Answer
The value of x in the equation $\frac{x-1}{5}-\frac{x-3}{2}=1-\frac{x}{4}$ is $-54$.
Work Step by Step
Consider the equation $\frac{x-1}{5}-\frac{x-3}{2}=1-\frac{x}{4}$
Multiplying both sides by 20 in the original equation we get,
$\begin{align}
& 20\left( \frac{x-1}{5}-\frac{x-3}{2} \right)=20\left( 1-\frac{x}{4} \right) \\
& 4\left( x-1 \right)-10\left( x-3 \right)=20-5x
\end{align}$
The fractions are cleared now; open the brackets and combine the x terms and the constant terms to get
$\begin{align}
& 4x-4-10x-30=20-5x \\
& 4x-10x+5x-4-30-20=0 \\
& -x-54=0
\end{align}$
Adding $54$ on both sides of the equation we get,
$\begin{align}
& -x-54+54=54 \\
& -x=54
\end{align}$
Dividing by $-1$ on both sides of the equation we get
$\begin{align}
& \frac{-x}{-1}=\frac{54}{-1} \\
& x=-54
\end{align}$
Hence, $x=-54$
Substituting $x=-54$ in the original solution we get
Left-hand side:
$\begin{align}
& =\frac{-54-1}{5}-\frac{-54+3}{2} \\
& =\frac{-55}{5}-\frac{-51}{2} \\
& =-11+25.5 \\
& =14.5
\end{align}$
Right-hand side:
$\begin{align}
& =1-\frac{-54}{4} \\
& =1-\left( -13.5 \right) \\
& =1+13.5 \\
& =14.5
\end{align}$
Hence, the left-hand side is equal to the right-hand side.
The value of x is true for the original equation.
Hence, the value of x in the equation $\frac{x-1}{5}-\frac{x-3}{2}=1-\frac{x}{4}$ is $-54$.
