Inequalities and Linear Programming: practice questions

6 questions · 12 marks · free, no account · Read the notes first

1. Question 1Multiple choice · 1 mark

   Solve $x^2 - x - 6 < 0$.

   A

   $x < -2$ or $x > 3$

   B

   $-2 < x < 3$

   C

   $-3 < x < 2$

   D

   $x < -3$ or $x > 2$

2. Question 2Multiple choice · 1 mark

   Solve $3 - 2x > 7$.

   A

   $x > -2$

   B

   $x < 2$

   C

   $x > 2$

   D

   $x < -2$

3. Question 3Multiple choice · 1 mark

   Solve $x^2 \geq 9$.

   A

   $x \geq 3$

   B

   $-3 \leq x \leq 3$

   C

   $x \leq -3$ or $x \geq 3$

   D

   $x \geq \pm 3$

4. Question 4Multiple choice · 1 mark

   A shaded region (including the boundary) has vertices $(0, 0)$, $(4, 0)$, $(3, 3)$ and $(0, 4)$. If $(x, y)$ is a point in the shaded region, find the greatest value of $3x + 2y$.

   A

   $15$

   B

   $12$

   C

   $8$

   D

   $18$

5. Question 53 marks

   Solve the compound inequality $2x - 1 < 5$ and $3x + 4 \geq x$. Hence find the number of integers satisfying it.

   Show worked solution

   $2x - 1 < 5 \Rightarrow x < 3$ 1A

   $3x + 4 \geq x \Rightarrow x \geq -2$ 1A

   So $-2 \leq x < 3$, and the integers are $-2, -1, 0, 1, 2$: there are 5 integers. 1A

6. Question 65 marks

   A bakery makes $x$ trays of cookies and $y$ trays of cakes each day, subject to

   $$x + 2y \leq 16, \quad 3x + 2y \leq 24, \quad x \geq 0, \quad y \geq 0$$

   The profit is $40 per tray of cookies and $50 per tray of cakes. Find the number of trays of each that gives the greatest profit, and the greatest profit.

   Show worked solution

   Let the profit be $$P$. Then $P = 40x + 50y$. 1A

   The lines $x + 2y = 16$ and $3x + 2y = 24$ meet where $2x = 8$, i.e. at $(4, 6)$. 1M + 1A

   Vertex	$(0, 0)$	$(8, 0)$	$(4, 6)$	$(0, 8)$
   $P$	$0$	$320$	$460$	$400$

   1M for testing the vertices

   The greatest profit is $460, from 4 trays of cookies and 6 trays of cakes. 1A