College Algebra, Chapter 2, 2.3, Section 2.3, Problem 60

Find the solutions of the inequality $0.5x^2 + 0.875 x \leq 0.25$ by drawing appropriate graphs. State each answer correct to two decimal places.


Based from the graph, the solutions are approximately
$\displaystyle -2 \leq x \leq \frac{1}{4}$

By using algebra,

$\begin{equation} \begin{aligned} 0.5 x^2 + 0.875x &\leq 0.25 \\ \\ 0.5x^2 + 0.875 x - 0.25 &\leq 0 && \text{Subtract } 0.25\\ \\ x^2 + \frac{7}{4}x - \frac{1}{2} &\leq 0 && \text{Divide both sides by } 0.5\\ \\ (x+2) \left( x - \frac{1}{4} \right) &\leq 0 && \text{Factor out} \end{aligned} \end{equation}$


The factors on the left hand side are $x+2$ and $x - \frac{1}{4}$. These factors are zero when $x$ is -2 and $\frac{1}{4}$, respectively. These factors divide the number line into intervals.
$(-\infty, -2]\left[ -2, \frac{1}{4} \right]\left[ \frac{1}{4}, \infty \right)$
Let's test the numbers at the intervals,



Thus, the solution set is...
$\displaystyle -2 \leq x \leq \frac{1}{4}$