Question

How do you solve `abs(x+10)=4x-8`?

Answer 1

`x = 6`

Explanation:

Your absolute value equation looks like this

`|x+10| = 4x-8`

Right from the start, you know that the solutions to this equation must satisfy the condition

`4x-8>0 <=> x>2`

That happens because the absolute value of a number, regardlesss if that number is positive or negative, is always positive.

`color(blue)( |a| = {(a",", "if " a>=0), (-a",", "if "a<0):})`

So, with this in mind, determine the equation's two possible solutions

• If `(x+10)>0`, you have

`|x+10| = x+10`

and the equation becomes

`x+10 = 4x - 8 => x = 18/3 = color(green)(6)`

• If `(x+10)<0`, you have

`|x+10| = -(x+10) = -x-10`

This will get you

`-x-10 = 4x+8 => x = color(red)(-18/5)`

This solution, `x=-18/5`, will be an extraneous solution because it does not satisfy the contion `x>2`.

Therefore, `x=6` is the only solution to this equation.