To solve the equation given, we start with:
\[ \sqrt{x + 4} - 7 = 1 \]
First, we will isolate the square root by adding 7 to both sides:
\[ \sqrt{x + 4} = 1 + 7 \] \[ \sqrt{x + 4} = 8 \]
Next, we will square both sides to eliminate the square root:
\[ (\sqrt{x + 4})^2 = 8^2 \] \[ x + 4 = 64 \]
Now, we will solve for \( x \) by subtracting 4 from both sides:
\[ x = 64 - 4 \] \[ x = 60 \]
Thus, the solution is:
\[ \boxed{60} \]
Verifying our solution by substituting \( x = 60 \) back into the original equation:
\[ \sqrt{60 + 4} - 7 = 1 \] \[ \sqrt{64} - 7 = 1 \] \[ 8 - 7 = 1 \]
This holds true, confirming that the solution \( x = 60 \) is correct.
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