A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 3: Equations and inequalities · Lesson 2

Linear and quadratic inequalities

74 marks

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Date

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Quick questions to start.

A1
Needed today
Factorise $3 x^{2} + 4 x - 15$ .
[2 marks]
Answer
A2
Needed today
The equation $2 x^{2} - 6 x + k = 0$ has equal roots. Find $k$ .
[2 marks]
Answer $k = \dots\dots$
A3
Last lesson
Solve the simultaneous equations $y = 2 x - 1$ and $x y = 3$ .
[2 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$
A4
From chapter 1
Write $\frac{12}{3 + 3}$ in the form $a + b 3$ , where $a$ and $b$ are integers.
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Find the set of values of $x$ for which $7 - 2 (x + 4) > 3 x + 9$ .

Hint Expand, collect the

x

terms on one side, and turn the sign round if you divide by a negative.

B1
Find the set of values of $x$ for which $\frac{x - 1}{3} - \frac{2 x + 1}{4} < 1$ .
[3 marks]
Answer $\dots\dots\dots\dots$

Example 2

Find the set of values of $x$ for which $4 + 7 x - 2 x^{2} < 0$ .

Hint Solve

4 + 7 x - 2 x^{2} = 0

to get the critical values. The curve is an upside-down U: where is it below the

x

-axis?

B2
Find the set of values of $x$ for which $x (11 - 3 x) \geq 6$ .
[4 marks]
Answer $\dots\dots\dots\dots$

Example 3

The equation $2 x^{2} + (k - 1) x + 2 = 0$ , where $k$ is a constant, has no real roots. Find the set of possible values of $k$ .

Hint No real roots means

b^{2} - 4 a c < 0

. That gives a quadratic inequality in

k

B3
The equation $(k + 1) x^{2} - 4 x + (k - 2) = 0$ , where $k$ is a constant, has two distinct real roots. Find the set of possible values of $k$ .
[5 marks]
Answer $\dots\dots\dots\dots$

Example 4

Find the set of values of $x$ for which $\frac{12}{x - 1} > 4$ , where $x \neq = 1$ .

Hint You do not know if

x - 1

is positive or negative, so multiply both sides by

(x - 1)^{2}

, which is positive.

B4
Find the set of values of $x$ for which $\frac{10}{x + 3} \leq 2$ , where $x \neq = - 3$ .
[4 marks]
Answer $\dots\dots\dots\dots$

Practice

A-Level

C1
Find the set of values of $x$ for which both $2 (x + 3) > 9 - x$ and $2 x^{2} + 3 x - 20 \leq 0$ .
[5 marks]
Answer $\dots\dots\dots\dots$

C2
Find the set of values of $x$ for which $(x + 4) (x - 2) > 3 x + 2$ . Give your critical values in surd form.
[4 marks]
Answer $\dots\dots\dots\dots$
C3
Find the set of values of $x$ for which $3 x - 1 < x - 7$ or $4 (x - 2) > 2 x + 5$ . Write your answer in set notation.
[3 marks]
Answer $\dots\dots\dots\dots$

Exam-style questions

Edexcel exam style

D1
Leo writes: $(2 x - 7) (x + 2) = 0$ gives critical values $\frac{7}{2}$ and $- 2$ , so $x < - 2$ or $x > \frac{7}{2}$ .
Leo is finding the set of values of $x$ for which $14 + 3 x - 2 x^{2} > 0$ .
(a)
Explain the mistake Leo has made. You may use a value of $x$ to help you.
[2 marks]
(b)
Write down the correct set of values of $x$ .
[1 mark]
Answer $\dots\dots\dots\dots$
(c)
Hence find the set of values of $x$ for which both $14 + 3 x - 2 x^{2} > 0$ and $3 (x + 2) > 7 x - 2$ .
[3 marks]
Answer $\dots\dots\dots\dots$
(d)
Write down the integers that satisfy both inequalities in part (c).
[2 marks]
Answer $\dots\dots\dots\dots$
Total for D1: 8 marks

D2
The equation $2 x^{2} + (k - 4) x + k - 1 = 0$ , where $k$ is a constant, has two distinct real roots.
(a)
Show that $k^{2} - 16 k + 24 > 0$ .
[3 marks]
(b)
Hence find the set of possible values of $k$ . Give your answer in surd form.
[4 marks]
Answer $\dots\dots\dots\dots$
(c)
$k$ is a positive integer less than $20$ . Find all the possible values of $k$ .
[2 marks]
Answer $\dots\dots\dots\dots$
Total for D2: 9 marks
D3
The fuel cost of a car journey, $C$ pence per km, at a steady speed of $v$ km/h is modelled by $C = 0.01 v^{2} - 1.2 v + 50$ , for $20 \leq v \leq 110$ .
(a)
Show that $C < 18$ when $v^{2} - 120 v + 3200 < 0$ .
[2 marks]
(b)
Hence find the range of speeds for which $C < 18$ .
[3 marks]
Answer $\dots\dots\dots\dots$
(c)
On one road the greatest speed allowed is $70$ km/h. Write down the speeds on this road for which $C < 18$ .
[1 mark]
Answer $\dots\dots\dots\dots$
(d)
Find the least value of $C$ , and the speed at which it happens.
[2 marks]
Answer $\dots\dots$ pence per km $v = \dots\dots$
Total for D3: 8 marks
D4
In this question $x \neq = 3$ .
(a)
Find the set of values of $x$ for which $\frac{2 x + 1}{x - 3} > 1$ .
[4 marks]
Answer $\dots\dots\dots\dots$
(b)
Hence find the set of values of $x$ for which both $\frac{2 x + 1}{x - 3} > 1$ and $x^{2} - 2 x - 48 \leq 0$ .
[3 marks]
Answer $\dots\dots\dots\dots$
Total for D4: 7 marks

Extension

Stretch

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

E1
The equation $x^{2} - 2 k x + 2 k + 15 = 0$ , where $k$ is a constant, has two distinct positive roots. Find the set of possible values of $k$ .
[6 marks]
Hint 1 · What to try
Two distinct real roots means the discriminant is positive.
Hint 2 · The first line
Sketch $y = x^{2} - 2 k x + 2 k + 15$ . For both roots to be positive, the curve must cross the $y$ -axis above $O$ and its turning point, at $x = k$ , must be to the right of the $y$ -axis.
Hint 3 · The full method
$4 k^{2} - 4 (2 k + 15) > 0$ gives $(k - 5) (k + 3) > 0$ . Now add $2 k + 15 > 0$ and $k > 0$ .
Answer $\dots\dots\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve a linear inequality.
I can solve a quadratic inequality with a sketch.
I can solve an inequality with $x$ in the denominator.
I can find the values of $k$ for a number of roots.

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