# How do you solve (2x)/(x+5)<=0?

Dec 16, 2016

$- 5 < x \le 0$

#### Explanation:

$\textcolor{red}{\text{There is a trap in this question in that there are excluded}}$$\textcolor{red}{\text{values for } x}$

$\textcolor{b l u e}{\text{Step 1}}$
Solve disregarding the excluded values.

Multiply both sides by $\left(x + 5\right)$

$2 x \le 0 \times \left(x + 5\right)$

$2 x \le 0$

$\textcolor{b l u e}{\text{Condition 1: } x \le 0}$

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$\textcolor{b l u e}{\text{Step 2}}$
An equation or expression becomes 'undefined' if you have division by 0.

$\implies \left(x + 5\right) \ne 0$

$\textcolor{b l u e}{\text{Condition 2: } x \ne - 5}$

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$\textcolor{b l u e}{\text{Step 3}}$

Can we have x< -5color(white)(.)?

Suppose $x = - 6$ then we have:

$\frac{2 \left(- 6\right)}{- 6 + 5} = \frac{- 12}{-} 1 = + 12$

This is not an excluded value but does not satisfy $\frac{2 x}{x + 5} \le 0$
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$\textcolor{b l u e}{\text{Putting it all together using proper notation}}$

$- 5 < x \le 0$