Exercises: Linear inequalities and changing the subject

Solve each inequality for $x \in \mathbb{R}$.

1. a) $x + 5 > 9$
2. b) $3x \le 18$
3. c) $2x - 1 < 7$
4. d) $-x \ge 4$

Solve each inequality for $x \in \mathbb{N}$ and list the solutions.

1. a) $x - 2 < 3$
2. b) $2x \le 9$
3. c) $x + 1 \ge 4$ and $x < 7$

Make $x$ the subject of each formula.

1. a) $y = x + 7$
2. b) $y = 4x$
3. c) $y = 3x - 5$

Solve each inequality for $x \in \mathbb{R}$ and graph the solution on a number line.

1. a) $5x - 3 \le 2x + 9$
2. b) $\frac{x}{3} + 2 > 4$
3. c) $4 - 3x < 13$

Solve $-3 < 2x + 1 \le 9$ for $x \in \mathbb{Z}$, list the integer solutions and show them on a number line.

Make $r$ the subject of each formula.

1. a) $A = \pi r^2$, where $r > 0$
2. b) $C = 2\pi r$
3. c) $V = \frac{1}{3}\pi r^2 h$, where $r > 0$

A taxi in Dublin charges a fixed fee of €4 plus €1.20 per kilometre. Ciara has at most €30 to spend on a journey.

1. a) Write an inequality for the number of kilometres $k$ she can travel.
2. b) Solve it and state the greatest distance, correct to one decimal place.
3. c) A second company charges €2 fixed plus €1.50 per kilometre. Find the values of $k$ for which the first company is cheaper.

The formula $s = ut + \frac{1}{2}at^2$ gives the distance travelled.

1. a) Make $a$ the subject.
2. b) Make $u$ the subject.
3. c) Find $a$ when $s = 100$, $u = 5$ and $t = 8$.

Make $x$ the subject of each formula, and state any restriction on the variables.

1. a) $y = \frac{2x + 3}{x - 1}$
2. b) $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ (make $v$ the subject)
3. c) $T = 2\pi\sqrt{\frac{l}{g}}$ (make $l$ the subject)