Let's solve the equation [tex]\(\sqrt{x-5} + 7 = 11\)[/tex] step by step.
1. Isolate the square root term:
[tex]\[
\sqrt{x-5} + 7 = 11
\][/tex]
Subtract 7 from both sides of the equation to isolate the square root term:
[tex]\[
\sqrt{x-5} = 11 - 7
\][/tex]
Simplify the right-hand side:
[tex]\[
\sqrt{x-5} = 4
\][/tex]
2. Eliminate the square root:
Square both sides of the equation to eliminate the square root:
[tex]\[
(\sqrt{x-5})^2 = 4^2
\][/tex]
Simplify:
[tex]\[
x-5 = 16
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 5 to both sides:
[tex]\[
x = 16 + 5
\][/tex]
Simplify:
[tex]\[
x = 21
\][/tex]
4. Verify the solution:
To ensure that [tex]\(x = 21\)[/tex] is not an extraneous solution, substitute [tex]\(x = 21\)[/tex] back into the original equation:
[tex]\[
\sqrt{21-5} + 7 = 11
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{16} + 7 = 11
\][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex], we get:
[tex]\[
4 + 7 = 11
\][/tex]
This simplifies to:
[tex]\[
11 = 11
\][/tex]
The original equation holds true, so the solution [tex]\(x = 21\)[/tex] is correct and it is not extraneous.
Thus, the correct answer is:
[tex]\[ x = 21, \text{ solution is not extraneous } \][/tex]
