Answer
$a\le-\dfrac{13}{2} \text{ or } a\ge\dfrac{3}{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Solve the given inequality, $
|2a+5|+1\ge9
,$ by isolating first the absolute value expression. Then use the definition of a greater than (greater than or equal to) absolute value inequality. 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, the given expression is equivalent to
\begin{array}{l}\require{cancel}
|2a+5|+1\ge9
\\\\
|2a+5|\ge9-1
\\\\
|2a+5|\ge8
.\end{array}
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the given inequality is equivalent to
\begin{array}{l}\require{cancel}
2a+5\ge8
\\\\\text{OR}\\\\
2a+5\le-8
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
2a+5\ge8
\\\\
2a\ge8-5
\\\\
2a\ge3
\\\\
a\ge\dfrac{3}{2}
\\\\\text{OR}\\\\
2a+5\le-8
\\\\
2a\le-8-5
\\\\
2a\le-13
\\\\
a\le-\dfrac{13}{2}
.\end{array}
Hence, the solution set is $
a\le-\dfrac{13}{2} \text{ or } a\ge\dfrac{3}{2}
.$
