# How do you solve (x+26)/(x+5)>=0?

Aug 6, 2016

$x \in \left(- \infty , - 26\right] \cup \left(- 5 , \infty\right)$

#### Explanation:

If $x = - 5$ then the denominator of the rational expression is zero and the quotient is undefined. So $- 5$ is not part of the solution set.

$\textcolor{w h i t e}{}$
Case $\boldsymbol{x < - 5}$

If $x < - 5$ then $\left(x + 5\right) < 0$

Multiply both sides of the inequality by $\left(x + 5\right)$ and reverse the inequality (since $\left(x + 5\right) < 0$) to get:

$x + 26 \le 0$

Subtract $26$ from both sides to get:

$x \le - 26$

So $\left(- \infty , - 26\right]$ is part of the solution set.

$\textcolor{w h i t e}{}$
Case $\boldsymbol{x > - 5}$

If $x > - 5$ then $\left(x + 5\right) > 0$

Multiply both sides of the inequality by $\left(x + 5\right)$ to get:

$x + 26 \ge 0$

Subtract $26$ from both sides to get:

$x \ge - 26$

Since $x > - 5$ this is already true.

So $\left(- 5 , \infty\right)$ is part of the solution set.

$\textcolor{w h i t e}{}$
Conclusion

The solution set is:

$\left(- \infty , - 26\right] \cup \left(- 5 , \infty\right)$