To solve the equation [tex]\(\sqrt{x + 9} - 4 = 1\)[/tex] for [tex]\(x\)[/tex], follow these detailed steps:
1. Isolate the Square Root Term:
[tex]\[
\sqrt{x + 9} - 4 = 1
\][/tex]
Add 4 to both sides of the equation to isolate the square root term:
[tex]\[
\sqrt{x + 9} = 1 + 4
\][/tex]
Simplify the right-hand side:
[tex]\[
\sqrt{x + 9} = 5
\][/tex]
2. Square Both Sides to Eliminate the Square Root:
[tex]\[
(\sqrt{x + 9})^2 = 5^2
\][/tex]
This reduces to:
[tex]\[
x + 9 = 25
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Subtract 9 from both sides of the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 25 - 9
\][/tex]
Simplify the right-hand side:
[tex]\[
x = 16
\][/tex]
4. Check for Extraneous Solutions:
Substitute [tex]\(x = 16\)[/tex] back into the original equation to ensure it satisfies the equation:
[tex]\[
\sqrt{16 + 9} - 4 = 1
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{25} - 4 = 1
\][/tex]
Since [tex]\(\sqrt{25} = 5\)[/tex], we have:
[tex]\[
5 - 4 = 1
\][/tex]
This simplifies correctly to:
[tex]\[
1 = 1
\][/tex]
The solution [tex]\(x = 16\)[/tex] satisfies the original equation, so it is not extraneous.
Therefore, the correct answer is:
[tex]\[
\boxed{x = 16, \text{ solution is not extraneous}}
\][/tex]
