If 1/x - 1/(x+1) (OG16)

boomgoesthegmat wrote:If 1/x - 1/(x+1) = 1/(x+4), then x could be

A) 0
B) -1
C) -2
D) -3
E) -4

OA: C

1/x - 1/(x+1) = 1/(x+4)
You can do this question both by testing values and algebraically.
I will solve it algebraically

x + 1 - x / (x)(x + 1) = 1/(x + 4)
1/x*(x + 1) = 1/(x + 4)

x + 4 = x^2 + x
x^2 = 4

x = 2 or -2

Correct Option: C

boomgoesthegmat wrote:If 1/x - 1/(x+1) = 1/(x+4), then x could be

A) 0
B) -1
C) -2
D) -3
E) -4

We must first eliminate the fractions in the equation 1/x - 1/(x+1) = 1/(x+4).

Thus, we will multiply the entire equation by the least common multiple of the denominators, which is:

x(x+1)(x+4)

We are now left with:

(x+1)(x+4) - x(x+4) = x(x+1)

x^2 + 5x + 4 - x^2 - 4x = x^2 + x

x + 4 = x^2 + x

4 = x^2

√4 = √x^2

x = 2 or x = -2

Alternate Solution:

Another option is to backsolve, substituting the answer choices into the given equation:

First, we can eliminate choices A, B and E because any one of them will make one of the denominators equal to 0 and we can’t have denominator = 0. Choice A will make the denominator of the first fraction on the left hand side of the equation equal to 0; choice B will make the denominator of the second fraction on the left hand side of the equation equal to 0; and choice E will make the denominator of the fraction on the right hand side of the equation equal to 0.

So we only need to test choices C and D.

C. -2

1/(-2) – 1/(-2+1) = 1/(-2+4) ?

-1/2 - 1/(-1) = 1/2 ?

-1/2 + 1 = 1/2 ?

1/2 = 1/2 (Yes!)

We see that C is the correct choice, but let’s show that D will not be the correct choice.

D. -3

1/(-3) – 1/(-3+1) = 1/(-3+4) ?

-1/3 - 1/(-2) = 1/1 ?

-1/3 + 1/2 = 1 ?

1/6 = 1 (No!)

Answer: C