College Algebra, Chapter 2, 2.3, Section 2.3, Problem 4

Below are the graphs of $y = 5x - x^2$ and $y =4$.




a.) Find the solutions of the equation $5x - x^2 = 4$
b.) Find the solutions of the inequality $5x - x^2 > 4$

a.)

$
\begin{equation}
\begin{aligned}
5x - x^2 &=4 && \text{Model}\\
\\
-5x + x^2 &= -4 && \text{Divide both sides by } -1 \\
\\
x^2 - 5x + \frac{25}{4} &= -4 + \frac{25}{4} && \text{Complete the square: Add } \left( \frac{-5}{2} \right)^2 = \frac{25}{4}\\
\\
\left( x - \frac{5}{2} \right)^2 &= \frac{9}{4} && \text{Perfect square}\\
\\
x &= \pm \sqrt{\frac{9}{4}} && \text{Take the square root}\\
\\
x &= \frac{5}{2} \pm \frac{3}{2} && \text{Add } \frac{5}{2}
\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}
x &= \frac{5}{2} + \frac{3}{2} &&\text{and}& x &= \frac{5}{2} - \frac{3}{2} && \text{Solve for } x\\
\\
x &=4 &&\text{and}& x &=1 && \text{Simplify}
\end{aligned}
\end{equation}
$


b.)

$
\begin{equation}
\begin{aligned}
5x - x^2 &> 4 && \text{Model}\\
\\
-5x + x^2 &< -4 && \text{Divide both sides by } -1 \\
\\
x^2 - 5x + \frac{25}{4} &< -4 + \frac{25}{4} && \text{Complete the square: Add } \left( \frac{-5}{2} \right)^2 = \frac{25}{4}\\
\\
\left( x - \frac{5}{2} \right)^2 &< \frac{9}{4} && \text{Perfect square}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}
x - \frac{5}{2} &< \sqrt{\frac{9}{4}} &&\text{and}& x - \frac{5}{2} &> - \sqrt{\frac{9}{4}} && \text{Take the square root}\\
\\
x &< \frac{5}{2} + \sqrt{\frac{9}{4}} &&\text{and}& x &> \frac{5}{2} - \sqrt{\frac{9}{4}} && \text{Add } \frac{5}{2}\\
\\
x &< \frac{5}{2} + \frac{3}{2} &&\text{and}& x &> \frac{5}{2} + \frac{3}{2} && \text{Solve for } x\\
\\
x &< 4 &&\text{and}& x &> 1 && \text{Simplify}
\end{aligned}
\end{equation}
$


The solution is $1 < x < 4$