College Algebra, Chapter 1, 1.7, Section 1.7, Problem 44

Solve the inequality $\displaystyle 2\left| \frac{1}{2}x + 3 \right| +3 \leq 51$. Express the answer using interval notation.

$\begin{equation} \begin{aligned} 2\left| \frac{1}{2}x + 3 \right| +3 &\leq 51\\ \\ 2\left| \frac{1}{2}x + 3 \right| &\leq 48 && \text{Subtract 3}\\ \\ \left| \frac{1}{2}x + 3 \right| &\leq 24 && \text{Divide by 2} \end{aligned} \end{equation}$



We have,


$\begin{equation} \begin{aligned} \frac{1}{2}x + 3 &\leq 24 && \text{and}& -\left( \frac{1}{2}x + 3 \right) &\leq 24 && \text{Divide each side by -1}\\ \\ \frac{1}{2}x + 3 &\leq 24 && \text{and}& \frac{1}{2}x + 3 &\geq -24 && \text{Subtract 3}\\ \\ \frac{1}{2}x &\leq 21 && \text{and}& \frac{1}{2}x &\geq - 27 && \text{Multiply by 2}\\ \\ x &\leq 42 && \text{and}& x &\geq -54 \end{aligned} \end{equation}$


The solution set is $[-54,42]$