Solving inequalities

Solving inequalities

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Solving inequalities

In a nutshell

Inequalities show when one expression is greater than or less than another expression. Inequalities with more complex expressions can be solved in a similar way to equations. The answers can be represented algebraically or on a number line.


Inequalities involve using symbols to describe the relationship between two expressions.




less than


less than or equal to


greater than


greater than or equal to

Note: The arrow points to the smaller number.

Number lines

x3x \le 3 means that xx is less than or equal to 33. It can be represented by the number line

Maths; Algebra; KS4 Year 10; Solving inequalities

use \circ  for >or<\gt or \lt

use \bullet for or\ge or \le

Solving inequalities

You can use knowledge of rearranging equations to solve the inequalities. 

Basic Inequalities

Rearrange like equations to solve. 

Example 1

3x+10223x12x4\begin {aligned}3x + 10 &\le 22 \\3x &\le 12 \\ \end {aligned}\\ \qquad \underline{x \le 4}​​

Inequalities with negatives

If you multiply or divide by a negative number, reverse the inequality sign.

Example 2

2x<10÷2x>5\begin {aligned}-2x &\lt 10 \\ & \qquad \div -2\\ \end {aligned}\\ \quad \underline {x \gt -5}​​

Inequalities in two parts

Like solving an equation, do the same to each of the 3 parts of the inequality.

Example 3

21<4x3<41+324<4x<44÷46<x<11\begin {aligned}21 &\lt 4x-3 &\lt 41 \\&&& +3 \\24 &\lt \quad 4x &\lt 44 \\&&& \div 4 \\\end {aligned}\\ \quad \underline{6 \lt x \lt 11}​​

Complex Inequalities in two parts

Split the inequality up into 2 separate questions, solve each separately then recombine the answers.

Example 4

3x19<5x3<4x+23x19<5x35x3<4x+28<xx<5\begin {aligned}&&3x-19 &\lt 5x-3 &\lt 4x+2 \\3x-19 &\lt 5x-3 &&& 5x-3 &\lt 4x+2\\-8 &\lt x &&& x &\lt 5\\\end {aligned}

8<x<5\underline {-8 \lt x \lt 5}

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FAQs - Frequently Asked Questions

What is a number line in inequalities?

How do you solve inequalities?

What are inequalities?


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