Question

How do you solve `-2<=x-7<=11`?

Answer 1

`x in [5, 18]`

Explanation:

The first thing to do here is get `x` alone in the middle of the compound inequality by adding `7` to all three sides

`-2 + 7 <= x - color(red)(cancel(color(black)(7))) <= 11 + 7`

`color(white)(aaaaa)5 <= color(white)(aa)xcolor(white)(aa) <= 18`

You now know that in order to be part of the solution interval, a value of `x` must satisfy two conditions

`x >= color(white)(1)5 ->` the left side of the compound inequality

`x <= 18 ->` the right side of the compound inequality

For the first condition, you need `x` to be greater than or equal to `5`. In interval notation, this is written as

`x in [5, +oo)`

For the second condition, you need `x` to be smaller than or equal to `18`. In interval notation, this is written as

`x in (-oo, 18]`

This means that the solution interval for the compound inequality must have `x` greater than or equal to `5` and smaller than or equal to `18`.

Thsi is written as

`x in (-oo, 18] nn [5, +oo) implies color(green)(|bar(ul(color(white)(a/a)color(black)( x in [5, 18])color(white)(a/a)|)))`