Exam-Style Questions on Inequalities

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

Worked Solution

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2.	GCSE Higher

(a) Write down the integer values of \(x\) that satisfy the inequality \(–3 \le x \lt 1 \).

(b) Solve the compound inequality \( -4 \leq 2y + 1 \lt 6 \) and write down the integer solutions for \(y\).

(c) Find the single digit integers for \(z\) which satisfy the inequality \( -3z \leq 9 \).

Worked Solution

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3.	GCSE Higher

(a) Solve 9m < 15m − 12

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

Worked Solution

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4.	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 \(f(x) \le 0\).

Worked Solution

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5.	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 \( x \ge 0.5\)
  
  

Worked Solution

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6.	GCSE Higher

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

Worked Solution

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7.	GCSE Higher

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

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

Worked Solution

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8.	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.

Worked Solution

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9.	GCSE Higher

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

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

Worked Solution

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10.	GCSE Higher

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

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

Worked Solution

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11.	GCSE Higher

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

$$-8 \le x \le 6$$ $$2y \ge x + 1$$ $$2y+x \le 12$$

(a) By shading the unwanted regions show the region on the grid below.

(b) Mr Mushnik says that the point with coordinates (-8, 10) does not satisfy all the inequalities because it does not lie in the region. Is Mr Mushnik correct? You must give a reason for your answer.

Worked Solution

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12.	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

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

Worked Solution

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13.	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 \( p \gt r \)

Worked Solution

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14.	GCSE Higher

Solve \(2x^2 + 7x - 15 <0\)

Worked Solution

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