Mathematics • Year 10 • Unit 2 • Lesson 11
Linear Inequalities — Mixed Challenge
Pull together every idea from Lesson 11: solve inequalities, FLIP the sign when needed, graph on a number line and write interval notation. Choose the right tool for each problem, spot another student's mistake, then design an inequality of your own that hits a target solution.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 11. Decide which tool applies before you start writing. Show your working. 3 marks each
1.1 Solve 4x − 7 ≤ 13 and write the answer in interval notation.
1.2 Solve 9 − 4x > 1. Show the FLIP and give the interval.
1.3 Draw the number-line graph for x ≥ −2.
1.4 Solve 3(x − 2) < 2(x + 4) and give the interval.
1.5 Convert each into interval notation: (i) x < 7, (ii) x ≥ −3, (iii) −1 ≤ x ≤ 4.
1.6 A gym charges a $40 sign-up plus $15 per visit. Maya has at most $250 to spend. Write and solve an inequality for the maximum whole-number visits v she can make.
2. Find the mistake
Another Year 10 student has tried to solve −5x + 3 ≥ 18. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — solve −5x + 3 ≥ 18:
Line 1: −5x + 3 ≥ 18
Line 2: −5x ≥ 15
Line 3: x ≥ −3
Line 4: Interval: [−3, ∞)
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected interval notation.
Stuck? Revisit lesson § "Spot the Trap" — dividing by a negative number must reverse the inequality sign.3. Open-ended challenge — design an inequality with a target solution
This question has many valid answers. Be creative but show every step. 4 marks
3.1 Design a single-variable linear inequality that uses all four of: a constant on each side, a coefficient ≥ 3, at least one set of brackets to be expanded, and at least one step where you must FLIP the sign (because you multiply or divide by a negative). The inequality must have x < −2 as its solution.
In your submission, include:
(i) Your inequality, written cleanly.
(ii) Every working step (expand, gather, isolate, FLIP).
(iii) The final solution shown to equal x < −2, plus the interval notation.
Bonus: Test x = −5 (should satisfy) and x = 0 (should not satisfy) in your original inequality.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Solve 4x − 7 ≤ 13
Add 7: 4x ≤ 20. Divide by 4: x ≤ 5. Interval: (−∞, 5].
1.2 — Solve 9 − 4x > 1
Subtract 9: −4x > −8. Divide by −4 and FLIP: x < 2. Interval: (−∞, 2).
1.3 — Number line for x ≥ −2
Closed circle at −2 (because ≥ includes the endpoint), shade to the right. Interval form: [−2, ∞).
1.4 — Solve 3(x − 2) < 2(x + 4)
Expand: 3x − 6 < 2x + 8. Subtract 2x: x − 6 < 8. Add 6: x < 14. Interval: (−∞, 14).
1.5 — Convert to interval notation
(i) x < 7 → (−∞, 7).
(ii) x ≥ −3 → [−3, ∞).
(iii) −1 ≤ x ≤ 4 → [−1, 4].
1.6 — Gym visits inequality
40 + 15v ≤ 250 → 15v ≤ 210 → v ≤ 14. Maya can make at most 14 visits. Check: 40 + 15(14) = 40 + 210 = $250 ✓ (exactly at the cap).
2 — Find the mistake (solve −5x + 3 ≥ 18)
(a) The mistake is on Line 3.
(b) When dividing both sides by −5, the inequality sign must FLIP from ≥ to ≤. The student divided correctly in magnitude but did not reverse the sign.
(c) Corrected working:
−5x + 3 ≥ 18
−5x ≥ 15 (subtract 3)
x ≤ −3 (divide by −5 AND flip)
Interval: (−∞, −3].
Lesson § "Spot the Trap" flags this exact misconception.
3 — Open-ended challenge (sample solution)
Many valid answers — here is one construction that has x < −2 as its solution and uses every required element.
Sample inequality: −3(x + 1) > 3
Step 1: expand the brackets. −3x − 3 > 3.
Step 2: add 3 to both sides. −3x > 6.
Step 3: divide by −3 and FLIP. x < −2. ✓
Interval: (−∞, −2).
Bonus check:
Test x = −5: −3(−5 + 1) = −3(−4) = 12. Is 12 > 3? ✓ Yes — satisfies original.
Test x = 0: −3(0 + 1) = −3. Is −3 > 3? ✗ No — does not satisfy, as expected.
Marking: 1 for an inequality containing all four required elements; 1 for clean expand + gather steps; 1 for correctly executing the FLIP when dividing by a negative; 1 for arriving at x < −2 with correct interval. Full marks for any valid construction.