If x - y = 8, which of the following must be true?

Hi All,

Roman Numeral questions are relatively rare on Test Day (you'll probably see just 1 in the Quant section), and they're typically designed to take longer to solve than average. However, there is usually at least one built-in design "shortcut" so that you can avoid doing some of the work.....

Here, we're told that X-Y = 8. We're asked which of the following MUST be true (which really means "which of the following is ALWAYS true no matter how many different examples we can come up with?"). In these situations, it's usually easier to try to prove that a Roman Numeral is NOT true, so that we can eliminate it from the answer choices...

If X - Y = 8

I. Do BOTH X and Y have to be POSITIVE?

IF...
X = 8, then Y = 0
Then Roman Numeral 1 is NOT true.
Eliminate Answers A and D.

II. If when X is POSITIVE, does Y have to be POSTIVE?

We can use the same example that we used in Roman Numeral 1....
IF....
X = 8, then Y = 0
Then Roman Numeral 2 is NOT true.
Eliminate Answers B and E.

There's only 1 answer remaining, so we don't even have to check Roman Numeral 3.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

If x-y=8, which of the following must be true?

I. Both x and y are positive
II. If x is positive, y must be positive
III. If x is negative, y must be negative

1. x and y can be negative, for instance, -2 - (-10) = 8 =>Eliminate A and D
2. x can be positive and y can be negative, for instance, 2 - (-6) = 8 => Eliminate B and E

A) I only
B) II only
C) III only
D) I and II
E) II and III

Here's an alternate approach to evaluating the three statements:

I. Both x and y are positive

x- y = 8
=> y = x - 8

It's clear that for some positive values of x (like x = 2), y will be negative. Since we have found a case where St. I doesn't hold true, it is not a MUST BE TRUE statement.

II. If x is positive, y must be positive
Again, by the same example as above (when x = 2), we see that St. II is not always true either.

Since we know by now that St. I and II are not MUST BE TRUE statements, we are sure that the correct answer is D. Still, for the sake of completeness, I'll elaborate how we can evaluate St. III using this algebraic approach.

III. If x is negative, y must be negative
Again, using y = x - 8

We can see that if x is negative, then y = (negative) - 8 = (negative)

Hence St. III is a MUST BE TRUE statement.

Hope this was useful!

Best Regards

Japinder

We are given that x - y = 8 and need to determine which of statement(s) must be true. Let’s analyze each Roman numeral answer choice.

I. Both x and y are positive

Both x and y DO NOT have to be positive. For instance if x = -1 and y = -9, then:

-1 - (-9) = -1 + 9 = 8

II. If x is positive, y must be positive

If x is positive, y DOES NOT have to be positive. For instance, if x = 7 and y = -1, then:

7 - (-1) = 7 + 1 = 8

III. If x is negative, y must be negative

If x is negative, it is true that y must also be negative to produce a positive difference between x and y. For instance, if x = -1 and y = -9, then:

-1 - (-9) = -1 + 9 = 8

Answer: C

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