# How do you solve #abs(2x)<=abs(x-3)#?

##### 1 Answer

When dealing with moduli, it is often helpful to split into cases at values where the sign of the enclosed value changes.

For our example,

(a)

(b)

(c)

(d)

(e)

In case (a):

So the original inequality is equivalent to

Adding

Since this is case (a), we have

In case (b):

So the inequality

So

In case (c):

So the original inequality is equivalent to

Add x to both sides and divide both sides by 3 to get:

Since this is case (c), we also require

In case (d):

In case (e);

So the original inequality is equivalent to

Subtracting

Since this is case (e), we also require

The union of our solutions from cases (a)-(c) gives us: