Answer
Solution set: $\left(-\infty,\frac{1}{2}\right)$
Work Step by Step
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.
$2x^3-x^2 \lt9-18x$,
$2x^3-x^2+18x-9 \lt 0$,
$2x^3-x^2+18x-9 \lt 0$,
$x^2(2x-1)+9(2x-1) \lt 0$,
$(x^2+9)(2x-1) \lt 0$,
$f(x)=(x^2+9)(2x-1)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x^2+9)(2x-1)=0$
$x=\frac{1}{2}$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \lt 0 ? \\
& &(a^2+9)(2a-1)& \\
(-\infty,\frac{1}{2}) & 0 & (+)(-) & T\\
(\frac{1}{2},\infty) & 4 & (+)(+) & F
\end{array}$
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: $\left(-\infty,\frac{1}{2}\right)$
