Question

How do you solve `- 5| x + 1| = -10`?

Answer 1

See below.

Explanation:

Firstly we can simplify this to:
`|x+1|=2`

Then there's two ways of doing it:

Case analysis (the generally most preferred method for simple ones like this):

Consider if `x+1` is positive:
Then: `x+1=2` so `x=1`
Consider if `x+1` is negative:
Then: `-(x+1)=2` so `x=-3`
(We take the negative of the left side since we want it to be positive and we've assumed it to already be negative, and two negatives make a positive, which is necessary for the absolute symbols since they always give a positive value by definition).

An alternative method, my personal favourite since it's more rigid:
`|x+1|=2`
We can square both sides, since the left side will definitely be positive after it has been squared (by definition of real numbers):
`(x+1)^2=4`
`x^2+2x+1=4`
`x^2+2x-3=0`
`(x+3)(x-1)=0`
`x=1 or -3`