# Solving inequalities involving modulus sign: |f(x)| > |g(x)| or |f(x)| < |g(x)|

To solve an inequality involving |f(x)| > |g(x)| or |f(x)| < |g(x)| , we can make use of the results:

• if |a| > |b|, then a² > b²

## Examples of Solving inequality involving |f(x)| > |g(x)| or |f(x)| < |g(x)|

Question 1: Solve for x given |x-3| > |x+5|.

Solutions

|x-3| > |x+5|

(x-3)² > (x+5)²

x² – 6x + 9 > x² +10x + 25

-16>16x

x < -1

Do note that we do not need to see | | at both sides of the inequality sign to use this method. We could use this method as long as you have || on one side, and the other side of the inequality sign is always more than or equal to 0.

Question 2: Solve for x given |x-3| < 5.

|x-3| < 5

5 is a number that is always greater than or equal to 0. Also, we know that |5| = 5

Hence, we can square both sides of the inequalities.

(x-3)² < 25

x² – 6x + 9 – 25 < 0

x² – 6x – 16 < 0

(x-8)(x+2)<0

Hence, -2< x< 8

Note, for example 2, you can also use the method discussed in this post here.

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