# Thread: Inequalities from GMAT question

1. ## Inequalities from GMAT question

Hi, I hope I am in the right forum. I came across this question while preparing for my GMAT exam. [Note : I am not good in maths since high school.] Appreciate if some1 can explain to me since I have left school 10 years ago.

This relates to Data Sufficiency (DS) problem:

e, d are Real Numbers. If e>d，is ed<0?

Statement 1: (e^2)d < e(d^2)

Statement 2 : e+d < 0

Are those Statements sufficient to answer the question: If e>d，is ed<0?

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) together are NOT sufficient.

Thank you in advance.

2. ## Re: Inequalities from GMAT question

Why don't you post your thoughts to start a discussion?

3. ## Re: Inequalities from GMAT question

For Statement 1: (e^2)d < e(d^2) :
[Reminder: In the question, it stated that e >d.]

Analysis:
From Statement 1, we can rewrite : (e^2)d < e(d^2) → e*(ed)<d*(ed),

Given that e>d (given in question)，we can deduce the following:
e *(ed) > d*(ed) [Multiply both sides with “ed”]

[Note : the above deduced inequality is only VALID provided “ed” is POSITVE or ed > 0.]

If you compare above derived inequality with Statement 1: e*(ed)<d*(ed), the only difference is the inequality sign: “>” vs “<”.

Again, from e>d (given in question) ，
Assume “ed” is NEGATIVE or ed < 0, we multiply both sides with “ed”

e > d
e * (ed) < d * (ed) [Multlply both sides with “ed” and flip the sign since ed<0 (i.e negative)]

The above deduced inequality MATCHES Statement 1.

As such, we come to the following conclusion:

In order to derive to Statement 1: (e^2)d < e(d^2) from e> d, “ed” must be NEGATIVE or ed<0.

From the above analysis, the Statement 1 is sufficient to answer the question.

For Statement 2: e+d<0:
[Reminder: In the question, it stated that e >d.]

Analysis:
If e+d< 0 and e>d, the following scenarios can exist:

Scenario 1: both variables “e” and “d” <0 [Eg, “e” = -1, “d”= -2]
This will result in ed > 0.

If this is the case, the above only satisfies the inequalities: e+d<0 and e>d, but DOES NOT meet the requirement for ed<0 (the stem of the question).

Scenario 2: variable “d”<0 and variable “e”>0 [Eg, “d”= -2, “e”=1]
This will result in ed < 0.

If this is the case, the variables “d” and “e” can satisfy ALL the inequalities : e+d<0, e>d and ed<0 (the stem of the question).

From the above, the Statement 2 is NOT sufficient to answer the question since Scenario 1 does not satisfy ed<0.

I do not know the correct answer. But, from the above, it should be “A": Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Let me know your thoughts on this. It's kind of confusing when dealing with Data Sufficiency question...

4. ## Re: Inequalities from GMAT question

Originally Posted by bestguy1977
For Statement 1: (e^2)d < e(d^2) :
[Reminder: In the question, it stated that e >d.]

Analysis:
From Statement 1, we can rewrite : (e^2)d < e(d^2) → e*(ed)<d*(ed),

Given that e>d (given in question)，we can deduce the following:
e *(ed) > d*(ed) [Multiply both sides with “ed”] This is not valid
It is not valid because it may be that $ed<0$.

However, given $e>d~\&~e^2d that tells us $e-d>0$.

From $e^2d we get
\begin{align*}e^2d-ed^2&<0 \\ed(e-d)&<0\end{align*}

Because $e-d>0$ that forces $ed<0.$

You have given a good counterexample to show II. does not work in any case.
So what is the correct answer?

5. ## Re: Inequalities from GMAT question

Originally Posted by bestguy1977
As such, we come to the following conclusion:

In order to derive to Statement 1: (e^2)d < e(d^2) from e> d, “ed” must be NEGATIVE or ed<0.

From the above analysis, the Statement 1 is sufficient to answer the question.
Why would you want to derive (e^2)d < e(d^2)? This statement is supposed to be used as an assumption. ("A is sufficient for B" means "A implies B.") You logic is probably ultimately right, but it seems confusing at first...

6. ## Re: Inequalities from GMAT question

Hi,

Thanks for your prompt reply.

Wow, you really made it so simple on the validity of statement I. Thanks for correcting my poor math reasoning.

The Official Answer is "A - Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient."