Intermediate Algebra, Chapter 2, 2.7 summary exercises, Section 2.7, Problem 40

Evaluate the inequality $\displaystyle \frac{1}{2} \leq \frac{2}{3}x \leq \frac{5}{4}$. Then give the solution in interval notation.

By using the property of Absolute value, we have

$
\begin{equation}
\begin{aligned}
5 - x &< 4 && \text{and} & 5 - x &> -4\\
\\
-x &< -4 && \text{and} & -x &> - 9
&& \text{Subtract each side by } 5\\
\\
x &> 1 &&& x &< 9 && \text{Divide each side by $-1$}
\end{aligned}
\\
\text{Remember that if you divide or multiply negative numbers, the inequality symbol reverses}
\end{equation}
$

Since the inequalities are joined with $and$, find the intersection of the two solution.
The intersection is shown and is written as $(1,9)$