Connect with knowledgeable individuals and find the best answers at IDNLearn.com. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.

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


Sagot :

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]
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. IDNLearn.com is your go-to source for dependable answers. Thank you for visiting, and we hope to assist you again.