Question

How do you solve `abs(6-x)=9`?

Answer 1

Remember the rule:
`|x|=a`
If `x<0`
`a_1= -x`
If `x>0`
`a_2=x`
So if `x`'s interval isn't determined we will have 2 results.

Explanation:

`6-x>0`

`|6-x|=6-x`
`6-x=9`
`x_1=-3`

`6-x<0`

`|6-x|=x-6`
`x-6=9`
`x_2=15`

Answer 2

`x = 15` and `x = -3`.

Explanation:

Absolute value equations have two solutions; since absolute values will always be positive, this makes sense. For example, in `abs(x) = 9`, `x` can equal 9 and -9; the absolute value of 9 equals 9 and the absolute value of -9 equals 9.

As such, we need two equations to find the two solutions. Our two equations are: `6-x = 9` and `6-x = -9`. You've probably noticed that these equations are extremely similar - except one equation equals 9, and the other -9. Always set up absolute value equations like this when you're ready to solve.

Let's get to the solving, starting with `6-x = 9`:

`6-x = 9` (original equation)
`-x = 3` (subtracting 6 from both sides)
`x = -3` (dividing by -1)

Alright, `x = -3` is one solution. Now, for `6-x = -9`:

`6-x = -9` (original equation)
`-x = -15` (subtracting 6 from both sides)
`x = 15` (dividing by -1)

And that's it. Our solutions are `x = 15` and `x = -3`.