year 8 maths

2I
Inequalities
103
10 At a certain school the following grades are awarded for different scores.
Score x ≥ 80 60 ≤ x < 80 40 ≤ x < 60 20 ≤ x < 40 x < 20
Grade A B C D E
a Convert the following scores into grades.
i 15 ii 79 iii 80 iv 60 v 30
b Emma got a B on one test, but her sister Rebecca got an A with just 7 more marks. What is the
possible range for Emma’s score?
c Hugh’s mark earned him a C. If he had scored half this mark, what grade would he have earned?
d Alfred and Reuben earned a D and a C, respectively. If their scores were added together, what
grade or grades could they earn?
e Michael earned a D and was told that if he doubled his mark he would have a B. What grade or
grades could he earn if he got an extra 10 marks?
11 Sometimes multiple inequalities can be combined to a simpler inequality.
a Explain why the combination x ≥ 5, x ≥ 7 is equivalent to the inequality x ≥ 7 .
b Simplify the following pairs of inequalities to a single inequality.
i x > 5, x ≥ 2 ii x < 7, x < 3 iii x ≥ 1, x > 1
iv x ≤ 10, x < 10 v x > 3, x < 10 vi x > 7, x ≤ 10
c Simplify the following pairs of inequalities to a single inequality.
i 3 < x < 5, 2 < x < 7 ii −2 ≤ x < 4, −2 < x ≤ 4
iii 7 < x ≤ 10, 2 ≤ x < 8 iv 5 ≤ x < 10, 9 ≤ x ≤ 11
12 Some inequalities, when combined, have no solutions; some have one solution and some have
infinitely many solutions. Label each of the following pairs using 0 , 1 or ∞ (infinity) to state how
many solutions they have.
a x ≥ 5 and x ≤ 5 b x > 3 and x < 10
c x ≥ 3 and x < 4 d x > 3 and x < 2
e −2 < x < 10 and 10 < x < 12 f −3 ≤ x ≤ 10 and 10 ≤ x ≤ 12
g x > 2.5 and x ≤ 3 h x ≥ −5 and x ≤ −7
ENRICHMENT
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13, 14
Working within boundaries
13 If it is known that 0 ≤ x ≤ 10 and 0 ≤ y ≤ 10 , which of the following inequalities must be true?
Justify your answers.
a x + y ≤ 30 b 2x ≤ 20
c 10 ≤ 2y ≤ 20 d x × y ≤ 100
e 0 ≤ x − y ≤ 10 f x + 5y ≤ 100
14 If it is known that 0 ≤ a ≤ 10 , 0 ≤ b ≤ 10 and 0 ≤ c ≤ 10 , what is the largest value that the following
expressions could have?
a a + b + c b ab + c
c a(b + c) d a × b × c
e a − b − c f a − (b − c)
g 3a + 4
h a − bc
Cambridge Maths NSW
Stage 4 Year 8 Second edition
ISBN 978-1-108-46627-1 © Palmer et al. 2018
Cambridge University Press
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