Answer
Solution set: $ [0,1]$
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.
$-x^{2}+x \geq 0\qquad /\times(-1)$ ... the sign turns ...
$x^{2}-x \leq 0 \quad $...factor the trinomial...
$x(x-1) \leq 0$
$f(x)= x(x-1) \leq 0$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$x(x-1) = 0$
$x=0$ or $x=1$
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,
$\begin{array}{llll}
Intervals: & (-\infty, 0) & (0,1) & (1,\infty)\\
a=test.val. & -1 & 0.5 & 2\\
f(a) & (-1)(-2) & (0.5)(-0.5) & (2)(1)\\
f(a) \leq 0 ? & F & T & F
\end{array}$
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: $ [0,1]$