Question

Solve the non-linear inequality. Express the solution using interval notation and graph the solution set.

$x^5>x^2$

Short answer

The solution set of the given inequality in interval notation is $\left(1,\infty \right)$.

The graph of the solution is shown below:

Step-by-step solution

Step 1. Solve the inequality by moving all terms to one side.

$\begin{array}{c}{x}^5>x^2\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Giveninequality}\\ x^5-x^2>0\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Subtract}{x}^2\\ x^2\left(x^3-1\right)>0\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Factorout}{x}^2\end{array}$

Step 2. Find the intervals.

The factors of the left hand side are$\left(x^3-1\right)$ and $x^2$.

Notice that, these factors will be zero when$x=1,$ and $x=0$.

So, using these values divide the number line into sub intervals as follows:

$\left(-\infty ,0\right),\left(0,1\right),\left(1,\infty \right)$.

Step 3. Make a diagram.

Using the test points determine the sign of each factor in the interval, this can be shown in the following diagram:

Step 4. Conclusion.

Here, the inequality is $x^2\left(x^3-1\right)>0$.

From the above diagram, one can notice that the inequality is positive in the interval $\left(1,\infty \right)$.

Also, the inequality is $>$. This means, the endpoints does not satisfy the inequality.

So, the solution set of the given inequality is $\left(1,\infty \right)$.

Step 5. Graphical Interpretation:

The graph of the solution set is shown below: