5.4 Introducing Inequalities

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: inequality symbols, number lines, listing integers and solving simple inequalities. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Read and write the symbols , ≤ and ≥
• Show an inequality on a number line and read one from a number line
• List the integers that satisfy an inequality
• Solve simple linear inequalities using the same steps as equations

Key words

• Inequality: A statement that one quantity is bigger or smaller than another, e.g. x > 3.
• Integer: A whole number. It can be positive, negative or zero: …, −2, −1, 0, 1, 2, …
• Solution set: All the values that make an inequality true.
• Open circle: On a number line, shows the end value is not included ().
• Closed circle: A filled circle: the end value is included (≤ or ≥).

The four symbols

• <; Meaning is less than; Example x < 4; Read as x is less than 4
• >; Meaning is greater than; Example x > −1; Read as x is greater than −1
• ≤; Meaning is less than or equal to; Example x ≤ 6; Read as x is at most 6
• ≥; Meaning is greater than or equal to; Example x ≥ 2; Read as x is at least 2

Key fact: The narrow end of the symbol points to the smaller value: 3 3 say the same thing.

Take care with negatives. −5 is to the left of −2 on a number line, so −5 < −2.

Number lines

Draw a circle at the end value. Use an open circle for and a filled circle for ≤ or ≥. Then draw an arrow or a line over the values that work.

The last line shows two inequalities together: x is greater than −2 and at most 3. This is read as “x is between −2 and 3, including 3”.

Listing integers

Worked example: integers in a range

List the integers n such that −3 n ≤ 2.
−3 is not included (<). 2 is included (≤).

Answer: −2, −1, 0, 1, 2

Worked example: smallest integer

What is the smallest integer that satisfies x > 4.5?
4.5 is not an integer. The next integer above it is 5.

Answer: 5

Solving inequalities

Solve an inequality just like an equation: use inverse operations on both sides. Keep the inequality sign in place of the = sign.

Worked example: one step

(a) x + 4 > 9. Subtract 4: x > 5
(b) 3x ≤ 12. Divide by 3: x ≤ 4

Answer: (a) x > 5 (b) x ≤ 4

Worked example: two steps

Solve 2x − 5 < 7.
Add 5: 2x x < 6
The largest integer that works is 5.

Answer: x < 6

Worked example: in context

A lift can carry at most 600 kg. A man of 90 kg goes in with boxes of 30 kg each. How many boxes can he take?
90 + 30b ≤ 600. 30b ≤ 510. b ≤ 17

Answer: at most 17 boxes

Common mistakes

• Mixing up open and filled circles: x open circle at 3.
• Including the end value when the sign is : integers with n < 3 stop at 2.
• Thinking −5 > −2. On a number line −5 is further left, so it is smaller.
• Writing an answer as x = 5 when you solve x + 4 > 9. The answer is an inequality: x > 5.

Revision questions

Try these, then check your answers below each one.

Question 1. Which inequality means “x is greater than or equal to -2”? (Choose from: x > -2, x ≥ -2, x ≤ -2, x < -2.)

Answer: x ≥ -2

Working: x ≥ -2.

Question 2. A number line has a filled (solid) circle at 3 and an arrow pointing to the left. Which inequality does it show? (Choose from: x < 3, x ≤ 3, x > 3, x ≥ 3.)

Answer: x ≤ 3

Working: Filled circle at 3 and values to the left: x ≤ 3.

Question 3. Solve 4x - 7 < 13. What is the largest integer x can be?

Answer: 4

Working: 4x < 20, x < 5. The largest integer less than 5 is 4.

Related lessons

• Previous lesson: 5.3 Solving Equations with Brackets
• Next lesson: 5.5 More Inequalities
• In this chapter: 5.1 Solving Equations 1
• In this chapter: 5.2 Solving Equations 2
• In this chapter: 5.6 Using Formulae

Continue in the full course

This lesson is section 5.4 (EGCMF0504) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.