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Cambridge IGCSE™ and O Level
Additional Mathematics (0606/4037)
Student’s Book, Second Edition
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Using graphs to solve cubic inequalities
4 EQUATIONS, INEQUALITIES AND GRAPHS
Using graphs to solve cubic
inequalities
Solve the inequality 3(x + 2)(x − 1)(x − 7) −100 graphically.
Cubic graphs have distinctive shapes determined by the coefficient of x³.
Solution
Worked example
Because you are solving the inequality graphically, you will need to draw
the curve as accurately as possible on graph paper, so start by drawing up a
table of values.
Negative x3 term
Positive x3 term
y = 3(x + 2)(x − 1)(x − 7)
x
The centre part of each of these curves may not have two distinct
turning points like those shown above, but may instead ‘flatten out’
to give a point of inflection. When the modulus of a cubic function is
required, any part of the curve below the x-axis is reflected in that axis.
−3
−2
Look inside
View sample material
from our Student’s
Books
a
b
1
2
2
3
3
4
4
5
5
8
9
−1
0
5
6
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
−120
0
48
42
0
−60
−120
−162
−168
−120
0
210
2
7
7
8
−2
(x − 7)
x
1
6
6
0
−3
3
4
10
7
The solution is given by the values of x that correspond to the parts of the
curve on or below the line y = −100.
Sketch the graph of y = 3(x + 2)(x − 1)(x − 7). Identify the points where the
curve cuts the axes.
y
50
Sketch the graph of y = |3(x + 2)(x − 1)(x − 7)|.
–2.9
Solution
a
1
−1
−4
Worked example
You are asked for
a sketch graph,
so although it
must show the
main features,
it does not need
to be absolutely
accurate. You
may find it easier
to draw the curve
first, with the
positive x³ term
determining the
shape of the curve,
and then position
the x-axis so that
the distance
between the
first and second
intersections is
about half that
between the
second and third,
since these are
3 and 6 units,
respectively.
0
−1
(x + 2)
(x − 1)
2.6
–1 O
–7 –6 –5 –4 –3 –2
The curve crosses the x-axis at −2, 1 and 7. Notice that the distance
between consecutive points is 3 and 6 units, respectively, so the y-axis is
between the points −2 and 1 on the x-axis, but closer to the 1.
The curve crosses the y-axis when x = 0, i.e. when y = 3(2)(−1)(−7) = 42.
y = – 100
y
1
7
2
3
6.2
4
5
6
8 x
7
–100
y = 3(x + 2) (x – 1) (x – 7)
42
–2
1
–50
–150
x
–200
y = 3(x + 2) (x – 1) (x – 7)
From the graph, the solution is x −2.9 or 2.6 x 6.2.
b
Exercise 4.3
To obtain a sketch of the modulus curve, reflect any part of the curve that
is below the x-axis in the x-axis.
Remember: ( x )
means the positive
square root of x.
y
y = | 3(x + 2) (x – 1) (x – 7) |
42
–2
1
2
1
2 Use the substitution x = u 3 to solve the equation x 3 + 3 x 3 = 4 .
4
3
2
3 Use the substitution x = u 2 to solve the equation x 3 − 10 x 3 = −9.
4 Using a suitable substitution, solve the following equations:
a x − 7 x = −12
x
7
1 Where possible, use the substitution x = u ² to solve the following
equations:
b x+2 x =8
a x − 4 x = −4
c x − 2 x = 15
d x + 6 x = −5
c
2
b x −2 x +1= 0
1
x 3 + 3 x 3 = 10
4
5
Using graphs to solve cubic inequalities
4 EQUATIONS, INEQUALITIES AND GRAPHS
Past-paper questions
1
5 a Use the substitution x = u 2 to solve the equation x 4 − 5 x 2 + 4 = 0.
b Using the same substitution, show that the equation x 4 + 5 x 2 + 4 = 0
has no solution.
c Solve where possible:
1 (i) Sketch the graph of y = |(2x + 3)(2x − 7)|.
(ii) How many values of x satisfy the equation
|(2x + 3)(2x − 7)| = 2x?
4
2
x 3 + 5x 3 + 4 = 0
4
2
x 3 − 5x 3 + 4 = 0
i
ii
6 Sketch the following graphs, indicating the points where they cross
the x-axis:
a y = x(x – 2)(x + 2)
b y = |x(x – 2)(x + 2)|
c y = 3(2 x – 1)(x + 1)(x + 3)
d y = |3(2 x – 1)(x + 1)(x + 3)|
7 Solve the following equations graphically. You will need to use graph
paper.
b x(x + 2)(x − 3) −1
a x(x + 2)(x − 3) 1
c (x + 2)(x − 1)(x − 3) > 2
d (x + 2)(x − 1)(x − 3) < −2
2 (i)
Cambridge IGCSE Additional Mathematics 0606
Paper 23 Q6 November 2011
On a grid like the one below, sketch the graph of
y = |(x − 2)(x + 3)| for −5 x 4, and state the coordinates of
the points where the curve meets the coordinate axes.
[4]
y
8 Identify the following cubic graphs:
a
b
y
y
4
4
3
3
2
2
1
1
O
–3 –2 –1
–1
1
2
3
[4]
[2]
Cambridge O Level Additional Mathematics 4037
Paper 23 Q6 November 2011
x
–2
–2
–1
O
–1
1
x
–5
–4
–3
–2
–1
O
1
2
3
4
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x
–2
–3
–3
9 Identify these graphs. (They are the moduli of cubic graphs.)
a
b
y
y
12
12
12
11
11
11
10
10
10
9
9
9
8
8
8
7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
O
–3 –2 –1
–1
6
c
y
1
2
3
4
x
O
–5 –4 –3 –2 –1
–1
1
2
3
4
x –3 –2 –1 O
–1
–2
–2
–2
–3
–3
–3
(ii) Find the coordinates of the stationary point on the curve
y = |(x − 2)(x + 3)|.
[2]
(iii) Given that k is a positive constant, state the set of values of
k for which |(x − 2)(x + 3)| = k has 2 solutions only.
[1]
Cambridge O Level Additional Mathematics 4037
Paper 12 Q8 November 2013
Cambridge IGCSE Additional Mathematics 0606
Paper 12 Q8 November 2013
3 Solve the inequality 9x 2 + 2x − 1 < (x + 1)2 .
[3]
Cambridge O Level Additional Mathematics 4037
Paper 22 Q2 November 2014
Cambridge IGCSE Additional Mathematics 0606
Paper 22 Q2 November 2014
1
2
3
4
x
7
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