Question

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

$\frac{4}{x}<x$

Short answer

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

The graph of the solution is shown below:

Step-by-step solution

Step 1. Finding the interval.

Given inequality is

$\begin{array}{l}\frac{4}{x}<x\\ \frac{4}{x}-x<0\\ \frac{4-x^2}{x}<0\\ \frac{(2-x)(2+x)x}{x^2}<0\\ x(x-2)(x+2)>0\end{array}$

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

When$x=0,2,-2$

So, the numbers $0,2,-2$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, required solution is$(-2,0)\cup (2,\infty )$

Step 4. Plotting the graph.

Graph or the inequality is