Question

How do you solve `|x-4|>3`?

Answer 1

`color(blue)(x > 7)` or `color(green)(x< 1)`

Explanation:

Remember this:
if `abs(a) < 3`
then `color(blue)(a < 3)` and `color(green)(a > -3)`

if `abs(b) > 3`
then `color(blue)(b > 3)` or `color(green)(b <-3)`

If that doesn't make sense, draw it out on a number line to convince yourself that it does.

Using that rule we can do this to our inequality:

`abs(x-4) > 3`
`color(blue)(x-4 > 3)`
`color(green)(x-4 < -3)`

`color(blue)(x-4 > 3)`
`color(blue)(x > 7)`

`color(green)(x-4 < -3)`
`color(green)(x< 1)`

`color(blue)(x > 7)` or `color(green)(x< 1)`

Answer 2

`x>7 or x<1`

Explanation:

`|x-4|>3`
Solve the absolute value
We know
either `x -4 >3 or x -4<-3`

We have two possibilities. The first one is `x - 4>3` and the second one is `x -4<-3`
Solve the first one
`x - 4>3`
Add `color(red)(4)` to both sides
`x cancel(-4) cancelcolor(red)(+4)>3 color(red)(+4)`
`x >7`

Solve the second possibility
`x - 4 <-3`
Add `color(green)(4)` to both sides
`x cancel(-4) cancelcolor(green)(+4) < -3 color(green)(+4)`
`x <1`