Solve for all possible values of x: sqrt(x + 7) - 11 = 2.

Step-by-step solution

1. 1

   Isolate the radical first. Squaring both sides while $-11$ is still on the left would give $(\sqrt{x+7}-11)^2 = x + 7 - 22\sqrt{x+7} + 121$ — still radical, and worse than before. Add $11$ to both sides:

   $\sqrt{x+7} = 13$

2. 2

   Note that a solution can exist. The right side is $13 > 0$, and a principal square root is non-negative, so there is no immediate contradiction. (Had we reached $\sqrt{x+7} = -13$, we could stop and answer "no solution".)

3. 3

   Square both sides. With the radical alone, squaring is clean:

   $x + 7 = 169$

4. 4

   Solve the linear equation.

   $x = 169 - 7 = 162$

5. 5

   Check for an extraneous root. Squaring can create false solutions, so substitute back into the original equation:

   $\sqrt{162+7} - 11 = \sqrt{169} - 11 = 13 - 11 = 2 \;\checkmark$

   So $x = 162$ is genuine. For contrast, the distractor $x = 169$ gives $\sqrt{176} - 11 \approx 2.27 \neq 2$, and $x = 6$ gives $\sqrt{13} - 11 \approx -7.39$ — both come from mishandling the $-11$.