What is the solution set for absx - 1 < 4|x|−1<4?

1 Answer
Stefan V.
Aug 14, 2015

-5 < x < 5−5<x<5

Explanation:

To solve this absolute value inequality, first isolate the modulus on one side by adding 11 to both sides of the inequality

|x| - color(red)(cancel(color(black)(1))) + color(red)(cancel(color(black)(1))) < 4 + 1

|x| < 5

Now, depending on the possible sign of x, you have two possiblities to account for

  • x>0 implies |x| = x

This means that the inequality becomes

x < 5

  • x<0 implies |x| = -x

This time, you have

-x < 5 implies x> -5

These two conditions will determine the solution set for the absolute value inequality. Since the inequality holds true for x> -5, any value of x that's smaller than that will be excluded.

LIkewise, since x<5, any value of x greater than 5 will also be excluded. This means that the solution set to this inequality will be -5 < x < 5, or x in (-5, 5).

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