Question

How do you solve `abs(3-x)-1=abs(4x+2)`?

Answer 1

`color(red)(x = 0)`

Explanation:

We have two absolute conditions, so we must write four different equations without the absolute value symbols and solve for `x`.

Your original equation is

`|3-x|-1 = |4x+2|`

So these equations would be

(1): `(3-x) -1 = (4x+2)`
(2): `(3-x) - 1 = -(4x+2)`
(3): `-(3-x) - 1 = (4x+2)`
(4): `-(3-x) - 1 = -(4x+2)`

Solve Equation 1:

`(3-x) -1 = (4x+2)`
`3-x-1 = 4x+2`
`2-x = 4x+2`
`-x = 4x`
`0 = 5x`
`x = 0`

This is a possible solution, which we will check later.

Solve Equation 2.

`(3-x) -1 = -(4x+2)`
`3-x-1 = -4x-2`
`2-x = -4x-2`
`-x = -4x`
`0 = -3x`
`x = 0`

This is a possible solution, which we will check later.

Solve Equation 3.

`-(3-x) - 1 = (4x+2)`
`-3+x-1 = 4x+2`
`x-4 = 4x+2`
`3x = -6`
`x = -2`

This is a possible solution, which we will check later.

Solve Equation 4.

`-(3-x) - 1 = -(4x+2)`
`-3+x-1 = -4x-2`
`x-4 = -4x-2`
`5x = 2`
`x = 2/5`

This is a possible solution, which we will check later.

Check Solutions 1 and 2

If `x=0`

`|3-x|-1 = |4x+2|`
`|3-0|-1 = |4×0+2|`
`|3|-1 =|0+2|`
`3-1= |2|`
`2 = 2`

Check Solution 3

If `x = -2`

`|3-x|-1 = |4x+2|`
`|3-(-2)|-1 = |4(-2)+2|`
`|3+2| -1 = |-8 +2|`
`|5| -1 = |-6|`
`5-1 = 6`
`4 = 6`

This is an impossible result. `x = -2` is not a solution.

Check Solution 4

If `x = 2/5`

`|3-x|-1 = |4x+2|`
`|3-2/5|-1 = |4×2/5+2|`
`|15/5-2/5|-1 = |8/5+10/5|`
`|13/5|-1 = |18/5|`
`13/5 – 1 = 18/5`
`13/5- 5/5 = 18/5`
`8/5 = 18/5`

This is an impossible result. `x = 2/5` is not a solution.

The only solution is `x = 0`.