Lesson: Inequalities and Absolute Value

Comment on Inequalities and Absolute Value

Hi Brent, have a query in the below question

https://gmatclub.com/forum/if-4-7-x-3-which-of-the-following-must-be-true-168681.html

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive

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

While I got the equations right, i.e in st2: x<-5 or x>-1 and st 3 is always true, the question asks which of the statement must be true and since st2 yields 2 possible solutions for x, the answer for me is B. Really confused how people have taken st2 to be always true.
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-4-7-x-3-which-of-the-following-must-be-tru...

Great question, Jalaj!

From the given information, we know for certain that x < -5

Given that x < -5, must statement II be true?
From statement II, you determined that EITHER x < -5 OR x > -1
So, if x < -5, must it be true that EITHER x < -5 OR x > -1?
In order to answer YES to the above question we only need one of the parts to be true: x < -5 OR x > -1
Since we can be certain that x < -5.

Notice that the question does NOT read "If x < -5, must it be true that BOTH x < -5 AND x > -1?"

Here's an analogous question:
Let's say you have a rock in your pocket, and someone asks you "Is it true that you have EITHER a rock in your pocket OR an elephant in your pocket?
The answer to that question is YES, it is true that I have EITHER a rock in my pocket OR an elephant in my pocket.

Does that help?

Cheers,
Brent

Hi Brent, in the question below statement III has 2 possible solutions: x < -1 and x > 3.
Therefore, x can have a value of -2, which negates the inequality question |x| > 3.
Therefore my answer is B. Can you help me identify where am I going wrong in this?

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

https://gmatclub.com/forum/if-x-3-which-of-the-following-must-be-true-138652-40.html
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-3-which-of-the-following-must-be-true-13...

Be careful! We cannot say "statement III has 2 possible solutions x < -1 AND x > 3"
The value of x cannot be less than -1 AND greater than 3 at the same time.
Instead, we can say that EITHER x < -1 OR x > 3

This is very similar to your question above.

I should also point out the error in your comment that "Therefore, x can have a value of -2, which negates the inequality question |x| > 3."
You are approaching this in the wrong direction.
We are told that it is 100% true that |x| > 3, and our job is to identify other statements (from I, II and III) that are 100% true.

Instead, you are assuming that statement III is true and trying to determine whether the given information (|x| > 3) is true.

Does this help?

Cheers,
Brent

https://gmatclub.com/forum/the-dark-purple-region-on-the-number-line-above-is-shown-in-its-entire-229498.html
sir how to handle this question?

https://gmatclub.com/forum/if-x-3-which-of-the-following-must-be-true-138652.html

I am not understanding why statement 3 must be true.

According to the statement x is any number greater than +3 or less than -3. The third choice (which MUST be true) shows that x is any number greater than +3 or less than -1

Doesn't this mean that x could be - 2 which is NOT true?
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-3-which-of-the-following-must-be-true-13...

You're reversing the order of the IF-THEN statement.
For example, IF we know that x < 5, THEN it must also be true that x < 8
Likewise, IF we know that x < 0, THEN it must also be true that x < 20

Notice that if we reverse the order, we can something that is false.
For example: IF we know that x < 20, THEN it must also be true that x < 0 (not true!)
This is what you're inadvertently doing.

From the given information, we know that EITHER x > 3 OR x < -3
Let's focus on the fact that x < -3
IF x < -3, THEN can we also be certain that x < -1? YES.

In other words: IF we know that x < -3, THEN it must also be true that x < -1
This is what statement III is saying.

Does that help?

Cheers,
Brent

BTW, here's my solution: https://gmatclub.com/forum/if-x-3-which-of-the-following-must-be-true-13...

No I am still not following here.

The way I conceptualised it was looking at the number line:

The first option gives us:

x < - 3

-------- (-3) --------- 0
This is
where x could lie

Compared with:

------ -2 ------ 0
X lies here

OR

So the third choice in the question allows for x = -2.0000000001 which is greater than where x could lie when its x < - 3
Here, x must cannot be anywhere to the right of the number line after -3

That's my reasoning and I still can't seem to fathom how I got it wrong. Have I misinterpreted the number line?


gmat-admin's picture

You're still reversing the IF-THEN structure of the question.

Here's another analogy of what you are doing.

----------------------------
Consider this question:
Joe is taller than 150 centimeters. Which of the following MUST be true?
i) Joe is taller than 100 centimeters.

Must statement i be true?
Here's how your rationale would look: If Joe is taller than 100 centimeters, then Joe could be 101 centimeters tall, which contradicts the premise that Joe is taller than 150 centimeters.
Therefore statement i is false.

Do you see the problem here?
The given information tells us that Joe is taller than 150 centimeters.
This is 100% true. So, Joe could be 151 centimeters tall, or 152 centimeters tall, or 153 centimeters tall, or 154 centimeters tall, . . . etc.
Also, since Joe is taller than 150 centimeters, we know that Joe CANNOT be 101 cm tall. We also know that Joe CANNOT be 102 cm tall, or 103 cm tall, or 104 cm, or 105 cm etc.

Given that Joe can be 151 cm, 152 cm, 153 cm,...etc, can we be certain that Joe is taller than 100 centimeters?
Yes, absolutely.
So, statement i is true.

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

Here's one last analogy.
Consider this question:
Joe is a billionaire (i.e., Joe has more than $1,000,000,000). Which of the following MUST be true?
i) Joe has more than $1

Must statement i be true?
Here's how your rationale would look: If Joe has more than $1, then Joe could have $2, which contradicts the premise that Joe has more than $1,000,000,000.
Therefore statement i is false.

In actuality, if Joe has more than $1,000,000,000, then we can be certain that Joe has more than $1.
So, statement i is true.
------------------------------

Does that help?

Cheers,
Brent

gmat-admin's picture

When you say "So the third choice in the question allows for x = -2.0000000001 which is greater than where x could lie when its x < -3", you are reversing the order.

The GIVEN information tells us that x < -3.
So, it could be the case that x = -3.1, or x = -3.2, or x = -4, or x = -5, or x = -5.5, etc
Also, since x < -3, it CANNOT be the case that x = -2.5, or x = -2.0000000001, or x = -2, or x = -1, or x = 0, or x = 4, etc

Given all of this, must it be true that x < -2?
Yes, all of the possible x-values (-3.1, or -3.2, or -4, or -5, or -5.5, etc) are ALL less than -2

Cheers,
Brent

Ha ha. The second analogy made me see where I went wrong with this one. That and the subsequent explanation. Thanks for this.

Hi Brent, I would appreciate your help on this DS qs:

Is |XY| > X²Y²

1) 0 < X² < 1/4
2) 0 < Y² < 1/9

I understand this property from your lectures-

if |A| > A² ...... then A > A² OR A < -A²

thanks

gmat-admin's picture

Great question!
Here's my full solution: https://gmatclub.com/forum/is-xy-x-2-y-2-1-0-x-2-1-4-2-0-y-260816.html#p...

Cheers,
Brent

Hi Brent,

Earlier in the lesson whenever we solved equations containing absolute values we checked if the solutions obtained were actually extraneous roots by substituting the values obtained in the original equation (Concept from Video 25).

Would it not then be necessary to check if the roots obtained extraneous in the above scenario as well? (i.e concepts covered in video 38 and 39)


Thank you in advance for clarifying!
gmat-admin's picture

Solution link: https://gmatclub.com/forum/is-xy-x-2-y-2-1-0-x-2-1-4-2-0-y-260816.html#p...

Notice that, in my solution, I don't actually solve for x or y. That is, I don't say x must equal some particular value (e.g., x = -1/3). Instead, I made conclusions about certain relationships (e.g., x > x² and y > y²).

As such, I didn't need to check for extraneous roots (since I didn't actually calculate any roots).

Does that help?

Cheers,
Brent

Yes, Brent! Thank you so much!

Hi Brent,
Can you please explain the rationale behind this statement?
"We know that the absolute value of something will always be greater than or equal to zero"
Thanks
gmat-admin's picture

Consider these examples:
|-3| = 3, |7.1| = 7.1, |-4.65| = 4.65, |0| = 0, and |5.06| = 5.06

Notice that, when we find the absolute value of any number, the result is always greater than or equal to zero.

Does that help?

Hi Brent,

Question link - https://gmatclub.com/forum/if-x-is-an-integer-what-is-the-value-of-x-233844.html

Why can we not solve the 1st statement using the same technique as the 2nd? By factoring the eq and getting roots -5 and -3 which means -5<x<-3. This will leave us with only one sol which is -4
Thanks.
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-is-an-integer-what-is-the-value-of-x-233...

Good question.

You took the inequality x² - 8x + 15 < 0,
and factored it to get: (x - 3)(x - 5) < 0

How did you conclude that -5 < x < -3?
It should be 3 < x < 5

Cheers,
Brent

Could you please tell me why statement 3 is correct in the following question:

https://gmatclub.com/forum/if-4-7-x-3-which-of-the-following-must-be-true-168681.html

Thanks,
Aman
gmat-admin's picture

Hi Aman,

Here's my full solution: https://gmatclub.com/forum/if-4-7-x-3-which-of-the-following-must-be-tru...

Cheers,
Brent

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