To solve the equation [tex]\(\sqrt{x-5} + 7 = 11\)[/tex], we need to isolate [tex]\(x\)[/tex] step by step. Here is the detailed solution:
1. Isolate the Square Root Term:
[tex]\[
\sqrt{x-5} + 7 = 11
\][/tex]
Subtract 7 from both sides 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. Remove the Square Root:
Square both sides of the equation to eliminate the square root:
[tex]\[
(\sqrt{x-5})^2 = 4^2
\][/tex]
Simplify both sides:
[tex]\[
x - 5 = 16
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 5 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 16 + 5
\][/tex]
Simplify:
[tex]\[
x = 21
\][/tex]
4. Check for Extraneous Solutions:
Substitute [tex]\(x = 21\)[/tex] back into the original equation to verify:
[tex]\[
\sqrt{21-5} + 7 = 11
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{16} + 7 = 11
\][/tex]
Calculate the square root:
[tex]\[
4 + 7 = 11
\][/tex]
Simplify the left-hand side:
[tex]\[
11 = 11
\][/tex]
Since the left-hand side equals the right-hand side, the solution [tex]\(x = 21\)[/tex] is not extraneous.
Thus, the correct answer is:
[tex]\[
x = 21, \text{ solution is not extraneous}
\][/tex]
