Question

What is the solution set for `absx - 1 < 4`?

Answer 1

`-5 < x < 5`

Explanation:

To solve this absolute value inequality, first isolate the modulus on one side by adding `1` 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)`.