Answer
$\text{the interval }
\left( -\infty,\infty \right)
$
Work Step by Step
Using the properties of inequality, the solution to the given inequality, $
6x+1\gt5x+4 \text{ or } 1-x\gt-4
,$ is
\begin{array}{l}\require{cancel}
6x+1\gt5x+4
\\\\
6x-5x \gt 4-1
\\\\
x \gt 3
\\\\\text{ OR }\\\\
1-x\gt-4
\\\\
-x\gt-4-1
\\\\
-x\gt-5
\\\\
x\lt5
.\end{array}
Combining the two inequalities, in interval notation, the solution is $
\text{the interval }
\left( -\infty,\infty \right)
.$