Absolute value inequality: Find the interval for the set of x-values that satisfy | 3x + 1 | < 7.

First we can remove the absolute value signs if we consider the quantity being either positive or negative.

If (3x + 1) is positive, we can write (3x + 1) < 7.

If (3x + 1) is negative, we can write -(3x + 1) < 7.

If we multiply the latter result by -1, and change the direction of the inequality as required by Rule 4 for inequalities, we get (3x + 1) > -7.

We can combine these in -7 < (3x + 1) < 7.

By Rule 2 for inequalities, we can add -1 to both sides:

-7 -1 < (3x + 1) -1 < 7 -1, or

-8 < (3x) < 6.

By Rule 4, we can multiply by 1/3 and preserve the direction of the inequality:

, or . In interval notation this is

Test Problem: Find the interval for the set of x-values that satisfy |(2 + 5x)| < 4.

a)

b)

c)

d)