Q. 10 - Precalculus Enhanced with Graphing Utilities

Question

In problems 7-22, solve the inequality.   

$x^2+8x>0$

Step-by-Step Solution

Answer

The solution set of the given inequality is $(-\infty ,-8) or (0,\infty )$.

Step 1. Given Information

The given inequality is $x^2+8x>0$.

Step 2. Find the intercepts and vertex

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

$x(x+8)=0$

• Equate the factors to 0.

$\begin{array}{rcl}x& =& 0\\ x+8& =& 0\\ x& =& -8\end{array}$

• So, the x-intercepts are $(0,0),(-8,0)$.
• The vertex of the parabola exists at $\frac{-b}{2a}=\frac{-8}{2\left(1\right)}=-4$
• Find the value of function at $x=-4$.

$\begin{array}{rcl}f(-4)& =& {(-4)}^2+8(-4)\\ & =& 16-32\\ & =& -16\end{array}$

• So, the vertex is $(-4,-16)$.

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 $(-\infty ,-8) or (0,\infty )$, $y>0$.
• So, the solution set of the given inequality is $(-\infty ,-8) or (0,\infty )$.