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# Inequalities

Algebra Foundations · Equations and inequalities

<!-- Speaker note: 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). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 13 min · Practise and Make 12 min). -->

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## Today

**We are learning to** solve an inequality like an equation, doing the same to both sides, but when you multiply or divide by a negative, reverse the inequality sign. Show the answer on a number line: open circle for < and >, closed for ≤ and ≥. Check with a value.

- 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.

<!-- Speaker note: Before class: Do the Investigate board yourself first. Students need Solving harder equations, and Lines and regions (Linear relations) for the graph. The one idea: inequalities solve like equations, except that multiplying or dividing by a negative reverses the inequality sign, because it flips the order on the number line (2 < 5 but −2 > −5). The common mistake is keeping the inequality sign; it is planted in the Predict and in Mia's working, whose own check fails. Students show answers on the number line board (move P, choose the circle and the direction), so the board marks the solution, not the way it is written. -->

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## Why it matters

> 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.

<!-- Speaker note: An **inequality** compares two sides with $<$, $>$, $\le$ or $\ge$. Its solution is usually a whole range of values, such as $x < 5$, which you can show on a number line. You solve it like an equation, doing the same to both sides, with one new rule. This lesson finds the rule, and why it works. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Which is the solution of $-2x &gt; 6$?**

- $x &lt; 3$
- $x &gt; 3$
- $x &lt; -3$
- $x &gt; -3$

<!-- Speaker note: What happens: x &lt; −3. Check x = −4: −2 × (−4) = 8, and 8 &gt; 6. But x = 0 (which x &gt; −3 allows) gives 0, and 0 &gt; 6 is false. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands, then have students test x = 0 in −2x > 6. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Falgebra%2Finequalities%2Fstep-2&code=widget%3A+graph%0Ax%3A+-6%2C+4%0Ay%3A+-4%2C+12%0Agrid%3A+1%2C+2%0Aplot%3A+y+%3D+-2x%0Aplot%3A+y+%3D+6%2C+dashed%0Apoint+P%3A+2%2C+-4%2C+drag%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/algebra/inequalities/?full=step-2#step-2)

<!-- Speaker note: Where are all the points that work: left or right of $x = -3$? -->

<!-- Speaker note: Teacher: ask where all the working points are, and what that says about the solution. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* x &lt; −3. Check x = −4: −2 × (−4) = 8, and 8 &gt; 6. But x = 0 (which x &gt; −3 allows) gives 0, and 0 &gt; 6 is false.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Solve, show, check

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Falgebra%2Finequalities%2Fstep-3&code=widget%3A+numberline%0Arange%3A+-10%2C+10%0Apoint+P%3A+0%2C+drag%0Aray%3A+P%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/algebra/inequalities/?full=step-3#step-3)

<!-- Speaker note: What this step is for: Adding, subtracting, and multiplying or dividing by a positive keep the inequality sign; multiplying or dividing by a negative reverses it, because it flips the order on the number line. Mia keeps the inequality sign after dividing by −2. -->

<!-- Speaker note: Teacher: model the worked example, then the number-line argument for reversing (2 < 5, −2 > −5). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Mia solves $-2x &gt; 6$. 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.
```

* Answer: line 3

<!-- Speaker note: 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. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Divide by a negative

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

- $x \ge 4$
- $x \le -4$
- $x \ge -4$

* Answer: $x \ge -4$

<!-- Speaker note: Divide by −3 and reverse: x ≥ −4. Check x = 0: 0 ≤ 12. ✓ -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **A closed circle:** Each task changes one thing.
- **A negative x:** 
- **A fraction:** 
- **A range:** 
- **Why it reverses:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/algebra/inequalities/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a closed circle, a negative x term, a fraction, then a range between two values, and the edge case: why the inequality sign reverses. Hints, in order: 2x ≥ 4, so x ≥ 2: a closed circle (2 is included). / Add 1: x/2 < 4. / Multiply by 2 (positive): x < 8. For students who need more support: Check every answer by testing one value inside and one outside the solution. For early finishers: Solve −4 < 2x + 2 ≤ 10 (a double inequality: −3 < x ≤ 4) and show it on the line. When students get stuck: A student keeps the sign after dividing by a negative (Mia's mistake). Ask: Pick a value from your answer and check it in the first inequality. Fix: If it fails, the sign should have reversed. / A student chooses the wrong circle. Ask: Is the end number itself a solution? Fix: < and >: no, open circle. ≤ and ≥: yes, closed circle. / A student avoids the negative by moving x across. Ask: Does that work too? Fix: Yes: −2x > 6 → 0 > 6 + 2x → −6 > 2x → −3 > x. Same answer, no reversing needed. -->

<!-- Speaker note: Teacher: before 5 − x > 2, ask how to get x on its own when it has a minus sign. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Positive coefficient

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

- $x \ge 2$
- $x \le -2$
- $x \ge -2$

* Answer: $x \ge -2$

<!-- Speaker note: 4x ≥ −8, x ≥ −2. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Negative coefficient

**Solve $-5x &lt; 20$.**

- $x &gt; -4$
- $x &gt; 4$
- $x &lt; -4$

* Answer: $x &gt; -4$

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

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Read the line

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

- $x &lt; 5$
- $x &gt; 5$
- $x \le 5$

* Answer: $x &lt; 5$

<!-- Speaker note: Open circle at 5, shaded left: x < 5. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: equations

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

* Answer: 5

<!-- Speaker note: x + 2 = 7, x = 5. Check: 17 = 17. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: write one with a negative

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

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/algebra/inequalities/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Writing an inequality with a negative x term whose solution is x ≥ −2, then showing it, reverses the lesson: from the solution back to an inequality that needs the reversal. Hints, in order: Start from x ≥ −2 and multiply both sides by −3: the sign reverses. / −3x ≤ 6. Check x = 0: 0 ≤ 6. ✓ -->

<!-- Speaker note: Teacher: collect students' inequalities and check one value from each. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

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

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

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This completes AC9M8A02: graphing lines and regions, solving equations, solving inequalities. Where it leads: inequalities with two variables (regions) and linear programming in later years. -->