Question

How do you solve `-3abs(6-x)<-15`?

Answer 1

`x in (-oo, 1) uu (11, + oo)`

Explanation:

First, isolate the modulus on one side of the inequality. Notice that you can do that by dividing both sides of the inequality by `-3` - do not forget that the sign of the inequality changes when you miltiply or divide by a negative number.

`(color(red)(cancel(color(black)(-3))) * |6-x|)/color(red)(cancel(color(black)(-3))) color(blue)(>) (-15)/(-3)`

`|6 - x| > 5`

Next, use the fact that the expression inside the modulus can be positive or negative to find the solution intervals that would satisfy these possibilities.

• `6-x>0 implies |6-x| = 6-x`

When the expression inside the modulus is positive, you get

`6 -x > 5`

`-x > -1 implies x < 1`

• `6-x < 0 implies |6-x| = -(6 - x)`

When the expression inside the modulus is negative, you have

`-(6 - ) > 5`

`-6 + x > 5`

`x > 11`

So, the absolute value inequality will be true for any value of `x` that is smaller than `1` or for any value of `x` that is bigger than `11`.

This means that the solution set for this inequality will be `x in (-oo, 1) uu (11, + oo)`.