Question

How do you solve `abs(8x+8) = abs(9x-4)`?

Answer 1

Split into cases for intervals `(-oo, -1)`, `[-1, 4/9)` and `(4/9, oo)`, solving the linear equations that result to find:

`x=-4/17` or `x=12`

Explanation:

Split into cases:

Case 1: `x in (-oo, -1)`

`8x+8 < 0`, so `abs(8x+8) = -(8x+8) = -8x-8`
`9x-4 < 0`, so `abs(9x-4) = -(9x-4) = -9x+4`

Equation becomes:

`-8x-8 = -9x+4`

Add `9x+8` to both sides to get:

`x = 12`

This lies outside `(-oo, -1)` so is not valid for this case.

Case 2: `x in [-1, 4/9)`

`8x+8 >= 0`, so `abs(8x+8) = 8x+8`
`9x-4 < 0`, so `abs(9x-4) = -(9x-4) = -9x+4`

Equation becomes:

`8x+8 = -9x+4`

Add `9x-8` to both sides to get:

`17x=-4`

Divide both sides by `17` to get:

`x=-4/17`

This does lie in `[-1, 4/9)` so is a valid solution.

Case 3: `x in [4/9, oo)`

`8x+8 >= 0`, so `abs(8x+8) = 8x+8`
`9x-4 >= 0`, so `abs(9x-4) = 9x-4`

Equation becomes:

`8x+8 = 9x-4`

Add `-8x + 4` to both sides to get:

`x = 12`

This does lie in `[4/9, oo)` so is a valid solution.

Answer 2

`abs(u) = abs(v)` if and only if either `u=v` or `u=-v`.

Explanation:

Two numbers have the same absolute value when the numbers are equal or they are opposites (negatives) of each other.

`abs(8x+8) = abs(9x-4)`

if and only if:

`8x+8 = 9x-4` `color(white)"xx"` or `color(white)"xx"` `8x+8 = -(9x-4)`

Solving `8x+8 = 9x-4` , we get

`8+4 = 9x-9x` so `12=x`, that is: `x=12`

Solving `8x+8 = -(9x-4)`. we get

`8x+8 = -(9x-4)`, so `8x+8 = -9x+4`

and then `8x+9x = 4-8`, so `17x = -4`, and finally `x = -4/17`

The solutions are: `12` and `-4/17`