# How do you solve abs(6 - 5y )>2?

Apr 1, 2018

The solution is $y \in \left(- \infty , \frac{4}{5}\right) \cup \left(\frac{8}{5} , + \infty\right)$

#### Explanation:

There are $2$ solutions for absolute values.

$| 6 - 5 y | > 2$

$6 - 5 y > 2$ and $- 6 + 5 y > 2$

$5 y < 4$ and $5 y > 8$

$y < \frac{4}{5}$ and $y > \frac{8}{5}$

The solutions are

${S}_{1} = y \in \left(- \infty , \frac{4}{5}\right)$

${S}_{2} = y \in \left(\frac{8}{5} , + \infty\right)$

Therefore,

$S = {S}_{1} \cup {S}_{2}$

$= y \in \left(- \infty , \frac{4}{5}\right) \cup \left(\frac{8}{5} , + \infty\right)$

graph{(y-|6-5x|)(y-2)=0 [-5.55, 6.934, -0.245, 5.995]}

Apr 1, 2018

$\left(- \infty , \frac{4}{5}\right) \cup \left(\frac{8}{5} , + \infty\right)$

#### Explanation:

$\text{inequalities of the form } | x | > a$

$\text{always have solutions of the form}$

$x < - a \text{ or } x > a$

$\text{thus we have to solve 2 inequalities}$

$6 - 5 y < - 2 \text{ or } 6 - 5 y > 2$

$\textcolor{b l u e}{\text{first solution}}$

$6 - 5 y < - 2 \leftarrow \text{subtract 6 from both sides}$

$\Rightarrow - 5 y < - 8 \leftarrow \text{divide both sides by } - 5$

$\Rightarrow y > \frac{8}{5} \leftarrow \textcolor{b l u e}{\text{reverse direction of sign}}$

$\textcolor{b l u e}{\text{second solution}}$

$6 - 5 y > 2 \leftarrow \text{subtract 6 from both sides}$

$\Rightarrow - 5 y > - 4 \leftarrow \text{divide both sides by } - 5$

$y < \frac{4}{5} \leftarrow \textcolor{b l u e}{\text{reverse direction of sign}}$

$\text{combining the solutions gives}$

$\left(- \infty , \frac{4}{5}\right) \cup \left(\frac{8}{5} , + \infty\right)$