Answer
Option (B) $3$ in. is correct.
Work Step by Step
Perimeter $P=13$ inches.
Sum of all sides $=$ Perimeter.
$\Rightarrow x-5+\frac{x}{2}+6=P$
Substitute $13$ for $P$.
$\Rightarrow x-5+\frac{x}{2}+6=13$
Add like terms.
$\Rightarrow \frac{2x+x}{2}+1=13$
$\Rightarrow \frac{3x}{2}+1=13$
Subtract $1$ from each side.
$\Rightarrow \frac{3x}{2}+1-1=13-1$
Simplify.
$\Rightarrow \frac{3x}{2}=12$
Multiply each side by $\frac{2}{3}$.
$\Rightarrow \frac{2}{3}\cdot \frac{3x}{2}=\frac{2}{3}\cdot 12$
Simplify.
$\Rightarrow x=8$
The sides of the triangle are
$x-5=8-5=3$ inches.
$\frac{x}{2}=\frac{8}{2}=4$ inches.
Hence, the shortest side is $3$ inches.
