# How do you solve  abs(2 - 5x )>= 2 / 3?

Aug 19, 2017

$x \in \left(- \infty , \frac{4}{15}\right] \cup \left[\frac{8}{15} , + \infty\right)$

#### Explanation:

You're dealing with an absolute value inequality, so right from the start, you should know that you must look at two possible cases

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• $2 - 5 x \ge 0 \implies | 2 - 5 x | = 2 - 5 x$

In this case, you have

$2 - 5 x \ge \frac{2}{3}$

This will get you

$- 5 x \ge \frac{2}{3} - 2$

$- 5 x \ge - \frac{4}{3} \implies x \le \frac{4}{15}$

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• $2 - 5 x < 0 \implies | 2 - 5 x | = - \left(2 - 5 x\right)$

In this case, you have

$- 2 + 5 x \ge \frac{2}{3}$

This will get you

$5 x \ge \frac{2}{3} + 2$

$5 x \ge \frac{8}{3} \implies x \ge \frac{8}{15}$

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You can thus say that the solution interval for the original inequality will be

$x \in \left(- \infty , \frac{4}{15}\right] \cup \left[\frac{8}{15} , + \infty\right)$

This tells you that the original inequality is satisfied if $x$ is less than or equal to $\frac{4}{15}$ or greater than or equal to $\frac{8}{15}$. In other words, any value of

$x \in \left(\frac{4}{15} , \frac{8}{15}\right)$

is not a solution to the original inequality.