Answer
The solution is $x=\frac{3}{2}$.
Work Step by Step
The given equation is
$\Rightarrow |3x-4|=|3x-5|$
Write the two related linear equations.
$3x-4=3x-5$ ...... (1)
$3x-4=-(3x-5)$ ...... (2)
Solve equation (1).
$\Rightarrow 3x-4=3x-5$
Add $5-3x$ to each side.
$\Rightarrow 3x-4+5-3x=3x-5+5-3x$
Simplify.
$\Rightarrow 1=0$
False statement.
No solution.
Solve equation (2).
$\Rightarrow 3x-4=-(3x-5)$
Clear the parentheses.
$\Rightarrow 3x-4=-3x+5$
Add $3x+4$ to each side.
$\Rightarrow 3x-4+3x+4=-3x+5+3x+4$
Simplify.
$\Rightarrow 6x=9$
Divide each side by $6$.
$\Rightarrow \frac{6x}{6}=\frac{9}{6}$
Simplify.
$\Rightarrow x=\frac{3}{2}$
Check : $x=\frac{3}{2}$
$\Rightarrow |3x-4|=|3x-5|$
$\Rightarrow |3(\frac{3}{2})-4|=|3(\frac{3}{2})-5|$
$\Rightarrow |\frac{9}{2}-4|=|\frac{9}{2}-5|$
$\Rightarrow |\frac{9-8}{2}|=|\frac{9-10}{2}|$
$\Rightarrow |\frac{1}{2}|=|\frac{-1}{2}|$
$\Rightarrow \frac{1}{2}=\frac{1}{2}$
True.
Hence, the solution is $x=\frac{3}{2}$.
