Quadratic and two-variable inequalities

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 3 marks
Solve each inequality.
(a) Solve $x^2 - 9 < 0$ . (2)
(b) Solve $x^2 \ge 16$ . (1)
1. $x \le -4$ or $x \ge 4$
2. $x \ge 4$
3. $-4 \le x \le 4$
4. $x \ge \pm 4$
Question 2Non-calculator · 4 marks
$(x+2)(x-5)<0$ .
(a) Solve the inequality. (2)
(b) Find the number of integer solutions. (2)
Question 3Calculator · 3 marks
(a) x and y are positive integers. They satisfy x + 2y $\le$ 13 and 3x + y $\le$ 17. Find the greatest possible value of 4x + 5y. (3)

Answers and marks

(a) $-3 < x < 3$

(b) $x \le -4$ or $x \ge 4$

(a) $-2<x<5$

M1 The boundary roots are −2 and 5; the upward-opening quadratic is negative between them.
A1 Correct answer: $-2<x<5$

(b) $6$

(a) $37$

Quadratic and two-variable inequalities

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