Q. 7 - Precalculus Enhanced with Graphing Utilities

Question

In problems 7-22, solve the inequality.

$x^2-3x-10<0$

Step-by-Step Solution

Answer

The solution of the inequality is $(-2,5)$.

Step 1. Given information

The given inequality is $x^2-3x-10<0$.

Step 2. Find the intercepts and vertex

• The value of $f\left(0\right)=-10$.
• So, the y-intercept is $(0,-10)$.
• Factor the equation $f\left(x\right)=0$.
• $\begin{array}{rcl}{x}^2-3x-10& =& 0\\ (x-5)(x+2)& =& 0\end{array}$
• Equate the factors to 0.
• $\begin{array}{rcl}x-5& =& 0\\ x& =& 5\\ x+2& =& 0\\ x& =& -2\end{array}$
• So, the x-intercepts are $(-2,0),(5,0)$.
• The vertex of the parabola exists at $\frac{-b}{2a}=\frac{-(-3)}{2\left(1\right)}=\frac{3}{2}$
• Find the value of function at $x=\frac{3}{2}$.

$\begin{array}{rcl}f\left(\frac{3}{2}\right)& =& {\left(\frac{3}{2}\right)}^2-3\left(\frac{3}{2}\right)-10\\ & =& \frac{9}{4}-\frac{9}{2}-10\\ & =& \frac{9-18-40}{4}\\ & =& \frac{-49}{4}\end{array}$

• So, the vertex is $(\frac{3}{2},\frac{-49}{4})$.

Step 3. Plot the Graph

Plot the parabola on a graph using the vertex and intercepts calculated in the previous step.

Step 4. Find the solution set

• From the graph, it is observed that for x belonging to $(-2,5)$, $y<0$.
• So, the solution set of the given inequality is $(-2,5)$.