Chapter 9 linear inequaliTieS 339We use double inequalities to solve word problems where the solution is a
limited range of values. For example, if a student is paid $15 per hour and wants
to earn between $300 and $400 per week, we solve 300 ≤≤Pay 400 , or
300 15≤≤x 400 , if x represents the number of hours worked in a week.
EXAMPLE
A high school student earns $8 per hour in her summer job. She hopes to
earn between $120 and $200 per week. What range of hours will she need
to work so that her pay is in this range?
Let x represent the number of hours worked per week. Represent her
weekly pay by p = 8x. The student wants 120 < p < 200, so we want to solve
the inequality 120 < 8x < 200.120 8 200
120
88
8200
8
15 25≤≤≤≤
≤≤xx
xThe student would need to work between 15 and 25 hours per week for her
pay to range from $120 to $200 per week.
A manufacturing plant produces pencils. It has monthly overhead costs of
$60,000. Each gross (144) of pencils costs $3.60 to manufacturer. The com-
pany wants to keep total costs between $96,000 and $150,000 per month.
How many gross of pencils should the plant produce to keep its costs in
this range?
Let x represent the number of gross of pencils manufactured monthly. Pro-
duction cost is represented by 3.60x. Represent the total cost by c = 60,000 +
3.60x.
The manufacturer wants 96,000 ≤ c ≤ 150,000. This gives us the inequality
96,000 ≤ 60,000 + 3.60x ≤ 150,000.,,.,
,,96 000 60 000 360 150 000
60 000 60 000 60≤+≤
-- -x
,,
,. ,
,
..
.000
36 000 36090000
36 000
360360
36090≤≤
≤≤x
x ,,
.
,,000
360
10 000≤≤x 25 000EXAMPLE
A high school student earns $8 per hour in her summer job. She hopes to