Question

How do you solve `|x + 6| = 2x` and find any extraneous solutions?

Answer 1

`x=-2` if `x<-6`
`x=6` if`x>=-6`

Explanation:

Knowing the property of absolute value that says:

if`|x|=a` then
`x=-a , x<0`

`x=a,x>=0`

Applying the above property we have:

`x+6=-2x,x+6<0`. Eq(1)

`x+6=2x,x+6>=0`. Eq(2)

Solving the above equations we have:

Eq(1): if `x+6<0` that is `x<-6`
`x+2x=-6`
`3x=-6`
`x=-6/3=-2`
So,
`x=-2,x<-6`

Eq(2): if `x+6>=0` that is `x>=-6`

`x+6=2x`
`x-2x=-6`
`-x=-6`
`x=6` whenever `x>=-6`