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

May 11, 2018

$- 9 \le - 3 x + 15 \le 12$ when $1 \le x \le 8$

#### Explanation:

If we define $f \left(x\right) = - 3 x + 15$ 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 $1 \le x \le 8$ fulfills the inequality.

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

The lower:
$f \left(x\right) = - 3 x + 15 \le 12$
Add 3x-12 on both sides to move 3x to one side and the constants to the other:
$15 - 12 = 3 \le 3 x$
This gives $x \ge 1$

The values for x fulfilling the inequality is, therefore
$1 \le x \le 8$

Which is what we could read directly from the graph.