Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set.

$\frac{x}{x+1}>3$

Short answer

The solution set is $\left(-\frac{3}{2},-1\right)$

Step-by-step solution

Step 1. Finding the interval.

Given inequality is

$\begin{array}{c}\frac{x}{x+1}>3\\ \frac{x}{x+1}-3>0\\ \frac{x-3x-3}{x+1}>0\\ \frac{-2x-3}{x+1}>0\\ \frac{(2x+3)(x+1)}{{(x+1)}^2}<0\\ (2x+3)(x+1)<0\end{array}$

Factors on the left-hand side are $(2x+3),(x+1)$. These factors are zero

When $x=\left(-\frac{3}{2},-1\right)$

So, numbers $\left(-\frac{3}{2},-1\right)$divide the real line into three intervals $\left(-\infty ,-\frac{3}{2}\right),\left(-\frac{3}{2},-1\right),(-1,\infty )$

Step 2. Determining the sign of each factor.

To determine the sign of each factor on each of the intervals that we found, we use test values. We choose a number inside each intervals and check the sign of each factors.

Step 3. Making the table.

So, the solution set is $\left(-\frac{3}{2},-1\right)$

Step 4. Plotting the graph.