1. The problem states an irregular polygon with sides labeled as $x-2$, $x$, $x-7$, $x-1$, and $x-3$, and the total perimeter is 52 inches. 2. To find $x$, we set up an equation: the sum of all side lengths equals the perimeter. $$ (x-2) + x + (x-7) + (x-1) + (x-3) = 52 $$ 3. Combine like terms: $$ x - 2 + x + x - 7 + x - 1 + x - 3 = 52 $$ $$ 5x - (2 + 7 + 1 + 3) = 52 $$ $$ 5x - 13 = 52 $$ 4. Solve for $x$: $$ 5x = 52 + 13 $$ $$ 5x = 65 $$ $$ x = \frac{65}{5} = 13 $$ 5. Check that all sides are positive using $x=13$: - $13 - 2 = 11$ - $13$ - $13 - 7 = 6$ - $13 - 1 = 12$ - $13 - 3 = 10$ All are positive. 6. Final answer: $$ \boxed{13} $$