Q. 9 - Precalculus Enhanced with Graphing Utilities

Question

In problems 7-22, solve the inequality.  

$x^2-4x>0$

Step-by-Step Solution

Answer

The solution set of the inequality is $(-\infty ,0) or (4,\infty )$.

Step 1. Given Information

The given inequality is $x^2-4x>0$.

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 .

$x(x-4)=0$

• Equate the factors to 0.

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

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

$\begin{array}{rcl}f\left(2\right)& =& 2^2-4\left(2\right)\\ & =& 4-8\\ & =& -4\end{array}$

• So, the vertex is $(2,-4)$.

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 $(-\infty ,0) or (4,\infty )$, $y>0$.
• So, the solution set of the given inequality is $(-\infty ,0) or (4,\infty )$.