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Solving inequalities

Solving inequalities

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Explainer Video

Tutor: Bilal

Summary

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.



Symbols

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


SYMBOL

DESCRIPTION

​<\lt<​​

less than

​≤\le≤​​

less than or equal to

​>\gt>​​

greater than

​≥\ge≥​​

greater than or equal to

Note: The arrow points to the smaller number.



Number lines

​x≤3x \le 3x≤3 means that xxx is less than or equal to 333. It can be represented by the number line


Maths; Algebra; KS4 Year 10; Solving inequalities


use ∘\circ ∘ for >or<\gt or \lt>or<​

use ∙\bullet∙ for ≥or≤\ge or \le≥or≤



Solving inequalities

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


Basic Inequalities

Rearrange like equations to solve. 


Example 1

​3x+10≤223x≤12x≤4‾\begin {aligned}3x + 10 &\le 22 \\3x &\le 12 \\ \end {aligned}\\ \qquad \underline{x \le 4}3x+103x​≤22≤12​x≤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}−2x​<10÷−2​x>−5​​​

​

Inequalities in two parts

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


Example 3

​21<4x−3<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}2124​<4x−3<4x​<41<44​+3÷4​6<x<11​​​

​

Complex Inequalities in two parts

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


Example 4

​3x−19<5x−3<4x+23x−19<5x−35x−3<4x+2−8<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}3x−19−8​<5x−3<x​3x−19​<5x−3​<4x+25x−3x​<4x+2<5​

​−8<x<5‾\underline {-8 \lt x \lt 5}−8<x<5​


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Solving equations

Solving equations

Inequalities: Greater than or less than

Inequalities: Greater than or less than

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Solving inequalities

Solving inequalities

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FAQs - Frequently Asked Questions

What is a number line in inequalities?

A number line is a visual way to show the solution to an inequality.

How do you solve inequalities?

Inequalities can be solved in a similar way to solving equations by rearranging.

What are inequalities?

Inequalities show when one expression is greater than or less than another expression. They can be represented algebraically, or on a number line.

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