Answer
Solution set: $(-3,\displaystyle \frac{5}{2})$
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.
$2x^{2}+x<15$
$2x^{2}+x-15<0$
$f(x)= 2x^{2}+x-15$
factor the trinomial...
find factors of $2(-15)=-30$ that add to $+1:$
($6$ and $-5$ )
$2x^{2}+x-15= 2x^{2}+6x-5x-15= \quad$ ... in pairs ...
$=2x(x+3)-5(x+3)=(2x-5)(x+3)$
$f(x)=(2x-5)(x+3)<0$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(2x-5)(x+3)=0$
$x=\displaystyle \frac{5}{2}$ or $x=-3$
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,
$\begin{array}{llll}
Intervals: & (-\infty, -3) & (-3,\frac{5}{2}) & (\frac{5}{2},\infty)\\
a=test.val. & -5 & 0 & 5\\
f(a) & (-15)(-2) & (-5)(3) & (5)(8)\\
f(a) < 0 ? & F & T & F
\end{array}$
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: $(-3,\displaystyle \frac{5}{2})$
