Answer
Solution set: $ [0,\displaystyle \frac{5}{3}]$
Work Step by Step
Follow the "Procedure for Solving Polynomial lnequalities",\ p.412:
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$3x^{2}-5x \leq 0$
$f(x)= 3x^{2}-5x \quad $...factor the trinomial...
$f(x)=x(3x-5)$
$x(3x-5) \leq 0$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$x(3x-5) = 0$
$x=0$ or $x=\displaystyle \frac{5}{3}$
3. Locate these boundary points on a number line, thereby dividing the number line into intervals.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & (-\infty, 0) & (0,\frac{5}{3}) & (\frac{5}{3},\infty)\\
a=test.val. & -1 & 1 & 2\\
f(a) & (-1)(-8) & (1)(-2) & (2)(1)\\
f(a) \leq 0 ? & F & T & F
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $ [0,\displaystyle \frac{5}{3}]$