Is 1/p > r/(r^2 + 2) ? (1) p = r (2) r > 0

Great wayof approaching!Super clear..thanks!

Thanks Bunuel , your approach to the problem was very good. !

thanks

I'm happy to help with this one.

Is (1/x) > y/(y^2+3)

(1) y > 0
(2) x = y

Statement #1: y > 0

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

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.

Hope it's clear.

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.

Hope it helps.

Target question: Is 1/p > r/(r² + 2) ?

Statement 1: p = r
Take the target question and replace p with r to get: Is 1/r > r/(r² + 2) ?
Since r² + 2 is always POSITIVE, we can safely multiply both side of the inequality by r² + 2.
When we do this, we get: Is (r² + 2)/r > r ?

Now we can apply the following fraction property: (a + b)/c = a/c + b/c
We get: Is r²/r + 2/r > r ?
Simplify to get: Is r + 2/r > r ?
Subtract r from both sides to get: Is 2/r > 0 ?
At this point, we can see that statement 1 is not sufficient.
When r is POSITIVE, the answer to our revised target question is YES, it IS the case that 2/r > 0
When r is NEGATIVE, the answer to our revised target question is NO, it is NOT the case that 2/r > 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: r > 0
Since we have no information about p, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
From statement 1 we learned that . . .
When r is POSITIVE, the answer to our revised target question is YES, it IS the case that 2/r > 0
When r is NEGATIVE, the answer to our revised target question is NO, it is NOT the case that 2/r > 0

Statement 2 tells us that r is POSITIVE
So, we can be certain that the answer to our revised target question is YES, it IS the case that 2/r > 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

1) p=r

Then,
1/p > r/(r^2+2)
Multiplying p=r on both sides,
1>r^2/(r^2+2)
Square are always positive.
As x/(x+something) <1, for any positive integer x, should the answer not be A instead of C???

Can someone please explain.

You can remember the following rule -
Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.

In this scenario :
\(\frac{1}{r} > \frac{r}{r^2+2}\)
we can't cross multiply r from \(\frac{1}{r} \) as we don't know the sign of r . No matter whether p=r ,we need to know the sign to cross multiply.

Here we can either multiply \({r^2+2}\). Or simplify like below :

\(\frac{1}{r} - \frac{r}{r^2+2}\)>0

statement 1 P=r

let p=1 r=1

1/1>1/1+2 = 1>1/3 ------ yes

let p=-1 r=-1

-1/1>-1/3 ---------no
statement 1 insufficient

statement - 2 r>0
let r= 1
1/p>1/3
no information about p hence statement 2 insufficient

combining 1 & 2

r=1 p=1

1/1>1/3----yes

r=2 p=2

1/2>2/6------yes

answer is C

Bunuel is my approach correct ?

Bunuel I understand the logic of not cross multiplying given you don't know the value of r. However, I cross multiplied and just did two scenarios (1) if r was positive (2) if r was negative and deemed first question insufficient. The second question tells you its negative, so I can select the correct equation of the two and answer together is sufficient.

My confusion is just that cross-multiplying doesn't give the same formula. It gives 0>-2. or 0<-2 Is that a problem ? Is my method still a mistake?

Your question is a bit challenging to follow entirely, but here's my response:

Firstly, it's crucial to understand that during cross-multiplication, we multiply by the denominators of the fractions. So, when cross-multiplying \(\frac{1}{p} > \frac{r}{r^2 + 2}\), we multiply by r^2 + 2 and p, not by r.

Secondly, if p is positive, after cross-multiplication (and keeping the sign as is), the question evolves to "is \(r^2 +2 > pr\)?". Conversely, if p is negative, then after cross-multiplication (and flipping the sign), the question becomes "is \(r^2 +2 < pr\)?".

For (1) that says p = r, if p (or r, since they're equal) is positive, the question simplifies to "is \(2 > 0\)?", hence the answer would be YES. For negative p, the question simplifies to "is \(2 < 0\)?", thus the answer would be NO. However, as we lack information about the sign of p (or r), we cannot say which case we have. .

(2) is obviously useless alone, but when paired with (1), it supplies the missing information: r, and thereby, p is positive. So, the question simplifies to "is \(2 > 0\)?" to which the answer is clearly YES.

Generally, you should never multiply an inequality by a variable when its sign is unknown.