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0580/42 May/Jun 2022, Question 10: Linear Inequalities

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 7 marks

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The question, in short

Solve an inequality in n for positive integers, then use a shaded graph with region R bounded by lines to find its defining inequalities and maximise 3x + y over integer points.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

Find all positive integers n satisfying 3n − 8 > 5n − 15.

  1. 13n − 8 > 5n − 15
  2. 215 − 8 > 5n − 3n, so 7 > 2n
  3. 3n < 3.5
  4. 4Positive integers: 1, 2, 3

Answer1, 2, 3

Part (bi) · 3 marks

From the graph, determine the three inequalities whose solution is the unshaded region R.

  1. 1x = 4 is broken and R is to the right: x > 4
  2. 210y + 8x = 80 is solid and R is below it: 10y + 8x ≤ 80
  3. 32y = x − 4 is broken and R is above it: 2y > x − 4

Answer10y + 8x ≤ 80, x > 4, 2y > x − 4

Part (bii) · 2 marks

Given R as x > 4, 2y > x − 4 and 10y + 8x ≤ 80, find the largest integer value of 3x + y in R.

  1. 1Try the integer points in R furthest to the right
  2. 2x = 8: need y ≤ 1.6 and y > 2, no point; x = 7: 1.5 < y ≤ 2.4, so y = 2
  3. 33 × 7 + 2 = 23 (x = 6, y = 3 gives only 21)
  4. 4Largest value = 23

Answer23

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Also searched as: 0580/42/M/J/22 · 0580_s22_qp_42 · 0580 s22 42 · question 10.
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