# How do you solve the inequality –9(x – 2) < –72?

Dec 30, 2015

S = $\left(x \in \mathbb{R} | x > 10\right)$

#### Explanation:

$- 9$ multiplies the parenthesis content, so:
$- 9 \cdot x + \left(- 9\right) \cdot - \left(2\right)$
$- 9 x + 18 < - 72$.

$18$ passes to the other side and changes it sign:

$- 9 x < - 72 - 18$

$- 9 x < - 90$

Now, $- 9$ that multiplies $x$ passes to the other side, dividing $- 90$:

$x > \frac{- 90}{- 9}$

Notice that the direction of inequality has changed because either side was divided by a negative number.

Double negative equals positive, and $\frac{90}{9} = 10$, and the sense
of direction changes because as a rule if a negative number is multiplied or divided to both sides of an inequality the sense of direction changes.
$x > 10$.

Solution set: S = $\left(x \in \mathbb{R} | x > 10\right)$.

This can be checked by substituting any number greater than positive 10 into the original inequality, say 11, and we have

$- 9 \left(11 - 2\right) = - 81 < - 72$

which is true!