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|.
|x-3| > |x+5|
(x-3)² > (x+5)²
x² – 6x + 9 > x² +10x + 25
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
Hence, -2< x< 8
Note, for example 2, you can also use the method discussed in this post here.