# how to solve this ? |x+1|-|x+2|<=2

lfryerda | Certified Educator

In order to find the solution to the inequality, we need to find the intersection points of the related equation

`|x+1|-|x+2|=2`

and then determine for each region between those intersection points, which region satisfies the inequality.

There are two cases to consider when solving the inequality.

Case 1:

`x+1-x-2=2`

`-1=2`   there are no intersections from this case since this is not a valid equation.

Case 2:

`x+1+x+2=2`

`2x=-1`

`x=-1/2`

For the region to the left of the point `x=-1/2` , we see that by using a test point such as `x=-1` and substituting it into the inequality, we get:

`|-1+1|-|-1+2|`

`=0-1`

`=-1<=2`

which means that the region `x in (-infty, -1/2]` is a solution to the inequality.

From the region to the right of the point `x=-1/2` , by using a test point such as `x=1` , we get:

`|1+1|-|1+2|`

`=2-3`

`=-1<=2`

which means the region `x in [-1/2,infty)` is also a solution to the inequality

This means that all real numbers `x in R` are a solution to the inequality.