Q. 14 - Precalculus Enhanced with Graphing Utilities

Question

In problems 7-22, solve the inequality.   

$x^2+7x<-12$

Step-by-Step Solution

Answer

The solution set is $(-4,-3)$.

Step 1. Given Information

The given inequality is $x^2+7x<-12$.

Step 2. Find the intercepts and vertex

• Rewrite the given inequality.

$x^2+7x+12<0$

• The value of $f\left(0\right)=12$.
• So, the y-intercept is $(0,12)$.
• Factor the equation.

$\begin{array}{rcl}{x}^2+7x+12& =& 0\\ x^2+4x+3x+12& =& 0\\ x(x+4)+3(x+4)& =& 0\\ (x+3)(x+4)& =& 0\end{array}$

• Equate the factors to 0.

$\begin{array}{rcl}x+3& =& 0\\ x& =& -3\\ x+4& =& 0\\ x& =& -4\end{array}$

• So, the x-intercepts are $(-4,0),(-3,0)$.
• The vertex of the parabola exists at $\frac{-b}{2a}=\frac{-7}{2\left(1\right)}=\frac{-7}{2}$.
• Find the value of the function at $x=\frac{-7}{2}$.

$\begin{array}{rcl}f\left(\frac{-7}{2}\right)& =& {\left(\frac{-7}{2}\right)}^2+7\left(\frac{-7}{2}\right)+12\\ & =& \frac{49}{4}-\frac{49}{2}+12\\ & =& \frac{49-98+48}{4}\\ & =& \frac{-1}{4}\end{array}$

• So, the vertex is $(\frac{-7}{2},-\frac{1}{4})$.

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 $(-4,-3)$, $y<0$.
• So, the solution set of the given inequality is $(-4,-3)$.