College Algebra, Chapter 1, 1.6, Section 1.6, Problem 38

Solve the nonlinear inequality $\displaystyle x(2 - 3x) \leq 0$. Express the solution using interval notation and graph the solution set.

We have

$\begin{array}{ccccc}
x \leq 0 & \text{ and } & 2 - 3x & \leq & 0 \\
& & 2 & \leq & 3x \\
& & \frac{2}{3} & \leq & x
\end{array} $

The factors on the left hand side are $x$ and $2 - 3x$. These factors are zero when $x$ is 0 and $\displaystyle \frac{2}{3}$, respectively. The numbers and $\displaystyle \frac{2}{3}$ divide the real line into three intervals.

$\displaystyle (- \infty, 0], \left(0, \frac{2}{3} \right), \left[ \frac{2}{3}, \infty \right)$







From the diagram, the solution of the inequality $x(2 - 3 x) \leq 0$ are