Question

How do you solve `3 < absx – 5`?

Answer 1

`x in (-oo, -8) uu (8, +oo)`

Explanation:

Start by isolating the modulus on one side of the inequality. You can do this by adding `5` to both sides

`3 + 5 < |x| - color(red)(cancel(color(black)(5))) + color(red)(cancel(color(black)(5)))`

`8 < |x|`

This is of course equivalent to

`|x| > 8`

Now, you need to take into account the fact that `x` can be both positive or negative, which means that you get

• `x>0 implies |x| = x`

For positive values of `x`, the inequality will be

`x > 8`

• `x<0 implies |x| = -x`

For negative values of `x`, the inequality will be

`-x > 8`

Multiply both sides by `-1` to get `x` on the left side of the inequality - do not forget that the sign of the inequality changes when you multiply or divide by a negative number

`-1 * (-x) color(red)(<) 8 * (-1)`

`x < -8`

This means that your origininal inequality will be tru for any value of `x` that is smaller than `-8` or bigger than `8`. In other words, you need `x` to belong to two distinct intervals, `x<-8` and `x>8`, which can be written as `x in (-oo, -8) uu (8, +oo)`