College Algebra, Chapter 1, 1.6, Section 1.6, Problem 58

Solve the nonlinear inequality $\displaystyle \frac{2x+6}{x-2} < 0$. Express the solution using interval notation and graph the solution set.

$
\begin{equation}
\begin{aligned}
\frac{2x+6}{x-2} & < 0\\
\\
\frac{2(x+3)}{(x-2)} & < 0 && \text{Factor out } 2\\
\\
\frac{x+3}{x-2} & < 0 && \text{Divide by } 2
\end{aligned}
\end{equation}
$


The factors on the left hand side are $x+3$ and $x-2$. These factors are zero when $x$ is -3 and 2 respectively. These numbers divide the real line into intervals
$(-\infty, -3), (-3,2), (2,\infty)$




From the diagram, the solution of the inequality $\displaystyle \frac{x+3}{x-2} < 0$ are
$(-3,2)$

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