Intermediate Algebra, Chapter 2, 2.7 summary exercises, Section 2.7, Problem 12

Evaluate the inequality $|5x - 8| + 9 \geq 7$. Then give the solution in interval notation.

$
\begin{equation}
\begin{aligned}
|5x - 8| + 9 &\geq 0\\
\\
|5x - 8| &\geq -2 && \text{Subtract 9 on each side}
\end{aligned}
\end{equation}
$


By using the property of Absolute value, we have

$
\begin{equation}
\begin{aligned}
5x - 8 &\geq -2 && \text{or} & 5x - 8 &\leq - (-2)\\
\\
5x - 8 &\geq -2 && \text{or} & 5x - 8 &\leq 2\\
\\
5x &\geq 6 && \text{or} & 5x &\leq 10 && \text{Add 8 on each side}\\
\\
x &\geq \frac{6}{5} && \text{or} & x &\leq 2 && \text{Divide each side by $5$ and solve for $x$.}
\end{aligned}
\end{equation}
$



Since the inequalities are joined with $or$, find the union of two union of the two solution. The union is shown and is
written as $(-\infty,\infty)$