Question

A student solves the following equation for all possible values of x:

8/x+2 = 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: 4x – 16 = x + 2
Step 4: 3x = 18
Step 5: x = 6

He determines that 6 is an extraneous solution because the difference of the numerators is 6, so the 6s cancel to 0.
Which best describes the reasonableness of the student’s solution?

a) His solution for x is correct and his explanation of the extraneous solution is reasonable.
b) 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.
c) 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.
d) His solution for x is incorrect. When solved correctly, there are no extraneous solutions

Answer 1

the answer is C

Explanation:

Answer 2

The correct answer is:
C) 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.

Explanation:
To solve this rational equation, we cross multiply:
8*(x-4)=2*(x+2).

Using the distributive property, we have
8*x-8*4=2*x+2*2;
8x-32=2x+4.

Subtract 2x from each side:
8x-32-2x=2x+4-2x;
6x-32=4.

Add 32 to each side:
6x-32+32=4+32;
6x=36.

Divide both sides by 6:

`(6x)/(6)`=
`(36)/(6)`;
x=6.

Extraneous solutions are solutions that come about in the problem but are not valid solutions to the problem; the only values of x that would give this sort of answer are -2 and 4, since these are the two values that would make one of the denominators 0 (we cannot divide by 0).