---
title: "Inequalities and their solution sets"
description: "Understand inequalities as statements comparing expressions, represent solutions on a number line, and solve simple linear inequalities using the same inverse-operation methods as equations"
canonical: https://lightmysky.com/learn/mathematics/inequalities-and-their-solution-sets-mt_i2F1nWxJjv
source: https://lightmysky.com/learn/mathematics/inequalities-and-their-solution-sets-mt_i2F1nWxJjv.md
retrieved: 2026-09-02
---

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# Inequalities and their solution sets

Understand inequalities as statements comparing expressions, represent solutions on a number line, and solve simple linear inequalities using the same inverse-operation methods as equations

Subject: Mathematics · Area: Algebra · Ages 11 to 14
Page: https://lightmysky.com/learn/mathematics/inequalities-and-their-solution-sets-mt_i2F1nWxJjv

## Ready when they can

- Write an inequality from a worded constraint (e.g., 'must be at least 12' → x ≥ 12)
- Represent the solution set of an inequality on a number line with open or closed circles
- Solve a one-step or two-step inequality such as 3x + 1 < 10

## Lesson: Inequalities and the number line

You already know how to solve an equation like x + 3 = 10, where both sides end up exactly equal. An inequality compares two expressions without an equal sign. It uses one of four symbols: < (less than), > (greater than), ≤ (less than or equal to, or 'at most'), and ≥ (greater than or equal to, or 'at least').

**Example.** A club rule says, 'You must be at least 12 to join.' If x stands for your age, that rule becomes x ≥ 12. The words 'at least' turn into ≥, because 12 works and so does any age above it.

*(drawing: The dot at 3 is filled in, and the shading runs to the right.)*

The solutions of an inequality are a whole set of numbers, so a number line shows them as a shaded stretch instead of one point. The boundary is the number in the inequality. Fill the circle for ≤ or ≥, because the boundary is itself a solution. Leave the circle open for < or >, because the boundary is not. Then shade the side that makes the statement true.

**Example.** Solve 3x + 1 < 10 the same way you'd solve an equation: undo the operations in reverse order. Subtract 1 from both sides: 3x < 9. Divide both sides by 3: x < 3. Every number smaller than 3 makes the inequality true.

**Tip.** Check your answer by putting a number from your solution set back into the original inequality. If the statement comes out true, your boundary and your direction are both right.

**Recap.** An inequality compares expressions with <, >, ≤, or ≥; solve it the same way you solve equations, then show the answer as a range on a number line with an open or closed circle.

## Practice

13 questions on this page, each with its working shown.

## Needs first

- [Solving Linear Equations](https://lightmysky.com/learn/mathematics/solving-linear-equations-mt_QhFEDyIwSO)
- [Fractions on a number line (age 11+)](https://lightmysky.com/learn/mathematics/fractions-on-a-number-line-age-11-mt_uDJY0X0hgo)

## Opens up

- [Solving Linear Inequalities and Solution Sets](https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM)