Question

How do you solve `-3+2abs(n-9)=1`?

Answer 1

`n=7` or `n=11`

Explanation:

First, isolate the modulus on one side of the equation by adding `3` to both sides of the equation and dividing everything by `2`.

`- color(red)(cancel(color(black)(3))) + color(red)(cancel(color(black)(3))) + 2|n-9| = 1 + 3`

`(color(red)(cancel(color(black)(2))) |n-9|)/color(red)(cancel(color(black)(2))) = 4/2`

`|n-9| = 2`

Now, this equation can produce 2 distinct values for `n` depending on which condition is true

• If `(n-9)<0`, then you have

`|n-9| = -(n-9) = -n + 9`

This means that the equation becomes

`-n + 9 = 2 => n = color(green)(7)`

• If `(n-9)>0`, then you have

`|n-9| = n-9`

This means that `n` will be equal to

`n-9 = 2 => n = color(green)(11)`

Answer 2

There is a geometric way to think about this.

Explanation:

Starting with `-3+2abs(n-9) = 1`,

we can see that we must have:

`2abs(n-9) = 4`, so

`abs(n-9) = 2`

On the number line, this means that the distance between `n` and `9` is 2.
There are two numbers that are `2` away from `9`: one on the left of `9` at `9-2` and the other on the right at `9+2`.

So, we get:

`n = 9-2" "` or `" "n=9+2`

and

`n=7" "` or `" "n=11`