Linear Inequalities — Solutions

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1. Simple inequalities

   1. x + 4 > 9: x > 5. Open circle at 5, arrow right.
   2. y − 3 ≤ 5: y ≤ 8. Closed circle at 8, arrow left.
   3. 2x < 14: x < 7. Open circle at 7, arrow left.
   4. 3m ≥ −12: m ≥ −4. Closed circle at −4, arrow right.
   5. 5 − x < 2: −x < −3 → x > 3 (flip). Open circle at 3, arrow right.
   6. −2y > 8: y < −4 (flip). Open circle at −4, arrow left.
   7. −4x ≤ 20: x ≥ −5 (flip). Closed circle at −5, arrow right.
   8. 4 − 2x ≥ −6: −2x ≥ −10 → x ≤ 5 (flip). Closed circle at 5, arrow left.

2. Two-step inequalities

   1. 2x + 3 > 11: 2x > 8 → x > 4. No flip (divide by +2).
   2. 3y − 7 ≤ 14: 3y ≤ 21 → y ≤ 7. No flip.
   3. −5m + 4 ≥ −6: −5m ≥ −10 → m ≤ 2. Flip (divide by −5).
   4. 6 − 3x < 0: −3x < −6 → x > 2. Flip (divide by −3).
   5. −2k + 10 > −4: −2k > −14 → k < 7. Flip (divide by −2).
   6. 4(x − 1) ≤ 12: 4x − 4 ≤ 12 → 4x ≤ 16 → x ≤ 4. No flip.
   7. −3(2p + 1) ≥ 9: −6p − 3 ≥ 9 → −6p ≥ 12 → p ≤ −2. Flip (divide by −6).
   8. 5 − (x + 2) > 1: 3 − x > 1 → −x > −2 → x < 2. Flip (divide by −1).

3. Match inequality to number line

   1. Open circle at 4, arrow right: A: x > 4
   2. Closed circle at −2, arrow left: B: x ≤ −2
   3. Open circle at 3, arrow left: C: x < 3
   4. Closed circle at 5, arrow right: D: x ≥ 5
   5. Open circle at 0, arrow right: E: x > 0
   6. Closed circle at −1, arrow right: F: x ≥ −1
   7. Open circle at 7, arrow left: G: x < 7
   8. Closed circle at 6, arrow left: H: x ≤ 6

4. True or False

   1. −4x > 12 divided by −4 gives x > −3: False. Must flip: result is x < −3.
   2. x = 5 satisfies 3x − 4 > 11: 3(5)−4 = 11. 11 > 11 is false. False (need x > 5, i.e. strictly greater).
   3. Adding a negative number reverses the inequality sign: False. Adding (or subtracting) never reverses the sign. Only multiplying or dividing by a negative does.
   4. x < 3 and x ≤ 3 have different solution sets: True. x ≤ 3 includes x = 3 itself; x < 3 does not.
   5. x = −5 satisfies −4x ≤ 20: −4(−5) = 20. 20 ≤ 20 ✓ True.
   6. 2(x + 3) ≤ 14 gives x ≤ 4: 2x + 6 ≤ 14 → 2x ≤ 8 → x ≤ 4. True.

5. Minimum test score

   1. Inequality: (65 + 72 + 68 + s)/4 ≥ 70 → (205 + s)/4 ≥ 70
   2. 205 + s ≥ 280 → s ≥ 75. The student needs at least 75%.

6. Taxi fare

   1. Inequality: 3.50 + 2.20k ≤ 25
   2. 2.20k ≤ 21.50 → k ≤ 9.77. Maximum whole kilometres = 9 km.

7. Variables on both sides

   1. 3x + 2 > x + 10: 2x > 8 → x > 4. Two examples: 5, 6.
   2. 5y − 4 ≤ 2y + 8: 3y ≤ 12 → y ≤ 4. Two examples: 3, 4.
   3. 4m − 3 ≥ 7m + 9: −3m ≥ 12 → m ≤ −4 (flip). Two examples: −4, −5.

8. Compound inequality

   1. 1 ≤ 2x − 3 < 7:

      Add 3 throughout: 4 ≤ 2x < 10
      Divide by 2: 2 ≤ x < 5

   2. Closed circle at 2 (because ≤), open circle at 5 (because <), line segment between them.
   3. Integer values: 2, 3, 4

9. Consecutive integers (Problem Solving)

   1. Inequality and solution:

      n + (n+1) + (n+2) < 45 → 3n + 3 < 45 → 3n < 42 → n < 14

   2. Since n must be an integer and n < 14, the largest valid n = 13. Largest set: 13, 14, 15 (sum = 42 < 45 ✓).
   3. The inequality gives n < 14 for any real number. But only integer values of n produce valid sets of consecutive integers, so we must restrict n to integers (n ≤ 13).

10. Rectangle dimensions (Problem Solving)

    1. Width = w, length = 2w + 3. Perimeter = 2(w + 2w + 3) = 2(3w + 3) = 6w + 6 ≤ 54
    2. 6w ≤ 48 → w ≤ 8. Maximum width = 8 cm.
    3. Length = 2(8) + 3 = 19 cm. Perimeter check: 6(8) + 6 = 54 ≤ 54 ✓