If x is replaced with -1/x in the expression Q is obtained. Which of

Question

P = \frac{x^2−1}{x^2+1 }

If x is replaced with  \frac{-1}{x}  in the expression P, the expression Q is obtained. Which of the following is the correct relationship between P and Q?

A)  PQ=1

B)  P+Q=0

C)  \frac{P}{Q } = 1

D)  \frac{P}{Q^2} = -1

E)  \frac{P}{Q^2} = 1

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MarkGMATMentor · Answer

This question illustrates the importance of simplifying algebraic expressions on the GMAT. In particular, when you are trying to match up or compare two expressions, look to simplify before you do so.

First, let's replace x with  \frac{-1}{x} :
 Q=\frac{(\frac{-1}{x})^2-1}{(\frac{-1}{x})^2+1}

As is, it will be difficult to compare the two expressions.  At this point your goal should be to simplify the expression Q to make it look more like P.

So let's distribute the squares:
 Q=\frac{\frac{1}{x^2}-1}{\frac{1}{x^2}+1}

Now let's get rid of the fractions-within-fractions. You can do this by multiplying both the top and bottom by the common denominator of each , which is  x^2 . So, multiply by  \frac{x^2}{x^2} :
 Q=\frac{1-x^2}{1+x^2}

Now compare to the original expression  P = \frac{x^2−1}{x^2+1 }

The denominator is the same, since  x^2+1=1+x^2 
The numerator is equal to the original times  -1 , since  -1(x^2-1)=1-x^2

Now you can test the answer choices. Rather than calculating each one completely, see if there are any that work out easily. It should be clear that adding P and Q results in zero, as stated in choice B. They have the same denominators, so you can simply add the numerators, which cancel each other out.  Choice B is correct.

Try to simplify expressions before you compare them. On this problem, it's helpful to get rid of fractions-within-fractions by multiplying top and bottom by the common denominator.

onlyPlanA · Answer

P = \frac{x^2−1}{x^2+1 }

Q = (-1/x)^2 - 1 / (-1/x)^2 + 1

=(1 - x^2)/x^2 / (1 + x^2)/x^2 
  = 1- x^2/1+x^2 
 = -(x^2 - 1) / (x^2 + 1)

We can see that P = -Q
Thus P + Q = 0

IMO, B is the answer.

TheAdmissionHub · Answer

((-\frac{1}{x})^2-1)/((-\frac{1}{x})^2 +1)= (\frac{1}{x^2}-1)/(\frac{1}{x^2} +1) = \frac{1-x^2}{x^2}/\frac{1+x^2}{x^2} = \frac{1-x^2}{1+x^2}

Therefore we can observe that P+Q=0

Answer: B

If x is replaced with -1/x in the expression Q is obtained. Which of

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GMAT Problem Solving (PS) Questions

If x is replaced with -1/x in the expression Q is obtained. Which of

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