Answer
$x\gt5$
Work Step by Step
The given inequality, $
1+2x\lt3(2+x)\lt1+4x
,$ is equivalent to
\begin{array}{l}\require{cancel}
1+2x\lt3(2+x)
\text{ and }
3(2+x)\lt1+4x
.\end{array}
Solving the inequalities separately results to
\begin{array}{l}\require{cancel}
1+2x\lt3(2+x)
\\
1+2x\lt6+3x
\\
2x-3x\lt6-1
\\
-x\lt5
\\
x\gt-5
\\\\\text{ and } \\\\
3(2+x)\lt1+4x
\\
6+3x\lt1+4x
\\
3x-4x\lt1-6
\\
-x\lt-5
\\
x\gt5
.\end{array}
Since "and" is used, then the solution set is the intersection of the two inequalities. Hence, the solution set is $
x\gt5
.$
