Inequalities

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 13 min · Practise and Make 12 min

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.

Running the lesson

1. Predict: take a show of hands, then have students test x = 0 in −2x > 6.
2. Run: ask where all the working points are, and what that says about the solution.
3. Investigate: model the worked example, then the number-line argument for reversing (2 < 5, −2 > −5).
4. Modify: before 5 − x > 2, ask how to get x on its own when it has a minus sign.
5. Make: collect students' inequalities and check one value from each.

When students get stuck

Answers to the checks

1. a negative in front: Which is the solution of $-2x > 6$?
   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.
2. when is −2x more than 6?: $P$ is on the line $y = -2x$. Drag it to a point where $-2x$ is more than 6 (above the dashed line). Was your prediction right?
   
3. Solve, show, check: Solve $3x - 4 < 11$, then show the solution: move $P$, choose the circle and the direction.
   
   3x < 15, so x < 5: an open circle at 5, shaded left.
4. Find the mistake: Mia solves $-2x > 6$. Click the line where her working goes wrong.
   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.
5. Divide by a negative: Which is the solution of $-3x \le 12$?
   $x \ge -4$
   Divide by −3 and reverse: x ≥ −4. Check x = 0: 0 ≤ 12. ✓
6. A closed circle: Solve $2x + 3 \ge 7$ and show it.
   
   2x ≥ 4, x ≥ 2: closed circle at 2, shaded right.
7. A negative x: Solve $5 - x > 2$.
   $x < 3$
   −x > −3, then divide by −1 and reverse: x < 3. Check x = 0: 5 > 2. ✓
8. A fraction: Solve $\dfrac{x}{2} - 1 < 3$ and show it.
   
   x/2 < 4, x < 8: open circle at 8, shaded left.
9. A range: A range: how many integers $x$ have $-1 < x \le 4$?
   5
   0, 1, 2, 3, 4: five integers. −1 is left out (open), 4 is in (closed).
10. Why it reverses: Edge case: $2 < 5$. Multiply both sides by $-1$. Which is true?
    $-2 > -5$
    Multiplying by −1 reflects every number about 0, so the order flips: 2 < 5 but −2 > −5. That is why the inequality sign reverses.
11. Positive coefficient: Solve $4x + 1 \ge -7$.
    $x \ge -2$
    4x ≥ −8, x ≥ −2.
12. Negative coefficient: Solve $-5x < 20$.
    $x > -4$
    Divide by −5 and reverse: x > −4. Check x = 0: 0 < 20. ✓
13. Read the line: Which inequality does this number line show?
    $x < 5$
    Open circle at 5, shaded left: x < 5.
14. Review: equations: Review: solve $3x + 2 = 2x + 7$.
    5
    x + 2 = 7, x = 5. Check: 17 = 17.
15. write one with a negative: Write an inequality with $-3x$ in it whose solution is $x \ge -2$. Solve it on paper, then show the solution on the line.
    
    For example −3x ≤ 6, or 1 − 3x ≤ 7. Dividing by −3 reverses ≤ into ≥: x ≥ −2.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › A closed circle

Open task: any program that does what the step asks.

one change at a time › A negative x

Open task: any program that does what the step asks.

one change at a time › A fraction

Open task: any program that does what the step asks.

one change at a time › A range

Open task: any program that does what the step asks.

one change at a time › Why it reverses

Open task: any program that does what the step asks.

write one with a negative

Open task: any program that does what the step asks.

Stretch and support

Stretch: Solve −4 < 2x + 2 ≤ 10 (a double inequality: −3 < x ≤ 4) and show it on the line.

Support: Check every answer by testing one value inside and one outside the solution.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Inequalities".

The idea: Solving inequalities and using graphical methods: I can Students solve one-variable inequalities algebraically and verify using graphical methods; represent solutions on a number line.

What understanding looks like:
- Solves a one-variable inequality correctly including sign reversal where needed.
- Represents the solution on a number line with correct open/closed endpoint.
- Verifies the solution graphically or by substitution.

Mistakes students often make here:
- A student keeps the sign after dividing by a negative (Mia's mistake).
- A student chooses the wrong circle.
- A student avoids the negative by moving x across.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Solving linear equations algebraically), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Inequalities".

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.

What each step is for

• a negative in front: Committing to a solution of −2x > 6 shows whether they divide by −2 and keep the sign, the slip this lesson is built around.
• when is −2x more than 6?: Dragging P along y = −2x to where it is above y = 6 shows the solution lies to the left of −3: the sign turned round.
• solve, show, check: 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.
• one change at a time: 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.
• inequalities: Solving with and without reversal, reading a number line and the largest integer, mixed; the review keeps equations fresh.
• write one with a negative: 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.

Where it 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.