# Calculate: |(3)/(x-2)|=(1)/(2)

## Expression: $|\frac{ 3 }{ x-2 }|=\frac{ 1 }{ 2 }$

Determine the defined range

$\begin{array} { l }|\frac{ 3 }{ x-2 }|=\frac{ 1 }{ 2 },& x≠2\end{array}$

Use $|\frac{ a }{ b }|=\frac{ |a| }{ |b| }$ to transform the expression

$\frac{ |3| }{ |x-2| }=\frac{ 1 }{ 2 }$

The absolute value of any number is always positive

$\frac{ 3 }{ |x-2| }=\frac{ 1 }{ 2 }$

Simplify the equation using cross-multiplication

$6=|x-2|$

Swap the sides of the equation

$|x-2|=6$

Use the absolute value definition to rewrite the absolute value equation as two separate equations

$\begin{array} { l }x-2=6,\\x-2=-6\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=8,\\x-2=-6\end{array}$

Solve the equation for $x$

$\begin{array} { l }\begin{array} { l }x=8,\\x=-4\end{array},& x≠2\end{array}$

Check if the solution is in the defined range

$\begin{array} { l }x=8,\\x=-4\end{array}$

The equation has $2$ solutions

$\begin{array} { l }x_1=-4,& x_2=8\end{array}$

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