Answer
$-\dfrac{7}{2} \le a \le 6$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
7+|4a-5|\le26
,$ isolate first the absolute value expression. Then use the definition of a less than absolute value inequality. Use the properties of equality to isolate the variable. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Using the properties of inequality to isolate the absolute value expression, the given is equivalent to
\begin{array}{l}\require{cancel}
7+|4a-5|\le26
\\\\
|4a-5|\le26-7
\\\\
|4a-5|\le19
.\end{array}
Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-19 \le 4a-5 \le19
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-19 \le 4a-5 \le19
\\\\
-19+5 \le 4a-5+5 \le19+5
\\\\
-14 \le 4a \le24
\\\\
-\dfrac{14}{4} \le \dfrac{4a}{4} \le\dfrac{24}{4}
\\\\
-\dfrac{7}{2} \le a \le 6
.\end{array}
Hence, the solution set $
-\dfrac{7}{2} \le a \le 6
.$
