Responding to a pm:

|x| > 3 implies that x is a point whose distance from 0 is more than 3. So x could be greater than 3 or less than -3. Before you move further, think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3

Which of the following must be true?

I. x > 3

For every value that x can take, must x be greater than 3? No. e.g. if x takes -3.45, -4 etc, it will not be greater than 3 so this is not true.

II. X^2 > 9
This is the same as |x| > 3 so it must be true

III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take.
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If |x| > 3, then it must be true that EITHER x > 3 OR x < -3

(1) x > 3
This need not be true, since it's also possible that x < -3.
For example, x COULD equal -5
So, statement I need not be true.
ELIMINATE answer choice A, C and E

Important: The remaining two answer choices (B and D) both state that statement II is true.
So, we need not analyze statement II, since we already know it must be true.
That said, let's analyze it for "fun"
(2) x² > 9
This means that EITHER x > 3 OR x < -3
Perfect - this matches our original conclusion that EITHER x > 3 OR x < -3


(3) |x-1| > 2
Let's solve this further.
We get two cases:
case a) x - 1 > 2, which means x > 3 PERFECT
or
case b) x - 1 < -2, which means x < -1
Must it be true that x < -1?
YES.
We already learned that EITHER x > 3 OR x < -3
If x < -3, then we can be certain that x < -1
For example, if I tell you that the temperature is less than -3 degrees Celsius, can we be certain that the temperature is less than -1 degrees? Yes.
So, statement 3 must also be true.

Answer: D

Cheers,
Brent
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|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
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|x| > 3
In words:
The distance between x and 0 is greater than 3.
Thus, x must lie within the two colored ranges below:
<----------(-3)..........0..........3---------->

I: x > 3
If x lies within the red range, this statement is not true.
Eliminate A, C and E.

III: |x-1| > 2
In words:
The distance between x and 1 is greater than 2.
Every value within the two colored ranges is more than 2 places away from 1.
Thus, III must be true.
Eliminate B.


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

I. x > 3

II. x^2 > 9

III. |x - 1| > 2

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

|x| > 3 means that \(x < -3\) or \(x > 3\):

---------(-3)---------0---------(3)---------
So, x is somewhere in green segments. Notice that, x can be ONLY in green segments, for example, x cannot equal -2, or -1.5, or 0, or, 2.7, ...

The question asks which of the following options is ALWAYS true about x.

I. \(x > 3\):

We know that x could be less than -3 (for example, if say x = -10), so this option is not always true.

II. \(x^2 > 9\):

Take the square root: \(|x| > 3\). This is what was given in the stem in the first place, so this option is always true.

III. \(|x - 1| > 2\):

\(x - 1 < -2\) or \(x - 1 > 2\);
\(x < -1\) or \(x > 3\).

This options causes most problems for students. Consider this, for ANY x from the green segments above, it would be true to say that it is either less than -1 or greater than 3.

Check it yourself: can you pick any x, which satisfies |x| > 3, so which is in green segments, for which \(x < -1\) or \(x > 3\) won't be true? NO!!! ANY x which satisfies |x| > 3 (so ANY x which is in the green segments), will also satisfy \(|x - 1| > 2\).

So, no matter what x is (we know it's \(< -3\) or \(> 3\)) it would be true to say that it is either less than -1 or greater than 3. So, this statement is also always true.

Answer: D.

To understand the underline concept better practice other Trickiest Inequality Questions Type: Confusing Ranges.
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Original statement |x| > 3, which means either x>3 or x<-3
Now Check the options
Option1- x > 3 - not always true as x can be smaller than -3
Thus option A,C & E is ruled out. Only B & D are left

Option2- X^2 > 9 - Always true for x>3 or x<-3
To check - if x = 4,5,6,7.... or -4,-5,-6,-7
x^2>9

Option3-|x-1|>2,
which means (x-1)>2 ---> x>3 (if x-1>0) - True
it also means (x-1)<-2---->x<-1 (if x-1<0)

X<-1 satisfies x<-3. Thus true

Both 2 & 3 is true
Thus Answer D

Hope it helps

Let's solve this question differently

Here following is given
I x I >3

So, Either x>3 OR x<-3
If we plot the values in line we get following -


----(-4)----(-3)-----(-2)-----(-1)-----(0)------(1)-------(2)-------(3)------(4)
<------------I-----------------------------------------------------------I------------>
(x<-3)...............................................................................(x>3)

Now, let us take some value (-4 and 4) and plug into the options.

Option I with x=4 : \(4> 3\) Correct
Option I with x=-4 \(-4>3\) Not correct. So, options I is out.

Option II with x=4 : \(4^2> 9\) Correct
Option II with x=-4 \((-4)^2>3\) Correct. So, options II is right

Option III with x=4 : \(I 4-1I > 2\) Correct
Option III with x=-4 \(I (-4-1) I >2\) Correct. So, options III is right

Hence, correct answer is D

I just came across this one and it is clear that this is a tough cookie.

Given that |x| > 3, so that means x > 3 or x < -3. We can also test values here, only 3,4,5,6 or -3, -4, -5 etc work.

Now, let's tackle the statements:
I. x > 3

We know that x > 3 or x < -3, but it is not ALWAYS the case that x > 3. We do have values of x that are less than 3, i.e., when x is -4 or -5. So, I is false.

II. x^2 > 9.

This means x>3 or x < -3. That is what we have above so this is golden. True.

III. |x - 1| > 2

When x > 0
x -1 > 2
x > 3

x < 0
-x + 1 > 2
-x > 1
x < -1

So , III says IF x > 3 or x < -3, THEN x > 3 OR x < -1. True. Test values to prove this x > 3 or x < -3 means x can be -4 and that hits x < -1

i am clearly unclear how option 3 is true, since if i take the value of x as -2. It holds true for |x-1|>2. However the value of -2 does not hold true for |x| > 3. Please clarify.

The question reads "If |x| > 3, which of the following must be true?". The value of x=-2 doesn't subscribe to this condition in the first place. You have to filter all the possible values for x, BASED on the condition given in the problem.
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If some numbers confuse you, don't fixate on them. Go ahead and take some other easier examples.
Let's keep the wording of the question same but make it simple.

If n < 6, which of the following must be true?

I.

II.

III. n < 8

Can we say that III must be true? Yes!
If n is less than 6 then obviously it is less than 8 too.
If n is less than 6, it will take values such as -20, 2, 5 etc. All of these values will be less than 8 too.

Values 6 and 7 are immaterial because n cannot take these values. You are given that n is less than 6 so you only need to worry about values that n CAN take. Those should satisfy n < 8.

Similarly, your question says that x > 3 or x < -3

Then we can say that x > 3 or x < -1. All values that will be less than -3 will be less than -1 too.

Check out my post on a similar tricky question : http://www.veritasprep.com/blog/2012/07 ... -and-sets/
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Thanks WondedTiger,

Maybe I didn't present my question correctly. I didn't leave the 3rd option but came to the conclusion that it was wrong and chose my answer as B.
I just want to prevent that in timed conditions for difficult questions like these, which have subtle differences that makes an answer choice right.

Try using the number line for inequalities and absolute values.

|x| > 3 means distance of x from 0 is more than 3. So x is either greater than 3 or less than -3. So on the number line, it looks like this:

___________-3________0________3____________

The red part is the range where x will lie.

Is |x-1| > 2?
|x-1| > 2 represents that distance of x from 1 is more than 2. So x is either greater than 3 or less than -1.
Is x either greater than 3 or less than -1?

___________-3________0________3____________

All points on the red lines satisfy this. They are either greater than 3 or less than -1.

Hence option III must be true.
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III. |x-1|>2
This implies that the distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take. For each value, can be say that x is either greater than 3 or less than -1? Yes.
3.00001 - x is greater than 3
3.5 : x is greater than 3
4.2 : x is greater than 3
5.7 : x is greater than 3
67 : x is greater than 3
1000 : x is greater than 3
-3.45 : x is less than -1
-4 : x is less than -1
-8 : x is less than -1
-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1.Hence |x-1|>2 must be true for every value that x can take.

VeritasPrepKarishma Thank you for this! I was doing 'must be true' questions wrong!

Responding to a pm:



They are not the same: |x - 1| > 2 and |x|-1 > 2

|x - 1| > 2
means x > 3 or x < -1

|x| - 1 > 2
|x| > 3
means x > 3 or x < -3

But not what is given and what is asked.
We are GIVEN that |x| > 3
So we KNOW that x is either greater than 3 or it is less than -3. So valid values for x are 3.4, 4, 101, 2398675, -3.6, -5, -78 etc

Now the question is:
"Is |x - 1| > 2?"
"Is x always either greater than 3 or less than -1?"
All positive values of x are given to be greater than 3.
All negative values of x are given to be less than -3. So obviously they are less than -1 too.

Hence, |x - 1| > 2 is true.

Helps?
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So if there were 4th option saying x is integer or that x is real numbers , that would have been true also?

No. x is an integer will not always hold. We are given that |x| > 3. This means x > 3 or x < -3.
So x could be 4 or 5.6 or -3.87 or -100 etc. It is not necessary that x will be an integer.

But yes, x will be a real number because all these values are real numbers.
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hi

very beautiful explanation ....I must say ..

anyway ...I quote you " Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1. Hence |x-1|>2 must be true for every value that x can take."

For option # 3
III. |x-1|>2

we got "-1>x>3"

but the question stem says "-3>x>3"

So, according to the question stem, if we suppose x = -4, and according to the option #3, if you suppose x = -2, then these two values may not match, but the range provided by the option # 3, that is "-1>x>3" will cover the range provided by the question stem that is "-3>x>3", because -4 is certainly less than -1 and so is x...

please correct me if I am missing something you meant...

thanks in advance .....

Note that -1>x>3 is not correct.
x cannot be less than -1 and greater than 3 at the same time.
It should be x < -1 OR x > 3
The question stem tells us that x > 3 OR x < -3. So every valid value of x is either greater than 3 OR less than -3.

Now, every valid value that is greater than 3 is of course greater than 3.
Every valid value that is less than -3 (e.g. -4, -5.67 etc) is obviously less than -1 too. x cannot be - 2 since it must be less than -3.
That is why option III is always true.
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