To solve the equation \( x^2 = \frac{36}{169} \) by inspection, we can take the square root of both sides.
Starting with the equation:
\[ x^2 = \frac{36}{169} \]
Taking the square root of both sides gives:
\[ x = \sqrt{\frac{36}{169}} \quad \text{or} \quad x = -\sqrt{\frac{36}{169}} \]
Calculating the square root:
\[ \sqrt{\frac{36}{169}} = \frac{\sqrt{36}}{\sqrt{169}} = \frac{6}{13} \]
Thus, the solutions are:
\[ x = \frac{6}{13} \quad \text{and} \quad x = -\frac{6}{13} \]
So, the complete solution set for the equation \( x^2 = \frac{36}{169} \) is:
\[ x = \frac{6}{13}, \quad x = -\frac{6}{13} \]