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This question was posted in another thread just an hour ago, here is my reply to that post.

Is \(\frac{1}{p}>\frac{r}{r^2 + 2}\)?

(1) \(p=r\) --> \(\frac{1}{r}>\frac{r}{r^2+2}?\) --> as \(r^2+2\) is always positive, multiplying inequality by this expression we'll get: \(\frac{r^2+2}{r}>r?\) --> \(r+\frac{2}{r}>r?\) --> \(\frac{2}{r}>0?\). This inequality is true when \(r>0\) and not true when \(r<0\). Not sufficient.

(2) \(r>0\). Not sufficient by itself.

(1)+(2) \(r>0\), \(\frac{2}{r}>0\). Sufficient.

Answer: C.

tejal777 again when you simplifying \(\frac{1}{p} >\frac{r}{r^2 +2}\) to \(r^2 +2 > pr\), you are making a mistake. You can multiply inequality by r^2 +2 as this expression is always positive, BUT you can not multiply inequality by p, as we don't know the sigh of p.

\(p < \frac{r^2+2}{r}\), by taking the reciprocals on each side (The inequality sign switches here, note that!)

Statement 1: p = r

This means the RHS becomes: \(\frac{p^2 + 2}{p} = p + \frac{2}{p}\). Note that \(p + \frac{2}{p}\) will be greater than p (you're adding the term 2/p to it) as long as p > 0. If p < 0, then ultimately, you're subtracting a number from p, and hence it'll be lesser. Since there are two cases here, this is insufficient by itself.

So we are down to B, C and E.

Statement 2: r > 0

Does this say anything at all about the relationship between p and r? No. Insufficient.

So, if you combine the statements, then p = r and r > 0, which means we've answered the only condition we had while solving the first statement and hence the statements together are sufficient.

That tells us no information about x, so it's not sufficient by itself.

Statement #2: x = y

Let's say x = y = 1. Then the left is 1, the right side is 1/(1^2 + 3) = 1/4, and the left side is bigger.

BUT, if x = y = -1, then the left side is -1, and the right side is -1/4, and --- here's one of the really tricky things about negatives and inequalities --- the "less negative" number -1/4 is greater than -1, so the right side is bigger. It may be less confusing to think about that in terms of whole numbers ---- for example, 10 > 5, but -5 > -10: it's better to have $10 in your pocket rather than $5 in your pocket, but it's better to be $5 in debt than $10 in debt. Does that make sense?

You are perfectly right --- y^2 is positive whether y is positive or negative, and therefore the denominator (y^2 + 3) is the same whether y is positive or negative, but what's different are whether the fractions themselves are negative, and that's what can reverse the order of the inequality.

Without knowing whether x & y are positive and negative, we cannot determine the direction of the inequality. Statement #2 by itself is not sufficient

Combined: y > 0 AND x = y

Now, we are guaranteed that the fractions are both positive, so multiplying by x or y will not reverse the order of the inequality. Because x = y, we have (1/y) > y/(y^2+3). Cross-multiplying, we get y^2 + 3 > y^2, which is always true. Together, the statements are sufficient. Answer choice = C.

Does that make sense? Please let me know if you have any further questions.

Mike
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Mike McGarry Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

1. y>0. No info on x, insuff. 2. x=y. Substituting 1/y > y^2/(y^2 + 3). However, you can only multiply both sides and not change the inequality if x=y=positive. If negative, inequality changes. Insuff.

Together, it is given that y is positive. So, x=y=positive. So, multiplying both sides with no change in inequality is possible. This implies 1 > y^2/(y^2+3). LHS is obviously < 1, so this inequality is true. Suff. C
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I am the master of my fate. I am the captain of my soul. Please consider giving +1 Kudos if deserved!

DS - If negative answer only, still sufficient. No need to find exact solution. PS - Always look at the answers first CR - Read the question stem first, hunt for conclusion SC - Meaning first, Grammar second RC - Mentally connect paragraphs as you proceed. Short = 2min, Long = 3-4 min

C. I dont see why you need (2) to solve the problem. Using (1) alone, you know that r^2+2 must be positive so the inequality sign does not change when we simplify to (r^2+2)/r > r.

If r>0, than r^2+2 > r^2 which can be simplified further to 2>0, this is TRUE.

If r<0 than r^2+2 < r^2 which can be simplified further to 2 < 0 which impossible so r must be greater than 0

What am i doing wrong?

if P=0 then R=0. Zero divided by anything is not defined so we cannot solve the equation when p=0.

Re: Is 1/p > r/(r^2+2) ? (1) p = r (2) r > 0 [#permalink]

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03 May 2013, 17:17

Kind of incomplete question. It should be given that p cannot be equal to zero. The answer is C. If you know that you can cross multiple with a number only if you know the sign of that number than you can get to the answer C.

Kind of incomplete question. It should be given that p cannot be equal to zero. The answer is C. If you know that you can cross multiple with a number only if you know the sign of that number than you can get to the answer C.

We are given that p = r and r > 0, thus p does not equal 0.

mr bunuel how does r^2+2/r >r => r^2>0. shouldnt it be r^2+ 2 >r^2 instead

\(\frac{1}{r}>\frac{r}{r^2+2}?\) --> as \(r^2+2\) is always positive, multiplying inequality by this expression we'll get: \(\frac{r^2+2}{r}>r?\) --> \(r+\frac{2}{r}>r?\) --> \(\frac{2}{r}>0?\).

As for your solution: we cannot cross-multiply \(\frac{1}{r}>\frac{r}{r^2+2}\) since we don't know whether r is positive or negative.

1/p > r/r^2+2 => r^2 + 2 > pr => r^2 - pr > -2 => r(r - p) > -2 now if p=r than r-p becomes 0 so => 0>-2 true

Why not statement A is suffiecient

We cannot cross-multiply in this case.

We can multiply 1/p > r/(r^2+2) by r^2+2 because r^2+2=non-negative+positive=positive but we cannot multiply 1/p > r/(r^2+2) by p because we don't know its sign: If p is positive, then we'd get 1 > p*r/(r^2+2): keep the sign when multiplying by positive value; If p is negative, then we'd get 1 < p*r/(r^2+2): flip the sign of the inequality when multiplying by negative value.

Here we must realize that r^2 +2 is always positive so we take it to the numerator of the left hand side Now statement 1 is not sufficient as we dont know the signs of p and r if they are positive the answer is yes else the answer is no => not sufficient statement 2 although gives a sign of r but we dont know what p or r actually are => not sufficient combining them => P>0 AND THE ANSWER IS YES => C is sufficient
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