Let's solve the given equation step-by-step to find the value of [tex]\( x \)[/tex] and determine if the solution is extraneous or not.
The equation is:
[tex]\[ \sqrt{x - 5} + 7 = 11 \][/tex]
Step 1: Isolate the square root term.
Subtract 7 from both sides of the equation:
[tex]\[ \sqrt{x - 5} = 11 - 7 \][/tex]
[tex]\[ \sqrt{x - 5} = 4 \][/tex]
Step 2: Square both sides to eliminate the square root.
By squaring both sides, we get:
[tex]\[ (\sqrt{x - 5})^2 = 4^2 \][/tex]
[tex]\[ x - 5 = 16 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
Add 5 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 16 + 5 \][/tex]
[tex]\[ x = 21 \][/tex]
Step 4: Verify the solution to check if it is extraneous.
Substitute [tex]\( x = 21 \)[/tex] back into the original equation to see if it holds true:
[tex]\[ \sqrt{21 - 5} + 7 = 11 \][/tex]
[tex]\[ \sqrt{16} + 7 = 11 \][/tex]
[tex]\[ 4 + 7 = 11 \][/tex]
[tex]\[ 11 = 11 \][/tex]
Since the left-hand side equals the right-hand side, the solution is correct and not extraneous.
Thus, the solution is [tex]\( x = 21 \)[/tex], and it is not extraneous.
So, the correct answer is:
[tex]\[ x = 21, \text{ solution is not extraneous} \][/tex]
