just a number. It does not have to be whole.
    x < 8 cm

Check. Test a value inside the solution: x = 7 gives left side = 2 × 7 + 14 = 28, right side = 30. Is 28 < 30? Yes, it is true. ✓
Test a value outside it: x = 9 gives left side = 2 × 9 + 14 = 32, right side = 30. Is 32 < 30? No, it is false, so 9 is correctly left out.

Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.

Worked example 10.6.3
Tom scored 62, 70 and 58 in three tests. He wants his mean mark over four tests to be at least 65. What mark does he need in the fourth test?

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.

  1. Choose a letter. Let x be Tom’s mark in the fourth test.
  2. Write an inequality. The total of the four marks is 62 + 70 + 58 + x = (190 + x). The mean is the total divided by 4, and it must be at least 65.
    (x + 190) ÷ 4 ≥ 65
  3. Multiply both sides by 4 to get rid of the fraction. Multiplying by the denominator cancels the division. 4 is a positive number, so the inequality sign does not change.
    x + 190 ≥ 260
  4. Get the plain numbers on the other side. Subtract 190 from both sides. This is the inverse of adding 190.
    x ≥ 70
  5. Answer the question in words. Tom needs at least this mark in the fourth test.
    x ≥ 70 marks

Check. Test a value inside the solution: x = 71 gives left side = 71 + 1904 = 65.25, right side = 65. Is 65.25 ≥ 65? Yes, it is true. ✓
Test a value outside it: x = 69 gives left side = 69 + 1904 = 64.75, right side = 65. Is 64.75 ≥ 65? No, it is false, so 69 is correctly left out.

Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.

Worked example 10.6.4
Plan A costs £10 plus £4 per hour. Plan B costs £25 plus £1 per hour. For how many hours is Plan A cheaper than Plan B?

Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.

Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.

  1. Choose a letter. Let h be the number of hours.
  2. Write an expression for each plan. Plan A costs 10 + 4h pounds. Plan B costs 25 + h pounds. We want A to be cheaper, so A < B.
    4h + 10 < h + 25
  3. Get the h terms on one side. There are h terms on both sides. We remove the smaller one, so that the h term left over is positive and we do not need to divide by a negative number later.
    Subtract h from both sides:
    3h + 10 < 25
  4. Get the plain numbers on the other side. Subtract 10 from both sides. This is the inverse of adding 10.
    3h < 15
  5. Divide both sides by 3. 3h means 3 lots of h. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.
    h < 15 ÷ 3 = 5
  6. Answer the question in words. Plan A is cheaper for fewer than this number of hours.
    h < 5 hours

Check. Test a value inside the solution: h = 4 gives left side = 4 × 4 + 10 = 26, right side = 4 + 25 = 29. Is 26 < 29? Yes, it is true. ✓
Test a value outside it: h = 6 gives left side = 4 × 6 + 10 = 34, right side = 6 + 25 = 31. Is 34 < 31? No, it is false, so 6 is correctly left out.

Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.

Watch out

The answer must make sense. If the answer is a number of people, tickets or cakes it must be a whole number, and you must round the right way: n ≤ 6.29 means at most 6, not 7. Also, do not use the nearest whole number blindly: n ≥ 18.67 means at least 19.

Exam tip

Write down what your letter stands for in a short sentence at the start. Examiners award marks for defining the variable, and it helps you avoid mixing up quantities.

Exercise 10G  Inequalities in context

Non-calculator
1
FWrite an inequality for each situation. You do not need to solve it.
(a)Ali has £40. Pens cost £3 each. He buys n pens and a £5 book.
(b)A taxi charges £4 plus £2 per mile. A journey of m miles costs less than £30.
(c)I think of a number x, multiply it by 5 and subtract 8. The result is at least 22.
(d)The perimeter of a square with side s cm is greater than 50 cm.
(e)A rectangle has width 5 cm and length (x + 3) cm. Its perimeter is at most 36 cm.
2
PSAli has £40. Pens cost £3 each, and he also buys a £5 book. What is the greatest number of pens he can buy?
3
PSA taxi charges £4 plus £2 per mile. Rosie has less than £30 to spend on the fare. How far can she travel?
4
PSA lift can carry at most 200 kg. Sam weighs 80 kg and carries boxes that each weigh 12 kg into the lift. What is the greatest number of boxes he can carry?
5
PSA club sells cakes for £3 each. The ingredients cost £25 in total. They want a profit of at least £50. How many cakes must they sell, at least?
6
PSMaya has saved £120. She saves £15 each week. She wants to have at least £400. What is the smallest number of whole weeks she needs to save?
7
PSAli is x years old. His sister is 3 years older. The sum of their ages is less than 30. What is the greatest whole number of years Ali could be?
8
PSGym A charges £15 to join and £8 per month. Gym B charges £5 to join and £12 per month. After how many whole months is Gym A cheaper than Gym B? Give the smallest number of months.
9
PSZara scores 64, 71 and 58 in three tests. She wants a mean of at least 65 over four tests. What must she score in the fourth test?
10
PSI think of a whole number, double it, and subtract 7. The answer is greater than 15. What is the smallest number I could be thinking of?
11
PSAn isosceles triangle has two sides of length (2x + 1) cm and a third side of 9 cm. Its perimeter is less than 40 cm. Find the possible values of x.
12
RIn the cinema problem a student gets n ≤ 6.29 and answers “6.29 tickets”. Explain why this is wrong and what the answer should be.
13
RA student writes “each notebook costs £2 and I have £15, so 2n ≥ 15”. Explain the mistake and write the correct inequality.
14
RTwo sides of a triangle are 5 cm and 8 cm. The third side is x cm. In any triangle the sum of two sides must be greater than the third side. Write down the range of values of x.

Chapter summary

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