Key Concepts: Inequalities show relationships between quantities using symbols like <, >, ≤, ≥. They can be solved algebraically or graphically. Linear programming involves finding optimal solutions subject to constraints represented as inequalities.
Section A: Solving Inequalities
1. Solve the following inequalities: [3]
(a) 2x + 3 > 11
(b) 5 - 2x ≤ -3
(c) -3 < x + 2 < 5
2. Solve: 3(x - 2) ≥ 2(x + 1) [2]
Section B: Graphical Inequalities
3. On a coordinate grid, shade the region satisfying all three inequalities: [4]
x ≥ 0, y ≤ 4, x + y ≤ 6
4. Write down the three inequalities that define the shaded region bounded by the lines x = 0, y = 2, and y = x + 1. [3]
Section C: Linear Programming
5. A company makes chairs and tables. Each chair takes 2 hours to make and each table takes 3 hours. The profit per chair is £20 and per table is £30. There are 60 hours available. [4]
(a) Write the constraint inequality
(b) If the company makes 10 chairs and 10 tables, calculate the profit
(c) Suggest the combination that maximizes profit with the time constraint
6. Maximize P = 3x + 4y subject to: x ≥ 0, y ≥ 0, x + 2y ≤ 8, 2x + y ≤ 10. [3]
Total marks: 22
Mark Scheme
1. (a) 2x > 8, x > 4 [1]
(b) -2x ≤ -8, x ≥ 4 [1]
(c) -5 < x < 3 [1]
2. 3x - 6 ≥ 2x + 2 [1]
x ≥ 8 [1]
3. Correct region shaded - bounded by x = 0 (y-axis), y = 4 (horizontal line), and x + y = 6 (diagonal line) [4]
4. x ≥ 0 [1]
y ≥ 2 [1]
y ≤ x + 1 [1]
5. (a) 2c + 3t ≤ 60 [1]
(b) Profit = 20(10) + 30(10) = £500 [1]
(c) Make 0 chairs and 20 tables for maximum profit of £600 [2]
6. Test corner points: (0,0)→0, (0,4)→16, (2,3)→18, (4,2)→17, (5,0)→15 [2]
Maximum is P = 18 at (2,3) [1]