Answer
$t \le -\dfrac{13}{5} \text{ or } t \ge \dfrac{7}{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|-5t-3| \ge 10
,$ use the definition of a greater than (greater than or equal to) absolute value inequality and solve each resulting 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 $\ge.$
$\bf{\text{Solution Details:}}$
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 inequality above is equivalent to
\begin{array}{l}\require{cancel}
-5t-3 \ge 10
\\\\\text{OR}\\\\
-5t-3 \le -10
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
-5t-3 \ge 10
\\\\
-5t \ge 10+3
\\\\
-5t \ge 13
\\\\\text{OR}\\\\
-5t-3 \le -10
\\\\
-5t \le -10+3
\\\\
-5t \le -7
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-5t \ge 13
\\\\
t \le \dfrac{13}{-5}
\\\\
t \le -\dfrac{13}{5}
\\\\\text{OR}\\\\
-5t \le -7
\\\\
t \ge \dfrac{-7}{-5}
\\\\
t \ge \dfrac{7}{5}
.\end{array}
Hence, the solution set is $
t \le -\dfrac{13}{5} \text{ or } t \ge \dfrac{7}{5}
.$
