solve for x in a right triangle with angles 90, 60, and 7x-12

1. Identify the triangle properties: The image shows a right-angled triangle. A right angle is equal to $90^{\circ}$. The other two angles are given as $60^{\circ}$ and $(7x-12)^{\circ}$.

2. Set up the equation: The sum of the interior angles of any triangle is always $180^{\circ}$. Therefore, we can write the equation:
$$90 + 60 + (7x - 12) = 180$$

3. Simplify the equation: Combine the constant terms on the left side:
$$150 + 7x - 12 = 180$$
$$7x + 138 = 180$$

4. Solve for x: Subtract 138 from both sides of the equation:
$$7x = 180 - 138$$
$$7x = 42$$
Divide both sides by 7:
$$x = \frac{42}{7}$$
$$x = 6$$