College Algebra, Chapter 1, 1.6, Section 1.6, Problem 74

Solve the nonlinear inequality $\displaystyle x^5 > x^2 $. Express the solution using interval notation and graph the solution set.

$
\begin{equation}
\begin{aligned}
x^5 & > x^2 \\
\\
x^5 - x^2 & > 0 && \text{Subtract } x^2\\
\\
x^2(x^3-1) & > 0 && \text{Factor } x^2\\
\\
x^2 ( x- 1)(x^2 + x+1) & > 0 && \text{Difference of cubes}
\end{aligned}
\end{equation}
$


The factors on the left hand side are $x^2$ and $x-1$. These factors are zero when $x$ is 0 and 1 respectively. These numbers divide the real line into intervals
$(-\infty, 0), (0,1),(1,\infty)$




From the diagram, the solution of the inequality $\displaystyle x^5(x-1)(x^2+x+1) > 0 $ are
$(1, \infty)$

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