Let's solve the given equation step by step:
The equation is:
[tex]\[
\sqrt{x - 5} + 7 = 11
\][/tex]
First, isolate the square root term by subtracting 7 from both sides of the equation:
[tex]\[
\sqrt{x - 5} = 11 - 7
\][/tex]
Simplify the right side:
[tex]\[
\sqrt{x - 5} = 4
\][/tex]
Next, eliminate the square root by squaring both sides of the equation:
[tex]\[
(\sqrt{x - 5})^2 = 4^2
\][/tex]
Simplify both sides:
[tex]\[
x - 5 = 16
\][/tex]
Now, solve for [tex]\( x \)[/tex] by adding 5 to both sides:
[tex]\[
x = 16 + 5
\][/tex]
Thus, the solution is:
[tex]\[
x = 21
\][/tex]
Now, we need to check if this solution is extraneous. To do this, we 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]
Since the left side equals the right side, the solution [tex]\( x = 21 \)[/tex] is correct and not extraneous.
Therefore, the correct solution is:
[tex]\[
x = 21, \text{ solution is not extraneous}
\][/tex]
