To solve the equation \( x^2 = \frac{36}{169} \) by inspection, we can take the square root of both sides.
First, observe that we can find the square root of the fraction:
\[ x = \pm \sqrt{\frac{36}{169}} = \pm \frac{\sqrt{36}}{\sqrt{169}} = \pm \frac{6}{13} \]
Thus, the solutions to the equation are:
\[ x = \frac{6}{13} \quad \text{and} \quad x = -\frac{6}{13} \]
Therefore, the solution set is:
\[ \left{ \frac{6}{13}, -\frac{6}{13} \right} \]