Mathematics • Year 10 • Unit 2 • Lesson 11
Linear Inequalities — Skill Drill
Build fluency with the three tools from Lesson 11: solve inequalities using inverse operations, FLIP the sign when multiplying or dividing by a negative, and write the answer using a number-line graph plus interval notation. One worked example, one guided trace, then eight independent problems sized from foundation to extension.
1. I do — fully worked example
Read every step — each one has a reason underneath so you can see why, not just what.
Problem. Solve −2x + 8 ≥ 4 and write the answer in interval notation.
Step 1 — Subtract 8 from both sides.
−2x + 8 − 8 ≥ 4 − 8 → −2x ≥ −4
Reason: isolate the variable term. Adding/subtracting never flips the sign.
Step 2 — Divide both sides by −2 and FLIP the sign.
−2x ÷ (−2) ≤ −4 ÷ (−2) → x ≤ 2
Reason: dividing by a negative reflects values across zero, so ≥ becomes ≤. This is the FLIP rule.
Step 3 — Test a value to check.
Try x = 0: −2(0) + 8 = 8 ≥ 4. True. ✓
Reason: 0 is in the solution set x ≤ 2, and it satisfies the original inequality.
Step 4 — Write the interval and graph.
Interval: (−∞, 2]. Number line: closed circle at 2, shade left.
Reason: ≤ includes the endpoint, so use a square bracket and a closed circle. Infinity always uses a round bracket.
Answer: x ≤ 2, or in interval notation (−∞, 2].
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 5 marks
Problem. Solve 5 − 3x < 11 and write the answer in interval notation.
Step 1 — Subtract 5 from both sides:
5 − 3x − ____ < 11 − ____ → −3x < ____
Step 2 — Divide both sides by −3. Because the divisor is ________, we must ________ the sign:
x ____ ____ (the ≷ sign reverses)
Step 3 — Test x = 0 in the original:
5 − 3(0) = ____. Is ____ < 11? ____ (yes/no). So 0 ____ in the solution.
Step 4 — Write the interval and describe the graph:
Interval = ( ____ , ____ ). Number line: ________ circle at ____, shade ________.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single skill). The middle two are standard (combine two tools). The last two are extension.
Foundation — single skill
3.1 Solve 2x + 3 < 11. Write the answer as an inequality only. 1 mark
3.2 Solve −3x ≥ 12. Show the FLIP. 1 mark
3.3 Sketch the number line for x > −1. State whether the circle is open or closed and which side is shaded. 1 mark
3.4 Convert x ≤ 4 into interval notation. 1 mark
Standard — combine tools
3.5 Solve 5x − 2 > 3x + 8. Show both steps (gather x, isolate x) and give the interval. 2 marks
3.6 Solve 2(x + 1) ≤ 3x − 4. Expand first, then isolate. Give the interval. 2 marks
Extension — push your thinking
3.7 Solve 6 − 2x ≥ 10. Write the interval, sketch the number line, and write one sentence explaining why the inequality sign reversed at one of your steps. 3 marks
3.8 Solve the compound inequality −5 < 2x − 3 ≤ 7. Give the interval notation and describe the graph in one sentence. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded 5 − 3x < 11)
Step 1: 5 − 3x − 5 < 11 − 5 → −3x < 6.
Step 2: divisor is negative, so we must flip/reverse the sign. x > −2.
Step 3: 5 − 3(0) = 5. Is 5 < 11? Yes. So 0 is in the solution.
Step 4: Interval = (−2, ∞). Number line: open circle at −2, shade right.
3.1 — Solve 2x + 3 < 11
Subtract 3: 2x < 8. Divide by 2: x < 4.
3.2 — Solve −3x ≥ 12
Divide by −3 and FLIP: x ≤ −4. (The ≥ becomes ≤ because we divided by a negative.)
3.3 — Number line for x > −1
Open circle at −1 (because > does not include the endpoint), shade to the right (because we want values larger than −1).
3.4 — x ≤ 4 in interval notation
(−∞, 4]. Square bracket on 4 because ≤ includes the endpoint. Infinity always uses a round bracket.
3.5 — 5x − 2 > 3x + 8
Subtract 3x: 2x − 2 > 8. Add 2: 2x > 10. Divide by 2: x > 5. Interval: (5, ∞). No flip needed — we divided by positive 2.
3.6 — 2(x + 1) ≤ 3x − 4
Expand: 2x + 2 ≤ 3x − 4. Subtract 2x: 2 ≤ x − 4. Add 4: 6 ≤ x, i.e. x ≥ 6. Interval: [6, ∞).
3.7 — 6 − 2x ≥ 10
Subtract 6: −2x ≥ 4. Divide by −2 and FLIP: x ≤ −2. Interval: (−∞, −2]. Number line: closed circle at −2, shade left.
Why the flip: dividing by a negative reflects every number across zero, which reverses the order of inequalities.
3.8 — −5 < 2x − 3 ≤ 7
Left side: −5 < 2x − 3 → −2 < 2x → −1 < x.
Right side: 2x − 3 ≤ 7 → 2x ≤ 10 → x ≤ 5.
Combined: −1 < x ≤ 5. Interval: (−1, 5]. Graph: open circle at −1, closed circle at 5, shade between.