Question

How do you solve `abs(x+5)>12`?

Answer 1

`x in (-oo, -17) uu (7, + oo)`

Explanation:

You're dealing with the absolute value of an expression, which means that you need totake into account the fact that the absolute value of a real number returns a positive value regardless of the sign of said number.

`color(blue)(|x| = {(x", "x >=0),(-x", "x<0) :}, " "(AA)x in RR)`

This means that you have to look at two possible scenarios, one in which the expression inside the modulus is positive and one in which it's negative.

• `x+5 >0 implies |x+5| = x + 5`

The inequality takes the form

`x + 5 >12 implies x > 7`

• `x+5<0 implies |x+5| = -(x+5)`

This time, you get

`-(x+5) > 12`

`-x - 5 > 12`

`-x > 17 implies x < -17`

The inequality will be true for any value of `x` that is greater than `7` or smaller than `(-17)`. The solution set will thus be `x in (-oo, -17) uu (7, + oo)`.