College Algebra, Chapter 1, 1.7, Section 1.7, Problem 42

Solve the inequality $7|x+2| + 5 > 4$. Express the answer using interval notation.

$\begin{equation} \begin{aligned} 7|x+2| + 5 &> 4\\ \\ 7|x+5| &> -1 && \text{Subtract 5}\\ \\ |x+5| &> - \frac{1}{7} && \text{Divide by 7} \end{aligned} \end{equation}$



We have,


$\begin{equation} \begin{aligned} x+5 &> -\frac{1}{7} && \text{and}& -(x+5) &> - \frac{1}{7} && \text{Divide each side by -1}\\ \\ x+5 &> -\frac{1}{7} && \text{and}& x+5 &< \frac{1}{7} && \text{Subtract 5}\\ \\ x &> \frac{34}{7} && \text{and}& x &< -\frac{34}{7} \end{aligned} \end{equation}$


The solution set is $\displaystyle \left( -\infty, -\frac{34}{7}\right] \bigcup \left[ \frac{34}{7}, \infty \right)$