 # Exam-Style Questions on Inequalities

## Problems on Inequalities adapted from questions set in previous Mathematics exams.

### 1.

GCSE Higher

The red line in the diagram below shows an inequality. If the variable is $$x$$, which inequality best describes $$x$$? (a) $$-9 \le x \le 4$$

(b) $$-9 \lt x \le 4$$

(c) $$-9 \le x \lt 4$$

(d) $$-9 \lt x \lt 4$$

(e) $$-4 \le x \le 9$$

### 2.

GCSE Higher

(a) Solve 9m < 15m − 12

(b) On the number line below, show the set of values of $$x$$ for which $$-4 \le x-1 \lt 5$$ ### 3.

GCSE Higher

The graph of a quadratic function, $$y=f(x)$$ is shown drawn accurately in the following diagram. Write down all the integer solutions of $$f(x) \le 0$$. ### 4.

GCSE Higher

The diagram below is a sketch of $$y = f(x)$$ where $$f(x)$$ is a quadratic function.

The graph intersects the x-axis where $$x=-2$$ and $$x = 0.5$$. Which of the following is the solution of $$f(x) \le 0$$ ?

• (a) $$x \ge -2$$ or $$x \ge 0.5$$

• (b) $$-2 \ge x \ge 0.5$$

• (c) $$x \ge -2$$ and $$x \le 0.5$$

• (d) $$x \le -2$$ and $$x \ge 0.5$$

### 5.

GCSE Higher

Describe the unshaded (white) region by writing down three inequalities. ### 6.

GCSE Higher

On the grid below indicate the region that satisfies all three of these inequalities.

$$y>-2$$ $$x+y<5$$ $$y-1 \le \frac{x}{2}$$ ### 7.

GCSE Higher

A region on a coordinate grid is described by the following three inequalities:

$$x>-2$$ $$x+y<7$$ $$y \ge \frac{x}{3}+2$$

By shading the unwanted regions show the region on the grid below. ### 8.

GCSE Higher

By shading the unwanted regions, show the region satisfied by these three inequalities.

$$y \le x + 3$$ $$y> 4-x$$ $$x < 2.5$$ ### 9.

GCSE Higher

Solve the following inequalities then explain how the whole number solutions to A and B different.

$$A: 5 \le 5x \lt 30$$ $$B: 5 \lt 5x \le 30$$

### 10.

GCSE Higher

Show that you understand equations and inequalities by answering the following:

(a) Solve $$5x^2=80$$

(b) Solve $$8x + 2 \gt x + 7$$

(c) Write down the largest integer that satisfies $$8x - 2 \lt 25$$

(d) Solve the following pair of equations

$$3x + 5y = 21$$ $$8x - 5y = 1$$

### 11.

GCSE Higher

John times how long it takes him to run around Hazelnut Park each Friday afternoon. The last three weeks his times, rounded to the nearest minute are $$p, q \; \text{and} \; r$$.

He notices that $$q$$ is 5% more than $$p$$ and $$r$$ is 5% less than $$q$$.

Indicate which of the following statements are true:

(a) $$r \lt p$$ and $$r \lt q$$

(b) $$r \gt p$$ and $$r \gt q$$

(c) $$r = p$$ and $$r \lt q$$

(d) $$p \gt q$$ and $$p = r$$

(e) $$p =q$$ and $$p \gt r$$

### 12.

GCSE Higher

Solve $$2x^2 + 7x - 15 <0$$

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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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