Answer
$$[0,\frac{5}{3}]$$
Work Step by Step
Let $3x^2 - 5x \leq 0$ be a function $f$.
Find the $x$-intercepts by solving $3x^2 - 5x=0$
Factor: $3x^2 - 5x = x(3x-5)$
$x=0$ or $3x-5=0$
$x = 0$ or $x = \frac{5}{3}$
These $x$-intercepts serve as boundary points that separate the number line into intervals.
The boundary points of this equation therefore divide the number line into three intervals:
$(-∞, \frac{5}{3})(\frac{5}{3},0)(0, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, 0)$
Test Value = $-1$
$f=x(3x-5)$
$f=(-1)(3(-1)-5)$
$f=8$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, 0)$.
$(0,\frac{5}{3})$
Test Value = $1$
$f=x(3x-5)$
$f=(1)(3(1)-5)$
$f=-2$
Conclusion: $f (x) < 0$ for all $x$ in $(\frac{5}{3},0)$.
$(\frac{5}{3},+∞)$
Test Value = $2$
$f=x(3x-5)$
$f=(2)(3(2)-5)$
$f=2$
Conclusion: $f (x) > 0$ for all $x$ in $(0,+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x) \leq 0$, where $f (x) = 3x^2-5x$. Based on the solution above, $f(x)\lt0$ for all $x$ in $(\frac{5}{3},0)$. However, because the inequality involves $\leq$ (less than or equal to), we must also include the solutions of $3x^2-5x = 0$, namely $0$ and $\frac{5}{3}$, in the solution set.
Thus, the solution set of the given inequality is:
$$[0,\frac{5}{3}]$$
