1. **State the problem:** Solve the equation $\sqrt{6x+7} = 2$ for $x$. 2. **Recall the formula and rules:** To solve an equation involving a square root, we isolate the square root term and then square both sides to eliminate the root. Remember to check for extraneous solutions after squaring. 3. **Isolate and square both sides:** $$\sqrt{6x+7} = 2$$ Square both sides: $$\left(\sqrt{6x+7}\right)^2 = 2^2$$ $$6x + 7 = 4$$ 4. **Solve the resulting linear equation:** $$6x + 7 = 4$$ Subtract 7 from both sides: $$6x + \cancel{7} - \cancel{7} = 4 - 7$$ $$6x = -3$$ Divide both sides by 6: $$\frac{6x}{\cancel{6}} = \frac{-3}{\cancel{6}}$$ $$x = -\frac{1}{2}$$ 5. **Check for extraneous solutions:** Substitute $x = -\frac{1}{2}$ back into the original equation: $$\sqrt{6\left(-\frac{1}{2}\right) + 7} = \sqrt{-3 + 7} = \sqrt{4} = 2$$ This matches the right side, so $x = -\frac{1}{2}$ is a valid solution. **Final answer:** $$x = -\frac{1}{2}$$