---
title: "Inequalities"
subtitle: "Algebra Foundations · Equations and inequalities · Lesson 14"
---

# Inequalities

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** You need more than 50 points to pass, and a ride needs riders at least 120 cm tall. Not one answer: a whole range of them.


:::teacher-note

**In one sentence:** Students solve one-variable inequalities like equations, reversing the inequality sign when multiplying or dividing by a negative, show solutions on a number line with open or closed circles, and check graphically and by substitution (AC9M8A02, node 3).

**Students should be able to say:** I can solve an inequality, reversing the inequality sign when I multiply or divide by a negative. I can show a solution on a number line with the right circle (open or closed) and direction. I can check a solution by substituting a value, or with a graph.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

:::


## Predict

Pick a value from your answer and check it in $-2x > 6$.

:::keep-together

**Which is the solution of $-2x > 6$?**

☐ $x < 3$

☐ $x > 3$

☐ $x < -3$

☐ $x > -3$

My prediction, and why:

{{lines-3}}

:::


:::teacher-note

**What happens:** x < −3. Check x = −4: −2 × (−4) = 8, and 8 > 6. But x = 0 (which x > −3 allows) gives 0, and 0 > 6 is false.

:::


## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## Solve, show, check


:::keep-together

### Find the mistake

**Mia solves $-2x > 6$. Click the line where her working goes wrong.**

```
 1  Solve −2x > 6.
 2  Divide both sides by −2.
 3  x > −3.
 4  Check x = 0: −2 × 0 = 0, and 0 > 6 is false.
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

:::


:::teacher-note

**Answer:** line 3

Line 3: dividing by −2 reverses the inequality sign, so x < −3. Her check on line 4 fails, which is the clue: x = 0 should work if x > −3 were right.

:::


:::keep-together

### Divide by a negative

**Which is the solution of $-3x \le 12$?**

☐ $x \ge 4$

☐ $x \le -4$

☐ $x \ge -4$

:::


:::teacher-note

**Answer:** $x \ge -4$

Divide by −3 and reverse: x ≥ −4. Check x = 0: 0 ≤ 12. ✓

:::


:::typical

## One change at a time


**A closed circle.** Each task changes one thing.

**A negative x.** 

**A fraction.** 

**A range.** 

**Why it reverses.** 

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

:::


## Inequalities


:::keep-together

### Positive coefficient

**Solve $4x + 1 \ge -7$.**

☐ $x \ge 2$

☐ $x \le -2$

☐ $x \ge -2$

:::


:::teacher-note

**Answer:** $x \ge -2$

4x ≥ −8, x ≥ −2.

:::


:::keep-together

### Negative coefficient

**Solve $-5x < 20$.**

☐ $x > -4$

☐ $x > 4$

☐ $x < -4$

:::


:::teacher-note

**Answer:** $x > -4$

Divide by −5 and reverse: x > −4. Check x = 0: 0 < 20. ✓

:::


:::keep-together

### Read the line

**Which inequality does this number line show?**

☐ $x < 5$

☐ $x > 5$

☐ $x \le 5$

:::


:::teacher-note

**Answer:** $x < 5$

Open circle at 5, shaded left: x < 5.

:::


:::keep-together

### Review: equations

**Review: solve $3x + 2 = 2x + 7$.**

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** 5

x + 2 = 7, x = 5. Check: 17 = 17.

:::


## Make: write one with a negative

Swap with a partner: solve theirs, and check a value.

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

**Solve 7 − 2x ≥ 1 and show it on a number line.**

{{lines-3}}

:::


:::teacher-note

**Answer:** −2x ≥ −6, x ≤ 3 (reversed): closed circle at 3, shaded left. Check x = 0: 7 ≥ 1.

:::