Answered

Solve for [tex]$x$[/tex], given the equation [tex]\sqrt{x-5}+7=11[/tex].

A. [tex]$x=21$[/tex], solution is extraneous
B. [tex][tex]$x=21$[/tex][/tex], solution is not extraneous
C. [tex]$x=81$[/tex], solution is extraneous
D. [tex]$x=81$[/tex], solution is not extraneous



Answer :

GinnyAnswer
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]