Answer
See the explanation
Work Step by Step
a. A polynomial inequality is solved following the steps:
Step $1$
Move All Terms to One Side. Rewrite the inequality so that all nonzero terms appear on one side of the inequality symbol.
Step $2$
Factor the polynomial into irreducible factors, and find the real zeros of the polynomial.
Step $3$
Find the Intervals. List the intervals determined by the real zeros.
Step $4$
Make a Table or Diagram. Use test values to make a table or diagram of the signs of each factor in each interval. In the last row of the table determine the sign of the polynomial on that interval.
Step $5$
Solve. Determine the solutions of the inequality from the last row of the table. Check whether the endpoints of these intervals satisfy the inequality.
(This may happen if the inequality involves $\leq$ or $\geq$.)
b. Let $r(x)=\frac{P(x)}{Q(x)}$ be a rational function. The cut points of $r$ are the values of $x$ at which either $P(x)=0$ or $Q(x)=0$.
A rational inequality is solved like a polynomial inequality in which the intervals are determined using both zeros of the numerator and denominator.
c. 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.
$x^2-9 \leq 8x$,
$x^2-8x-9 \leq 0$,
$x^2+x-9x-9 \leq 0$,
$x(x+1)-9(x+1) \leq 0$,
$(x-9)(x+1) \leq 0$.
$f(x)=(x-9)(x+1)$
Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x-9)(x+1)=0$
$x=9$ or $x=-1$
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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \leq 0 ? \\
& & (a-9)(a+1) & \\
(-\infty,-1) & -5 & (-)(-) & F\\
(-1,9) & 0 & (-)(+) & T\\
(9,\infty) & 10 & (+)(+) & F
\end{array}$
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: $[-1, 9]$
