# How do you solve abs((5y+2)/(2))= 6?

Apr 2, 2015

The answer: $\left\{- \frac{14}{5} , 2\right\}$

Absolute value operation always return non-negative.

So,

$\mathmr{if} \frac{5 y + 2}{2} \ge 0 \to \left\mid \frac{5 y + 2}{2} \right\mid = \frac{5 y + 2}{2}$

$\mathmr{if} \frac{5 y + 2}{2} < 0 \to \left\mid \frac{5 y + 2}{2} \right\mid = \left(- 1\right) \cdot \left(\frac{5 y + 2}{2}\right)$

Since we don't know the value of $y$, we need to check all these possibilities.

First

Assume that $\frac{5 y + 2}{2} \ge 0$

We need to find the range of $y$

$2 \cdot \left(\frac{5 y + 2}{2}\right) \ge 2 \cdot 0$

$5 y + 2 \ge 0$

$A : y \ge - \frac{2}{5}$

We need to remember this.

Now, try to solve the given equation.

$\frac{5 y + 2}{2} = 6$

$5 y + 2 = 12$
$5 y = 10$
$y = 2$

Remember expression $A$, $y = 2$ satisfies the inequality in expression $A$. So $2$ is in the solution set.

Second

Assume that $\frac{5 y + 2}{2} < 0$

$B : y < - \frac{2}{5}$

Now try to solve the given equation again, but this time the absolute value operation will return $\left(- 1\right)$ times of the input since the input is assumed negative.

$\left(- 1\right) \cdot \frac{5 y + 2}{2} = 6$

$\frac{5 y + 2}{2} = - 6$

$5 y + 2 = - 12$

$5 y = - 14$

$y = - \frac{14}{5}$

$y = - \frac{14}{5}$ satisfies the inequality in the expression $B$ so $- \frac{14}{5}$ is also in our solution set.

Result: $\left\{- \frac{14}{5} , 2\right\}$