Answer
The solutions are $x=1$ and $x=5$.
Work Step by Step
The given equation is
$\Rightarrow |x-7|=|2x-8|$
Write related linear equations.
$\Rightarrow x-7=2x-8$ or $x-7=-(2x-8)$
Solve the first equation.
$\Rightarrow x-7=2x-8$
Add $8-x$ to each side.
$\Rightarrow x-7+8-x=2x-8+8-x$
Simplify.
$\Rightarrow 1=x$
Solve the second equation.
$\Rightarrow x-7=-(2x-8)$
Use distributive property.
$\Rightarrow x-7=-2x+8$
Add $-8-x$ to each side.
$\Rightarrow x-7-8-x=-2x+8-8-x$
Simplify.
$\Rightarrow -15=-3x$
Divide each side by $-3$.
$\Rightarrow \frac{-15}{-3}=\frac{-3x}{-3}$
Simplify.
$\Rightarrow 5=x$
Check $x=1$
$\Rightarrow |x-7|=|2x-8|$
$\Rightarrow |1-7|=|2(1)-8|$
$\Rightarrow |-6|=|2-8|$
$\Rightarrow |-6|=|-6|$
$\Rightarrow 6=6$
True.
Check $x=5$
$\Rightarrow |x-7|=|2x-8|$
$\Rightarrow |5-7|=|2(5)-8|$
$\Rightarrow |-2|=|10-8|$
$\Rightarrow |-2|=|2|$
$\Rightarrow 2=2$
True.
Hence, the solutions are $x=1$ and $x=5$.
