Lesson: Useful Contradictions

Comment on Useful Contradictions

I am quite new to algebra shouldn't it be -2x12 instead of -4x-6 around 2min.
gmat-admin's picture

Yes, you are correct. The purpose of the video is to demonstrate that the properties of DS questions can help you identify errors that test-takers may make. So, at 2:55, we use this property to identify our mistake and then correct our factorization.

There is a mistake in the mathematical calculation in the quadratic equation x^2 - 10x - 24 = 0; i think it was taken as x^2 - 10x + 24 = 0.. i request you to go through the video and rectify the same. Thank you
gmat-admin's picture

The "mistake" was made intentionally to show how a certain property of Data Sufficiency questions can help us identify errors we might make in our calculations.
So, in statement 1, we conclude that x = -2 or x = -3.
In statement 2, we conclude (incorrectly) that x = 4 or x = 6.
Since we arrive at CONTRADICTORY results, we know that we must have made a mistake in one of our calculations, and we can re-check our work to see where we erred.
So, around 2:50 in the video, we use this property to identify our mistake and correct our factorization.

But, it can also be the case that if both statements were contradicting, the answer is E.
gmat-admin's picture

The statements will NEVER contradict each other. So, if it seems like there's a contradiction, then we can be certain that the test-taker made an error.

Hello, excellent course! Just have a quick question:

If we can assume that both sentences are always true and they do not contradict each other, couldn't we conclude that since they're both ecuations in function of X with only variable they both must be sufficient to determine the value of X, independently of one another?

Thank you,
gmat-admin's picture

Not always.

Consider this example:

What is the value of x?
(1) 2x = 6
(2) x² = 9

Both statements are true, but only one statement (statement 1) is sufficient to answer the target question alone.

for your last comment 2nd sep 2017 X^2=9 is sufficient as we know 3^2=9.Plesae let me know if I am considering any thing wrong.
gmat-admin's picture

You're referring to the question:
What is the value of x?
(1) 2x = 6
(2) x² = 9

Statement 1 is sufficient because there is only one possible solution: x = 3

Statement 2 is not sufficient because there are two possible solutions: x = 3 and x = -3

Notice that 3² = 9 and (-3)² = 9

Does that help?

Cheers,
Brent

Wouldn't it be a waste of time to solve both equations? If we know that each statement provides us with 2 possible answers for x (eliminating A,B & D), and we also know that the 2 statements cannot contradict each other, can't we conclude that the correct answer is C? Unless you're saying there are cases where 2 different quadratic equations can give us the same 2 values of x (keeping answer "E" relevant) and the only way to find out is by solving both equations. If so, can you provide an example?
gmat-admin's picture

Great question, Max!

There are A few IF's in your proposed rule that we should examine, but, in short, we can say:

If each quadratic equation yields two DIFFERENT possible x-values, and if the two quadratic equations do not yield the SAME two possible x-values, then the correct answer must be C.

However, there are different things that can mess up this rule. Here are two examples:

If x > 0, what is the value of x?
(1) x² + x - 6 = 0
Solving this quadratic, we see that EITHER x = 2 OR x = -3
Since we're told x > 0, we can be certain that x = 2
So, statement 1 is sufficient on its own.

------------------------------------------------

What is the value of x?
(1) x² - 6x + 9 = 0
(2) x² + 2x - 15 = 0
Factoring (1), we get: (x - 3)(x - 3) = 0
So, it MUST be the case that x = 3
So, statement 1 is sufficient on its own.

Factoring (2), we get: (x - 3)(x + 5) = 0
So, EITHER x = 3 OR x = -5
So, statement 2 is NOT sufficient on its own.

-------------------------------------------------

I like you're proposed rule, just as long as you're aware of a few potential loopholes.

Cheers,
Brent

Hey Brett,

Is the combined statement sufficient because both results contain x= -2?
gmat-admin's picture

You're referring to the question that appears at 1:25 in the above video.

The answer to your question is yes.
Keep in mind that both statements are true.
So, since x = -2 is the only value the two statements have in common, it must be the case that x = -2

Cheers,
Brent - now with 50% less "t" :-)

Hi Brent,

I am new to Data sufficiency amd watching this video concerning the contradictions, I am confused.

WHen we tackle the Data sufficiency, is it true that we will solve each statement indvudually. So I think it would be possible that different statements (alternatives) will provide different result/outcome?

Why we need to compare the two results from the 2 statements at the end?

Thanks.

Stanley
gmat-admin's picture

Yes, you must definitely analyze each statement individually, but that is not what this video covers.

The key concept covered in this video is that both statement are 100% true.
Consider this question:

What is the value of x?
(1) 2x = 6
(2) x + 1 = 11
The above question would never appear on the GMAT, since it's impossible for both statements to be true.
That is, statement 1 tells us that x = 3, while statement 2 tells us that x = 10.

Now consider this example:
What is the value of x?
(1) x² = 16
(2) x² - x - 12 = 0
This COULD be an actual GMAT question, because both statements are 100% true.
Statement 1 tells us that EITHER x = 4 OR x = -4. So, statement 1 not sufficient.
IMPORTANT: It is 100% true that EITHER x = 4 OR x = -4

Upon factoring statement 2, we get: (x - 4)(x + 3) = 0. This tells us that EITHER x = 4 OR x = -3.
So, statement 2 not sufficient.
IMPORTANT: It is 100% true that EITHER x = 4 OR x = -3

Here, the correct answer is C because, when we COMBINE the statements, we know that x MUST equal 4 (since it is the ONLY value that satisfies both TRUE statements.

So, what do we do if we come across a DS question in which statement 1 tells us that EITHER x = 1 OR x = 2, and statement 2 tells us that EITHER x = 5 OR x = 6?
It appears that these two statements contradict each other.
That is, both statements cannot be true.
Since both statements are ALWAYS true in a DS question, we can be certain that we made a MISTAKE when we concluded that statement 1 tells us that EITHER x = 1 OR x = 2, and statement 2 tells us that EITHER x = 5 OR x = 6.
This "useful" contradiction tells us to go back and re-analyze the two statements.

Does that help?

Hi Brent,

Could I understand that this method would be applied in the statement that has a particular value ?

Thanks.
gmat-admin's picture

That's correct.

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