DEPARTMENT OF MATHEMATICS
Core 1
Topic 5 – Inequalities
1
Inequalities
Solving Linear Inequalities
Solving Quadratic
Inequalities
Finding a solution set to
two separate inequalities
Inequalities and the
Discriminant
Inequalities involving
𝑥 2 + 𝑏𝑥 + 𝑐
that factorise
Inequalities involving
a𝑥 2 + 𝑏𝑥 + 𝑐
that factorise
Inequalities involving
completing the square
2
Inequalities
Again, the start of this topic should be a recap of GCSE material. We will look at
finding solutions to linear inequalities.
An inequality literally denotes two things are not equal; unlike an equation, they
are not the same!
Solving linear inequalities is very similar to solving linear equations. There is one
thing we need to be careful of:
If we multiply or divide both sides of an inequality by a negative number,
the inequality sign changes direction.
(Note: this is not affected by adding or subtracting negative numbers!)
Worked Examples:
Solve the following linear inequalities:
a)
5𝑥 + 9 ≤ 3𝑥 + 1
b)
12 − 3𝑥 < 27
3
Notes:
4
Solving quadratic inequalities:
We also need to be able to solve quadratic inequalities. We have already learnt
how to sketch quadratic graphs – this skill will be crucial in solving quadratic
inequalities:
Worked Examples:
Solve the following quadratic inequalities:
a) Find the set of values of 𝑥 for which 𝑥 2 − 4𝑥 − 5 < 0
5
b) Find the set of values of 𝑥 for which 3 − 2𝑥 2 ≤ 5𝑥
6
Ones to be wary of:
A particularly common trap that students fall into involves the difference of 2
squares:
Worked Example:
a)
𝑥2 < 9
b)
2𝑥 2 ≥ 32
Find the set of values of 𝑥 for the following inequalities:
7
I’ve seen this asked once before – you need to use the difference of 2 squares
method, even though you’ll find a slight problem with your ‘square’ number (this
is easy to get around, thankfully!)
c)
3𝑥 2 − 30 ≥ 0
8
Notes:
9
Finding a solution set to 2 separate inequalities:
Occasionally, we’ll be asked to find the set of solutions to 2 separate inequalities
that are true at the same time. To do this, we do the following:
 Solve both inequalities separately
 Draw the set of solutions to each inequality on a number line. The
overlapping section is the solution set that is true for both
Worked Example:
a)
Find the set of solutions that satisfies both inequalities:
6 − 𝑥 − 𝑥 2 > 0 and 2𝑥 − 1 ≤ 0
10
Notes:
11
Inequalities and the Discriminant:
A common application of sketching quadratic inequalities comes from applying the
rules of the Discriminant. Let’s have a look at a couple of examples of this style of
question, taken from past exam papers/revision papers:
Worked Example:
1)
[Jan 2011]
The equation 𝑥 2 + (𝑘 − 3)𝑥 + (3 − 2𝑘) = 0, where 𝑘 is a constant,
has two distinct real roots.
a)
Show that 𝑘 satisfies
b)
Hence, find the set of possible values of 𝑘
𝑘 2 + 2𝑘 − 3 > 0
12
13
2)
[Jan 2010]
𝑓(𝑥 ) = 𝑥 2 + 4𝑘𝑥 + (3 + 11𝑘), where 𝑘 is a constant
a)
Express 𝑓(𝑥 ) in the form (𝑥 + 𝑝)2 + 𝑞 , where 𝑝 and 𝑞 are
constants to be found in terms of 𝑘
b)
Given that the equation 𝑓(𝑥 ) = 0 has no real roots,
find the set of possible values of 𝑘
c)
Given that 𝑘 = 1, sketch the graph of 𝑦 = 𝑓(𝑥 ), showing the
coordinates of any point where the graph crosses a coordinate
axis.
14
15
Notes:
16
Inequalities – Questions
Exercise 1:
Exercise 2:
17
Exercise 3:
1)
The equation
(k + 3) x2 + 6x + k = 5, where k is a constant,
has two distinct real solutions for x.
a) Show that k satisfies
k2 − 2k − 24 < 0
b) Hence find the set of possible values of k.
2)
The equation kx2 + 4x + (5 − k) = 0, where k is a constant, has 2 different real solutions for x.
a) Show that k satisfies
k2 − 5k + 4 > 0.
b) Hence find the set of possible values of k.
3)
Find the set of values of x for which
a) 3x − 7 > 3 – x
b) x2 − 9x ≤ 36
c) both 3x − 7 > 3 − x and x2 − 9x ≤ 36
4)
Find the set of values of x for which
a) 4x − 3 > 7 − x
b) 2x2 − 5x − 12 < 0
c) both 4x − 3 > 7 − x and 2x2 − 5x − 12 < 0
5)
Find the set of values of x for which
a) 3(x − 2) < 8 − 2x
b) (2x − 7)(1 + x) < 0
c) both 3(x − 2) < 8 − 2xand (2x − 7)(1 + x) < 0
18
Solutions to all exercises:
Exercise 1:
Exercise 2:
Exercise 3:
1.b) −4 < 𝑘 < 6
2.b) 𝑘 < 1 𝑜𝑟 𝑘 > 4
3.a) 𝑥 > 2.5
3.b) −3 ≤ 𝑥 ≤ 12
3
4.a) 𝑥 > 2
4.b) − < 𝑥 < 4
5.a) 𝑥 < 2.8
5.b) −1 < 𝑥 <
2
7
2
3.c) 2.5 < 𝑥 ≤ 12
4.c) 2 < 𝑥 < 4
5.c) −1 < 𝑥 < 2.8
19