Question

How do you solve `-9 <= -3x+15 <= 12`?

Answer 1

`-9<=-3x+15<=12` when `1<=x<=8`

Explanation:

Let us start with a graph of the inequality:

If we define `f(x)=-3x+15` we are interested in the value area for x that will give values of f(x) between -9 and 15, which is the shaded area on the graph.
Reading directly on the graph, we see that `1<=x<=8` fulfills the inequality.

Mathematically we would solve it by treating the two inequalities separately:
`-9<=f(x)=-3x+15`
Add `3x+9` on both sides:
`3x<=15+9=24`
`x<=8` This is the upper limit of the area.

The lower:
`f(x)=-3x+15<=12`
Add 3x-12 on both sides to move 3x to one side and the constants to the other:
`15-12=3<=3x`
This gives `x>=1`

The values for x fulfilling the inequality is, therefore
`1<=x<=8`

Which is what we could read directly from the graph.