Inequalities and Number Lines
An inequality compares two expressions using <, >, <= or >=, instead of an equals sign, and describes a whole range of possible values rather than a single answer. Inequalities can be shown on a number line, using an open circle for a strict inequality and a filled circle for one that includes the boundary value, and are solved like equations except that multiplying or dividing both sides by a negative number reverses the inequality sign.
Before you start
Make sure you're comfortable with these topics first:
Method
- Learn the meaning of each symbol: < means less than, > means greater than, <= means less than or equal to, >= means greater than or equal to.
- On a number line, use an open circle for < or > (the boundary value is not included) and a filled, closed circle for <= or >= (the boundary value is included).
- Solve a linear inequality using the same steps as solving an equation: do the same operation to both sides.
- If you multiply or divide both sides by a negative number, reverse the direction of the inequality symbol.
- Write the solution as an inequality (e.g. x > 3) and, if asked, draw it on a number line with an arrow continuing in the direction of the solution.
- Check your answer by substituting a value from your solution set back into the original inequality.
Worked example
Solve the inequality 5x + 3 <= 23. Then solve -3x + 4 > 19, explaining why the inequality sign changes direction.
- For 5x + 3 <= 23, subtract 3 from both sides: 5x <= 20.
- Divide both sides by 5: x <= 4.
- For -3x + 4 > 19, subtract 4 from both sides: -3x > 15.
- Divide both sides by -3, and reverse the inequality sign because we divided by a negative number: x < -5.
Practice questions
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Q1Solve x + 5 > 12.Show answer
Answer: x > 7
Q2Solve 3x <= 21.Show answer
Answer: x <= 7
Q3Solve 2x - 4 < 10.Show answer
Answer: 2x < 14, so x < 7
Q4Solve 4x + 1 >= 25.Show answer
Answer: 4x >= 24, so x >= 6
Q5Solve -2x < 8.Show answer
Answer: Divide both sides by -2 and reverse the sign: x > -4
Q6A number line shows an open circle at 3 with an arrow pointing right. Write down the inequality shown.Show answer
Answer: x > 3
Q7Is x = 5 a solution to the inequality 2x - 3 > 6? Explain your answer.Show answer
Answer: Yes, because 2(5) - 3 = 7, and 7 > 6 is true.
Q8Solve 15 - 3x >= 0.Show answer
Answer: Subtract 15: -3x >= -15. Divide by -3 and reverse the sign: x <= 5
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Solve the inequality 7x - 4 < 3x + 16.
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(a) Solve the inequality 12 - 2x > 4, giving your answer as an inequality. (b) Write down the smallest integer value of x that satisfies -2x + 7 <= 15.
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Free printable worksheet
Want more practice on paper? Download the inequalities and number lines worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 33 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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