# How do you solve (x+4)/x>0?

Jul 21, 2016

$x \in \left(- \infty , - 4\right) \cup \left(0 , \infty\right)$

#### Explanation:

If $x = 0$ then the denominator is $0$ so the quotient is undefined. So $x = 0$ is not part of the solution set.

That leaves two cases:

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Case $\boldsymbol{x > 0}$

Multiply both sides of the inequality by $x$ to get:

$x + 4 > 0$

Subtract $4$ from both sides to get:

$x > - 4$

Since $x > 0$, this holds already, so the whole of $\left(0 , \infty\right)$ is part of the solution set.

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Case $\boldsymbol{x < 0}$

Multiply both sides of the inequality by $0$ and reverse the inequality (since the multiplier is negative) to get:

$x + 4 < 0$

Subtract $4$ from both sides to get:

$x < - 4$

So $\left(- \infty , - 4\right)$ is part of the solution set.

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Conclusion

The solution set is the union of these two cases, namely:

$\left(- \infty , - 4\right) \cup \left(0 , \infty\right)$