Math Question
![A student solves the following equation for all possible values of \( x \) :
\[
\frac{8}{x+2}=\frac{2}{x-4}
\]
His solution is as follows:
Step 1: \( 8(x-4)=2(x+2) \)
Step 2: \( 4(x-4)=(x+2) \)
Step 3: \( 4 x-16=x+2 \)
Step 4: \( 3 x=18 \)
Step 5: \( x=6 \)
He determines that 6 is an extraneous solution because the difference of the numerators is 6, so the 6 s cancel to 0 . Which best describes the reasonableness of the student's solution?
His solution for \( x \) is correct and his explanation of the extraneous solution is reasonable.
His solution for \( x \) is correct, but in order for 6 to be an extraneous solution, both denominators have to result in 0 when 6 is substituted for \( x \).
His solution for \( x \) is correct, but in order for 6 to be an extraneous solution, one denominator has to result in 0 when 6 is substituted for \( x \).
His solution for \( x \) is incorrect. When solved correctly, there are no extraneous solutions.](https://d3hlidq5bf9vg0.cloudfront.net/app-server-student/2021/12/7/abd913f1-5237-449b-97d2-e85233716282.png "A student solves the following equation for all possible values of \( x \) :
\[
\frac{8}{x+2}=\frac{2}{x-4}
\]
His solution is as follows:
Step 1: \( 8(x-4)=2(x+2) \)
Step 2: \( 4(x-4)=(x+2) \)
Step 3: \( 4 x-16=x+2 \)
Step 4: \( 3 x=18 \)
Step 5: \( x=6 \)
He determines that 6 is an extraneous solution because the difference of the numerators is 6, so the 6 s cancel to 0 . Which best describes the reasonableness of the student's solution?
His solution for \( x \) is correct and his explanation of the extraneous solution is reasonable.
His solution for \( x \) is correct, but in order for 6 to be an extraneous solution, both denominators have to result in 0 when 6 is substituted for \( x \).
His solution for \( x \) is correct, but in order for 6 to be an extraneous solution, one denominator has to result in 0 when 6 is substituted for \( x \).
His solution for \( x \) is incorrect. When solved correctly, there are no extraneous solutions.")
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