#### Answer

$t\le6 \text{ or } t\ge8$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inqeuality, $
|t-7|+3\ge4
,$ isolate first the absolute value expression. Then 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 $\ge.$
$\bf{\text{Solution Details:}}$
Using the properties of inequality to isolate the absolute value expression results to
\begin{array}{l}\require{cancel}
|t-7|+3\ge4
\\\\
|t-7|\ge4-3
\\\\
|t-7|\ge1
.\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-7\ge1
\\\\\text{OR}\\\\
t-7\le-1
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
t-7\ge1
\\\\
t\ge1+7
\\\\
t\ge8
\\\\\text{OR}\\\\
t-7\le-1
\\\\
t\le-1+7
\\\\
t\le6
.\end{array}
Hence, the solution set is $
t\le6 \text{ or } t\ge8
.$