1. **State the problem:** Solve the equation $\sqrt{3x-2} = 5$ for $x$. 2. **Recall the formula and rules:** To solve an equation involving a square root, we square both sides to eliminate the square root. Remember, squaring both sides can introduce extraneous solutions, so we must check our answers. 3. **Square both sides:** $$\left(\sqrt{3x-2}\right)^2 = 5^2$$ $$3x - 2 = 25$$ 4. **Isolate $x$:** $$3x = 25 + 2$$ $$3x = 27$$ 5. **Divide both sides by 3:** $$x = \frac{27}{3}$$ $$x = 9$$ 6. **Check the solution:** Substitute $x=9$ back into the original equation: $$\sqrt{3(9) - 2} = \sqrt{27 - 2} = \sqrt{25} = 5$$ This matches the right side, so $x=9$ is a valid solution. **Final answer:** $$x = 9$$