Answer
$\color{blue}{(-3.5, -1)}$
Work Step by Step
Solve the equation $\dfrac{5}{x+1}=-2$ to obtain:
\begin{array}{ccc}
\require{cancel}
&\dfrac{5}{x+1}&=&-2
\\&(x+1)\cdot \dfrac{5}{x+1}&=&-2(x+1)
\\&\cancel{(x+1)}\cdot \dfrac{5}{\cancel{x+1}}&=&-2x-2
\\&5&=&-2x-2
\\&5+2&=&-2x-2+2
\\&7&=&-2x
\\&\frac{7}{-2} &= &\frac{-2x}{-2}
\\&-\frac{7}{2} &= &x
\\&-3.5&=&x
\end{array}
Find the number that will make the $\dfrac{5}{x+1}$ undefined.
Note that when $x=-1$ the denominator becomes $0$, making the expression undefined..
Thus, the numebrs $-3.5$ and $-1$ are the critical points of the given inequality.
These numbers divide the number line into three parts:
$(-\infty, -3.5)$, $(-3.5, -1)$, and $(-1, +\infty)$
Pick a test point from each interval/part, and test if it satisfies the given inequality.
For $(-\infty, -3.5)$, use the test point $-4$ and substitute it into the given inequality:
$\dfrac{5}{x+1}\lt -2
\\\dfrac{5}{-4+1} \lt -2
\\\dfrac{5}{-3} \lt -2
\\-1.\overline{6} \not \lt -2$
Thus, the numbers in this interval/part are not solutions to the given inequality.
For $(-3.5, -1)$, use the test point $-2$ and substitute it into the given inequality:
$\dfrac{5}{x+1}\lt -2
\\\dfrac{5}{-2+1}\lt -2
\\\dfrac{5}{-1} \lt -2
\\-5 \lt -2$
This statement is true therefore the numbers within this interval are solutions to the given inequality.
For $(-1, +\infty)$, use the test point $0$ and substitute it into the given inequality:
$\dfrac{5}{x+1} \lt -2
\\\dfrac{5}{0+1} \lt -2
\\\dfrac{5}{1}\lt -2
\\5 \not \lt -2$
Thus, the numbers in this interval are not solutions to the given inequality.
$-3.5$ is not a solution since the inequality involves strictly less than.
$-1$ is not a solution because it makes an expression undefined.
Therefore, the solution to the given inequality is: $\color{blue}{(-3.5, -1)}$.
