Question

In problems 7-22, solve the inequality.

$x^2+3x-10>0$

Short answer

The solution set of the inequality is$(-\infty ,-5)or(2,\infty )$.

Step-by-step solution

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 $(-5,0),(2,0)$.
• 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 $(-\infty ,-5)or(2,\infty )$, $y>0$.
• So, the solution set of the given inequality is $(-\infty ,-5)or(2,\infty )$.