Answer
$(-\infty,3]\cup[5,\infty)$.
The graph of the solution set is shown below.
Work Step by Step
The given compound inequality is
$x-4\geq1$ or $-3x+1\geq-5-x$.
Solve each inequality separately.
First $x-4\geq1$.
Add $4$ to both sides.
$\Rightarrow x-4+4\geq1+4$
Simplify.
$\Rightarrow x\geq5$
Second $-3x+1\geq-5-x$.
Add $x-1$ to both sides.
$\Rightarrow -3x+1+x-1\geq-5-x+x-1$
Simplify.
$\Rightarrow -2x\geq-6$
Divide both sides $-2$ and change the sense of inequality.
$\Rightarrow \frac{-2x}{-2}\leq\frac{-6}{-2}$
Simplify.
$\Rightarrow x\leq3$
First graph then take the union of the two inequalities.
We can write the compound inequality.
$x\geq5$ as $[5,\infty)$ or $x\leq3$ as $(-\infty,3]$
The union is
$(-\infty,3]\cup[5,\infty)$.
The combined graph is shown below.
