Algebra Demystified 2nd Ed

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