Answer
$t=\left\{ 4,10 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|t-7|+1=4
,$ isolate first the absolute value expression. Then use the definition of an absolute value equality. Finally, use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using the proeprties of equality to isolate the absolute value expression, the given is equivalent to
\begin{array}{l}\require{cancel}
|t-7|+1=4
\\\\
|t-7|=4-1
\\\\
|t-7|=3
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
t-7=3
\\\\\text{OR}\\\\
t-7=-3
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
t-7=3
\\\\
t=3+7
\\\\
t=10
\\\\\text{OR}\\\\
t-7=-3
\\\\
t=-3+7
\\\\
t=4
.\end{array}
Hence, $
t=\left\{ 4,10 \right\}
.$
