To find the values of \( x \) that will produce a square with an area less than or equal to 36 square inches, we start with the formula for the area of a square:
\[ \text{Area} = \text{side length}^2 \]
In this case, the side length of the square is \( 2x \) inches, so the area \( A \) is given by:
\[ A = (2x)^2 = 4x^2 \]
We want this area to be less than or equal to 36 square inches:
\[ 4x^2 \leq 36 \]
Next, we divide both sides of the inequality by 4:
\[ x^2 \leq 9 \]
Now we take the square root of both sides. Since \( x \) must be greater than 0, we only consider the positive root:
\[ x \leq 3 \]
Thus, the values of \( x \) that will produce a square with an area less than or equal to 36 square inches are:
\[ 0 < x \leq 3 \]
So, the value of \( x \) must be greater than 0 but less than or equal to 3 inches.