Answer
$t\le \dfrac{5}{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
t-2\le|t-3|
,$ use the definition of a greater than (greater than or equal to) absolute value equality.
$\bf{\text{Solution Details:}}$
Reading the original expression from right to left, the given expression is equivalent to
\begin{array}{l}\require{cancel}
|t-3|\ge t-2
.\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 inequality above is equivalent to
\begin{array}{l}\require{cancel}
t-3\ge t-2
\\\\\text{OR}\\\\
t-3\le -(t-2)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
t-3\ge t-2
\\\\
t-t\ge -2+3
\\\\
0\ge 1
\text{ (FALSE)}
\\\\\text{OR}\\\\
t-3\le -(t-2)
\\\\
t-3\le -t+2
\\\\
t+t\le 2+3
\\\\
2t\le 5
\\\\
t\le \dfrac{5}{2}
.\end{array}
Hence, $
t\le \dfrac{5}{2}
.$
