Question

How do you solve `abs(3x-9)-2<=7`?

Answer 1

`x in [0, 6]`

Explanation:

The first thing to do here is isolate the modulus on one side of the equation by adding `2` to both sides

`|3x-9| - color(red)(cancel(color(black)(2))) + color(red)(cancel(color(black)(2))) <= 7 + 2`

`|3x - 9| <= 9`

Now, you have two possible cases to deal with

• `3x - 9 >=0 implies | 3x-9| = 3x-9`

In this case, the inequality becomes

`3x-9 <= 9`

`3x <= 18 implies x <= 18/3 = 6`

• `3x - 9 < 0 implies |3x - 9| = - (3x-9)`

In this case, the inequality becomes

`-(3x-9) <= 9`

`-3x +9 <=9`

`-3x <= 0 implies x >= 0`

You thus know that any value of `x` that is smaller than or equal to `6` and greater than or equal to `0` will satisfy the original inequality. In interval notation, this is written as

`x in (-oo, 6] nn [0, + oo)`

This means that the solution interval will be

`x in [0, 6]`