#### Answer

The solution of the given expression is $41$.

#### Work Step by Step

We have to multiply both sides by $20$ to get the simplified form:
$\begin{align}
& 20\cdot \left( \frac{x+3}{4}-\frac{x+1}{10} \right)=20\cdot \left( \frac{x-2}{5}-1 \right) \\
& 20\cdot \frac{x+3}{4}-20\cdot \frac{x+1}{10}=20\cdot \frac{x-2}{5}-20\cdot 1 \\
& 5\left( x+3 \right)-2\left( x+1 \right)=4\left( x-2 \right)-20 \\
& 5x+15-2x-2=4x-8-20
\end{align}$
And combine the terms on both sides to get the equation:
$3x+13=4x-28$
And subtract $4x+13$ from both sides to get the value of x,
$\begin{align}
& 3x+13-\left( 4x+13 \right)=4x-28-\left( 4x+13 \right) \\
& 3x-4x=-28-13 \\
& -x=-41
\end{align}$
And divide both sides by -1 to get:
$\begin{align}
& \frac{-x}{-1}=\frac{-41}{-1} \\
& x=41
\end{align}$
Then, to check whether the solution is correct or not, put $x=41$ in the equation:
$\begin{align}
& \frac{41+3}{4}-\frac{41+1}{10}=\frac{41-2}{5}-1 \\
& \frac{44}{4}-\frac{42}{10}=\frac{39}{5}-1
\end{align}$
And multiply both sides by 20 to get:
$\begin{align}
& 20\left( \frac{44}{4}-\frac{42}{10} \right)=20\left( \frac{39}{5}-1 \right) \\
& 5\cdot 44-2\cdot 42=4\cdot 39-20 \\
& 220-84=156-20 \\
& 136=136
\end{align}$
Here, the left-hand side is equal to the right-hand side of the equation. When we have the left-hand side equal to the right-hand side of the equation, then the equation is satisfied.
Hence, the solution of the given expression is $41$.