336 algebra De mystif ieD
Applications of Double Inequalities
Before we work with applied problems, we need to learn how to solve inequal-
ities that involve two variables. We are given a range for one variable and we
want to find the corresponding range for the other variable, like the tempera-
ture example in the introduction to this chapter. In general, we make a substitu-
tion for the variable that is in the inequality.
EXAMPLE
Give the corresponding interval for x.
y = 3 x – 2
If 7 <y< 10, what is the corresponding interval for x? Because y = 3x – 2,
replace “y” with “3x –2.” “7 < y < 10” becomes “7 < 3 x – 2 < 10”.
73 210
222
93 12
9
3
3
3
12
3
34
≤-≤
+++
≤≤
≤≤
≤≤
x
x
x
x
Give the corresponding interval for x.
y = 4x + 1; –< 33 <y
If –3 < y < 3, the corresponding interval for x can be found by solving
–3 < 4x + 1 < 3.
-< +<
---
-< <
- <<
-< <
34 13
111
44 2
4
4
4
4
2
4
1 1
2
x
x
x
x
Give the corresponding interval for x.
y = 3 – x; 0 < y < 4
EXAMPLE
Give the corresponding interval for