Question

How do you solve the absolute value `4abs(3x+4) = 4x + 8`?

Answer 1

Firstly we need to get the understanding of an absolute value, which is that what ever is inside the sign, will always come out positive.

As for all positive `RR` numbers of `x` we will get that
`abs(-x) = x` and `absx = x`

beware where the `-` (negative) lies as
`-absx = -x`

now to look at the sum.

we got

`4abs(3x +4) = 4x + 8`

now we will divide all by `4` and get

`abs(3x + 4) = x + 2`

now here is where the "tricky" part of the absolute value comes in,
as now we break the sum up, to take into consideration the "negative" side of things. so now we have two sums, of which:

`color(red)(3x + 4 = x + 2)` or `color(blue)(3x+4 = -(x+2))`

`color(red)(3x - x = 2 -4)` or `color(blue)(3x + x = -2 -4)`

`color(red)(2x = -2)` or `color(blue)(4x = -6)`

`color(red)(x=-1)` or `color(blue)(x= -(3/2))`

thus you end up with two answers which satisfy the equation.