Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 78

Solve the inequality $|0.5x - 3.5| + 0.2 \geq 0.6$.


$\begin{equation} \begin{aligned} |0.5x - 3.5| + 0.2 \geq & 0.6 \\ |0.5x - 3.5| \geq & 0.4 \end{aligned} \end{equation}$


The absolute value inequality is rewritten as

$0.5x - 3.5 \geq 0.4$ or $0.5x - 3.5 \leq -0.4$,

because $0.5x - 3.5$ must represent a number that is more than $0.4$ units from on either side of the number line. We can solve the compound inequality.


$\begin{equation} \begin{aligned} 0.5x - 3.5 \geq & 0.4 && \text{or} &&& 0.5x - 3.5 \leq & -0.4 && \\ 5x - 35 \geq & 4 && \text{or} &&& 5x - 35 \leq & -4 && \text{Multiply each side by } 10 \\ 5x \geq & 39 && \text{or} &&& 5x \leq & 31 && \text{Add each side by } 35 \\ x \geq & \frac{39}{5} && \text{or} &&& x \leq & \frac{31}{5} && \end{aligned} \end{equation}$


The solution set is $\displaystyle \left( \frac{31}{5}, \frac{39}{5} \right)$.