College Algebra, Chapter 1, 1.6, Section 1.6, Problem 70

Solve the nonlinear inequality $\displaystyle \frac{1}{x+1} + \frac{1}{x+2} \leq 0 $. Express the solution using interval notation and graph the solution set.

$
\begin{equation}
\begin{aligned}
\frac{1}{x+1} + \frac{1}{x+2} & \leq 0 \\
\\
\frac{(x+2)+(x+1)}{(x+1)(x+2)} & \leq 0 && \text{Multiply the LCD } (x+1)(x+2)\\
\\
\frac{2x+3}{(x+1)(x+2)} & \leq 0 && \text{Simplify the numerator}
\end{aligned}
\end{equation}
$


The factors on the left hand side are $2x+3$, $x+1$ and $x+2$. These factors are zero when $x$ is $\displaystyle \frac{-3}{2}$, -1 and -2 respectively. These numbers divide the real line into intervals
$\displaystyle (-\infty, -2), \left( -2, - \frac{3}{2} \right], \left[ -\frac{3}{2}, -1 \right), (-1, \infty)$




From the diagram, the solution of the inequality $\displaystyle \frac{2x+3}{(x+1)(x+2)} \leq 0$ are
$(-\infty,-2) \bigcup \left[ -\frac{3}{2},-1 \right)$