Problem Solving

Hi Sid,

Certain Quant questions can be solved by using 'brute force' - you don't need to do any advanced math, you just need to do enough basic calculations to prove whatever pattern(s) the question is based on.

Here, we're dealing with Absolute Values, so you should be looking for at least 2 different 'groups' of numbers that fit the given inequality.

With |X| > 3, you likely see the 'obvious' group... X > 3. However, you also have to spot the other group of numbers that fit this inequality: X < -3.

Knowing those two options, you can quickly prove that Roman Numeral 1 is NOT always true but that Roman Numeral 2 IS always true. With Roman Numeral 3, you might not be sure, but you can prove that it IS always true with some brute force math...

From the information in the prompt, we can know that we have to consider X > 3 and X < -3. Plug as many of those values into Roman Numeral 3 as you like and then document the results. For example...

What happens when...
X = 4 or 5
X = 3.01

What happens when...
X = -4 or -5
X = -3.01

You'll find that with EVERY example that fits the given restrictions, Roman Numeral 3 IS true. That proof confirms that Roman Numeral 3 is part of the correct answer.

Final Answer: D

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

GMATsid2016 wrote:If |x|>3, which of the following must be true?

1) x>3

2) x^2>9

3) |x-1|>2

A) 1 only

B) 2 only

C) 1 and 2 only

D) 2 and 3 only

E) 1,2, and 3

OAD

|x| > 3 is equivalent to x > 3 OR x < -3. Let's now examine each answer choice.

1) x > 3

Since x > 3 or x < -3, x does not have to be greater than 3. Answer choice 1 is not necessarily true.

2) x^2 > 9

If x > 3, then x^2 > 9, and if x < -3, then x^2 > 9. Answer choice 2 is true.

3) |x - 1| > 2

Recall that |x| > 3 is equivalent to x > 3 or x < -3. Let's examine each of these two cases for the given inequality.

Case 1: x > 3

If x > 3, then x - 1 > 2. If a quantity is greater than a positive number, its absolute value is also greater than that positive number. Thus, |x - 1| > 2.

Case 2: x < -3

If x < -3, then x - 1 < -4. If a quantity is less than a negative number, its absolute value is actually greater than the absolute value of the negative number.

Thus |x - 1| > |-4|, i.e., |x - 1| > 4. Since |x - 1| > 4, that means |x - 1| > 2.

Therefore, we see that in either case, |x - 1| > 2. Answer choice 3 is also true.

Answer: D

Good catch, vs224!
I have edited my response accordingly (but the answer is still D)

Cheers and thanks,
Brent

GMATsid2016 wrote:If |x|>3, which of the following must be true?

1) x>3

2) x^2>9

3) |x-1|>2

A) 1 only

B) 2 only

C) 1 and 2 only

D) 2 and 3 only

E) 1,2, and 3

OAD

Please explain how come 3 is possible?

Thanks,

Sid

3) |x-1|>2 can be written as x-1 > 2 => x > 3 OR x-1 < -2 (notice the sign-reversal of the inequality)

=> x > 3 OR x-1 < -2 => x < -1

If x < -1, we can say x < -3 must be true.

Thus, x > 3 OR x > -3

=> |x| > 3.

Hope this helps!

-Jay
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Hi All,

When a question includes an absolute value, you have to keep your mind open to the idea that there are probably going to be multiple answers (or multiple GROUPS of answers) that fit the given information. In Roman Numeral questions, it helps to pay attention to how the answer choices are written (since you can often avoid some of the 'work' by noting which answers can be eliminated at certain points in the process).

Here, we're told that |X| > 3. This means that any number that is EITHER greater than 3 OR less than -3 will fit the given inequality. We have to consider both possibilities, since the question asks which of the following MUST be true.

I: X > 3

Since X could be -4, Roman Numeral I is NOT always true.
Eliminate Answers A, C and E.

At this point, we can tell from the remaining two answer choices that Roman Numeral II MUST be true (since it occurs in both of the remaining answers). However, here is the proof that it's true....

II: X^2 > 9

Squaring any number GREATER than 3 will result in a number that is GREATER than 9
Squaring any number LESS than -3 will result in a number that is GREATER than 9
Roman Numeral II IS always true.

III: |X - 1| > 2

IF.... X > 3, then |X - 1| will be greater than |2|.
IF... X < -3, then |X - 1| will be greater than |-4| = |4|.
Roman Numeral III IS always true.

Final Answer: D

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