AQA Level 2 Further Maths · Level 2 Certificate · Chapter 4: Algebra IV · Lesson 4

Linear and quadratic inequalities

About 80 minutesNo calculator72 marks

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Date

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Quick questions to start.

A1
Needed today
Factorise $3 x^{2} - 7 x - 6$ .
[2 marks]
Answer
A2
GCSE Higher
Solve $7 - 3 x > 19$ .
[2 marks]
Answer $\dots\dots\dots\dots$
A3
Last lesson
Show that $(x - 2)$ is a factor of $x^{3} + 4 x^{2} - 3 x - 18$ .
[2 marks]
A4
From chapter 3
$f (x) = 2 x - 5$ with domain $1 ⩽ x ⩽ 6$ . Work out the range of $f (x)$ .
[2 marks]
Answer $\dots\dots\dots\dots$

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Solve $3 (x - 4) > 7 x + 8$ .

Hint Work as for an equation. If you divide by a negative number, the inequality sign turns round.

B1
Solve $2 (3 - 2 x) ⩽ 5 x - 21$ .
[3 marks]
Answer $\dots\dots\dots\dots$

Example 2

Solve $- 7 < 5 - 3 x ⩽ 11$ . Write down the integers that satisfy it.

Hint Do the same to all three parts. Dividing by

- 3

turns both signs round, so rewrite it with the smaller number first.

B2
Solve $9 > 4 - 5 x ⩾ - 16$ . Write down the integers that satisfy it.
[3 marks]
Answer $\dots\dots\dots\dots$ ; integers $\dots\dots\dots\dots$

Example 3

$- 1 ⩽ a ⩽ 5$ and $2 ⩽ b ⩽ 6$ . Work out an inequality for $a - b$ and an inequality for $a^{2}$ .

Hint

a - b

is least when

a

is least and

b

is greatest. For

a^{2}

a

can be

0

, and the square of a negative number is positive.

B3
$- 4 < x < 3$ and $1 < y < 5$ . Work out an inequality for $2 x - y$ and an inequality for $x^{2}$ .
[4 marks]
Answer $\dots\dots < 2 x - y < \dots\dots \dots\dots x^{2} < \dots\dots$

Example 4

Solve (i) $x^{2} - 2 x - 15 ⩽ 0$ and (ii) $x^{2} - 2 x - 15 > 0$ .

Hint Factorise to find the critical values. Sketch the U-shaped graph: it is below the axis BETWEEN them and above the axis OUTSIDE them.

B4
Solve $2 x^{2} + x - 6 > 0$ .
[3 marks]
Answer $\dots\dots\dots\dots$

Example 5

Solve $14 - 5 x > x^{2}$ .

Hint Collect everything on the side where

x^{2}

is positive first:

0 > x^{2} + 5 x - 14

B5
Solve $6 + x - 2 x^{2} < 0$ .
[4 marks]
Answer $\dots\dots\dots\dots$

Practice

Level 2

C1
Solve $\frac{3 x - 2}{4} ⩾ x - 3$ .
[3 marks]
Answer $\dots\dots\dots\dots$
C2
The sketch shows the graph of $y = 10 + 3 x - x^{2}$ . It crosses the $x$ -axis at $A$ and $B$ .
Not drawn accurately
(a)
Work out the coordinates of $A$ and $B$ .
[2 marks]
Answer $A (\dots, \dots) B (\dots, \dots)$
(b)
Hence solve $10 + 3 x - x^{2} < 0$ .
[2 marks]
Answer $\dots\dots\dots\dots$
Total for C2: 4 marks

C3
$a < - 2$ and $b > 3$ . For each statement, write ALWAYS TRUE, SOMETIMES TRUE or NEVER TRUE.
(i) $ab < - 6$ (ii) $a + b > 0$ (iii) $a^{2} > 4$ (iv) $b - a < 5$
[4 marks]
C4
The perimeter of the rectangle is greater than the perimeter of the square. All lengths are in centimetres. Work out the values of $x$ for which this is true.
[3 marks]
Not drawn accurately
Answer $\dots\dots\dots\dots$
C5
A rectangle measures $(x + 2)$ cm by $(x - 1)$ cm. Its area is less than $40$ cm². Work out the values of $x$ for which this is true.
[4 marks]
Not drawn accurately
Answer $\dots\dots\dots\dots$
C6
How many integers satisfy $x^{2} - 2 x - 24 ⩽ 0$ ?
[3 marks]
Answer

Exam-style questions

AQA exam style

D1
Circle the solution of $9 - x^{2} < 0$
$- 3 < x < 3 x < - 3$ or $x > 3 x > 3 x < - 9$ or $x > 9$
[1 mark]

D2
Max says, "To solve $x^{2} ⩾ 6 x$ , I divide both sides by $x$ . So the solution is $x ⩾ 6$ ."
Max is wrong.
(a)
Give a value of $x$ that satisfies $x^{2} ⩾ 6 x$ but is not in Max's answer.
[1 mark]
Answer $x = \dots\dots$
(b)
Solve $x^{2} ⩾ 6 x$
[3 marks]
Answer $\dots\dots\dots\dots$
Total for D2: 4 marks
D3
A ball is thrown upwards. After $t$ seconds its height is $h$ metres, where
$h = 35 t - 5 t^{2}$
(a)
Work out the values of $t$ for which the ball is more than $50$ m high.
[3 marks]
Answer $\dots\dots\dots\dots$
(b)
For how many seconds is the ball more than $50$ m high?
[1 mark]
Answer $\dots\dots$ seconds
Total for D3: 4 marks
D4
Work out the values of $x$ that satisfy both
$2 x + 1 > 7 and x^{2} - 4 x - 21 < 0$
You must show your working.
[5 marks]
Answer $\dots\dots\dots\dots$
D5
$n$ is an integer and $17 n > 6 (3 n - 4)$ . Work out the greatest possible value of $n$ .
[3 marks]
Answer
D6
The area of the triangle is less than the area of the rectangle. All lengths are in centimetres. Work out the values of $x$ for which this is true.
[4 marks]
Not drawn accurately
Answer $\dots\dots\dots\dots$

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The equation $x^{2} + (k - 1) x + k + 2 = 0$ has two different real roots. Work out the possible values of $k$ .
[5 marks]
Hint 1 · What to try
Two different real roots means the discriminant $b^{2} - 4 a c$ is positive.
Hint 2 · The first line
$(k - 1)^{2} - 4 (k + 2) > 0$ . Expand and collect into a quadratic inequality in $k$ .
Hint 3 · The full method
$k^{2} - 6 k - 7 > 0$ , so $(k - 7) (k + 1) > 0$ . The critical values are $- 1$ and $7$ ; choose the outer regions.
Answer $\dots\dots\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve linear and double inequalities.
I can combine two ranges of values.
I can solve a quadratic inequality.
I can form an inequality from a problem.

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