Question

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

$x^3-4x>0$

Short answer

The solution set is$(-2,0)\cup (2,\infty )$

Step-by-step solution

Step 1. Finding the interval.

Given inequality is

$\begin{array}{d}{x}^3-4x>0\\ x(x^2-4)>0\\ x(x-2)(x+2)>0\end{array}$

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

When $x=0,2$, and $-2$.

So, the numbers $0,-2$, and 2 divide the real line into four intervals$(-\infty ,-2)(-2,0),(0,2),(2,\infty )$

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 required solution of or the inequality is$(-2,0)\cup (2,\infty )$

Step 4. Plotting the graph.

Graph of the inequality is