division

[Guest]

#8) Is pq = 1?

(1) pqp = p

(2) qpq = q

Why can't we simply divide both sides of the equation by p to obtain pq = 1. Solution says we must factor this equation.


#19) Where as in the following problem,

Is p^2q > pq^2?

(1) pq < 0

(2) p < 0

Solution:
The question can first be rewritten as "Is p(pq) > q(pq)?"
If pq is positive, we can divide both sides of the inequality by pq and the question then becomes: "Is p > q?"
If pq is negative, we can divide both sides of the inequality by pq and change the direction of the inequality sign and the question becomes: "Is p < q?"




In solution to question #8, we are dividing both side of the inequality by pq, whereas in question #19 it says we can't divide. How is it different, or I am missing something here. Please clarify



From MGMAT Question Bank: Equations

[Guest]

Correction:

In solution to question #19, we are dividing both side of the inequality by pq, whereas in question #8 it says we can't divide. How is it different, or I am missing something here. Please clarify

[esledge]

Quick note: the problems referenced here are actually #8 and #17 (typo in post above)

Dividing by variables in equations is DANGEROUS because the variables might be zero. Dividing by zero is forbidden (it yields an undefined value), and doing so generally will cause you to miss a possible solution (zero). Review #8 to see what I mean. pq might be 1, but another way for statement (1) to be true is for p (and thus pq) to be zero.

Dividing by variables in inequalities is EVEN WORSE because you not only have to worry about whether they could be zero, but you have to worry about their sign. If you multiply or divide an inequality by a negative number you have to flip the sign, but you don't flip the sign when you multiply or divide by a positive number. So that is why the rephrase in #17 became two questions: one for the positive scenario, one for the negative scenario (with the sign flipped).

It seems to me that you still should be wary of dividing by pq even to do the rephrase in #17--after all, they have accounted for the positive and negative signs, but not the zero possibility. Statement (1) clearly indicates that pq is not zero so no worries there, but statement (2) still leaves open the possibility that pq is zero. Maybe it doesn't affect the answer--I'll look into it tomorrow to be sure.

[christiancryan]

By the way, just to close the loop -- we've inserted an upfront condition stating that pq ≠ 0 in problem #17, in order to remove any ambiguity.

[Guest]

So is the answer for #8 - E?


And is the answer for #17 E also?

[unique]

for #17 I think the answer should be C

the question turns out
if pq> 0 then is p>q
if pq<0 then is p<q

1. pq<0 which p<0 or q<0 it is not stated so INSUFFICIENT

2. p<0 but q=? INSUFFICIENT

TOGETHER pq<0 and p<0 so q>0 so p<q SUFFICIENT

[dbernst]

Unique, that is correct. The statements together tell us that p<0 and q>0, giving us sufficient information to answer the question.

The correct answer is C.

-dan