# How do you solve  3 abs(5g+2) -8 ≥ 7 and find any extraneous solutions?

Dec 15, 2017

This inequality will hold true if the $g$ values are greater than or equal to $\frac{3}{5}$, or less than or equal to $- \frac{7}{5}$ The solution is not extraneous.

#### Explanation:

First, we isolate $\left\mid 5 g + 2 \right\mid$
$3 \left\mid 5 g + 2 \right\mid - 8 \ge 7$
$3 \left\mid 5 g + 2 \right\mid \ge 15$
$\left\mid 5 g + 2 \right\mid \ge 5$

Since the variable side is larger than the number side, we know this is a disjunction. Therefore, we write down the two possibilities.
$5 g + 2 \ge 5$
$5 g + 2 \le - 5$

Solve each one.
$5 g + 2 \ge 5$
$5 g \ge 3$
$g \ge \frac{3}{5}$

$5 g + 2 \le - 5$
$5 g \le - 7$
$g \le - \frac{7}{5}$

This means that this inequality will hold true if the $g$ values are greater than or equal to $\frac{3}{5}$ ,or less than or equal to $- \frac{7}{5}$ These values are not extraneous, since they use "or" between the two different situations instead of "and". If this somehow were a conjunction, this would have been extraneous.