ABSOLUTE VALUE (Modulus) https://gmatclub.com/forum/download/file.php?id=10390 This post is a part of GMAT MATH BOOK ] created by: walker edited by: bb , Bunuel Definition The absolute value (or modulus) |x| of a real number x is x's numerical value without regard to its sign. For example, |3| = 3 ; |-12| = 12 ; |-1.3|=1.3 Graph: https://gmatclub.com/forum/download/file.php?id=10105 Important properties: |x|\geq0 |0|=0 |-x|=|x| |x|+|y|\geq|x+y| How to approach equations with moduli It's not easy to manipulate with moduli in equations. There are two basic approaches that will help you out. Both of them are based on two ways of representing modulus as an algebraic expression. 1) |x| = \sqrt{x^2} . This approach might be helpful if an equation has × and /. 2) |x| equals x if x>=0 or -x if x<0. It looks a bit complicated but it's very powerful in dealing with moduli and the most popular approach too (see below). 3-steps approach: General approach to solving equalities and inequalities with absolute value: 1. Open modulus and set conditions. To solve/open a modulus, you need to consider 2 situations to find all roots: Positive (or rather non-negative) Negative For example, |x-1|=4 a) Positive: if (x-1)\geq0 , we can rewrite the equation as: x-1=4 b) Negative: if (x-1) 1) and all points left are second condition (x x=5 b) -x+1=4 --> x=-3 3. Check conditions for each solution: a) x=5 has to satisfy initial condition x-1>=0 . 5-1=4>0 . It satisfies. Otherwise, we would have to reject x=5. b) x=-3 has to satisfy initial condition x-1 x = -1 . We reject the solution because our condition is not satisfied (-1 is not less than -8) b) -8 \leq x x = -15 . We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.) c) -3 \leq x x = 9 . We reject the solution because our condition is not satisfied (9 is not within (-3,4) interval.) d) x \geq 4 . (x+3) + (4-x) = (8+x) --> x = -1 . We reject the solution because our condition is not satisfied (-1 is not more than 4) (Optional) The following illustration may help you understand how to open modulus at different conditions. https://gmatclub.com/forum/download/file.php?id=10488 Answer : 0 Example #2 Q.: |x^2-4| = 1 . What is x? Solution: There are 2 conditions: a) (x^2-4)\geq0 --> x \leq -2 or x\geq2 . x^2-4=1 --> x^2 = 5 . x e { -\sqrt{5} , \sqrt{5} } and both solutions satisfy the condition. b) (x^2-4) -2 x^2 = 3 . x e { -\sqrt{3} , \sqrt{3} } and both solutions satisfy the condition. (Optional) The following illustration may help you understand how to open modulus at different conditions. https://gmatclub.com/forum/download/file.php?id=10489 Answer : -\sqrt{5} , -\sqrt{3} , \sqrt{3} , \sqrt{5} Tip & Tricks The 3-steps method works in almost all cases. At the same time, often there are shortcuts and tricks that allow you to solve absolute value problems in 10-20 sec. I. Thinking of inequality with modulus as a segment at the number line. For example, Problem: 1<x<9. What inequality represents this condition? https://gmatclub.com/forum/download/file.php?id=10106 A. |x|<3 B. |x+5|<4 C. |x-1|<9 D. |-5+x|<4 E. |3+x|<5 Solution: 10sec. Traditional 3-steps method is too time-consume technique. First of all we find length (9-1)=8 and center (1+8/2=5) of the segment represented by 1 3 is equal to x e (-inf,-6)&(0,+inf) III. Thinking about absolute values as the distance between points at the number line. For example, Problem: A<X<Y<B. Is |A-X| <|X-B|? 1) |Y-A|<|B-Y| Solution: https://gmatclub.com/forum/download/file.php?mode=view&id=10107 We can think of absolute values here as the distance between points. Statement 1 means than the distance between Y and A is less than that between Y and B. Because X is between A and Y, |X-A| https://gmatclub.com/forum/which-of-the ... 44267.html 2. https://gmatclub.com/forum/if-y-is-an-i ... 39867.html 3. https://gmatclub.com/forum/on-the-numbe ... 16027.html 4. https://gmatclub.com/forum/if-k-2-m-2-w ... 07128.html 5. https://gmatclub.com/forum/if-y-0-which ... 09621.html 6. https://gmatclub.com/forum/which-of-the ... 08204.html 7. https://gmatclub.com/forum/the-function ... 96171.html 8. https://gmatclub.com/forum/4-272003.html 9. https://gmatclub.com/forum/how-many-int ... 88429.html 10. https://gmatclub.com/forum/if-x-is-a-ne ... 64088.html Medium: 1. https://gmatclub.com/forum/what-is-the- ... 79892.html 2. https://gmatclub.com/forum/if-x-y-1-whi ... 22118.html 3. https://gmatclub.com/forum/if-x-y-0-whi ... 86959.html 4. https://gmatclub.com/forum/for-how-many ... 68675.html 5. https://gmatclub.com/forum/given-the-in ... 18973.html 6. https://gmatclub.com/forum/hot-competit ... 33254.html 7. https://gmatclub.com/forum/around-the-w ... 15784.html 8. https://gmatclub.com/forum/around-the-w ... 15984.html 9. https://gmatclub.com/forum/if-q-is-the- ... 31226.html 10. https://gmatclub.com/forum/what-is-the- ... 96176.html Hard: 1. https://gmatclub.com/forum/if-y-x-5-x-5 ... 73626.html 2. https://gmatclub.com/forum/if-x-0-and-x ... 43572.html 3. https://gmatclub.com/forum/if-x-3-4-and ... 10071.html 4. https://gmatclub.com/forum/what-is-the- ... 98596.html 5. https://gmatclub.com/forum/how-many-roo ... 79379.html 6. https://gmatclub.com/forum/which-of-the ... 54341.html 7. https://gmatclub.com/forum/if-x-2-x-5-0 ... 24552.html 8. https://gmatclub.com/forum/what-is-the- ... 48153.html 9. https://gmatclub.com/forum/what-is-the- ... 99579.html 10. https://gmatclub.com/forum/what-is-the- ... 20033.html lineAXYZ.png line1x9.png graph_modulus.png Math_icon_absolute_value.png Math_abs_example1.png Math_abs_example2.png Math_abs_example0.png

In the following example of 3-steps approach for complex problems Example #1 Q.: |x+3|−|4−x|=|8+x||x+3|−|4−x|=|8+x|. How many solutions does the equation have? Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions: a) x x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not less than -8) Can any one please explain me how x =-8 for both |8+x| and -|8+x| b) −8≤x x=−15x=−15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.) c) −3≤x x=9x=9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.) d) x≥4x≥4. (x+3)+(4−x)=(8+x)(x+3)+(4−x)=(8+x) --> x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

IN Example #1 Q.: |x+3|−|4−x|=|8+x||x+3|−|4−x|=|8+x|. How many solutions does the equation have? C condition has typo error c) −3≤x x=9x=9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.) its not -15 but 9

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Math: Absolute value (Modulus)

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onyx12102

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How frequent or how many absolute value questions can we approximately expect on the GMAT? I understand there may not be a sure way to know or guess but just looking for an estimate based on past and present experiences.

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I'd say 1 or 2.

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onyx12102

Not more than a couple. The problem is that often GMAT likes to combine topics. So, you might have a geometry question which will make you use some basic absolute value concept such as area bounded by the graph of |x| + |y| = 4. So a basic understanding of all topics is a good idea even if you plan to ignore some topics.

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Quote:

2. Solve new equations:
a) x−1=4x−1=4 --> x=5
b) −x+1=4−x+1=4 --> x=-3

3. Check conditions for each solution:
a) x=5x=5 has to satisfy initial condition x−1>=0x−1>=0. 5−1=4>05−1=4>0. It satisfies. Otherwise, we would have to reject x=5.
b) x=−3x=−3 has to satisfy initial condition x−1<0x−1<0. −3−1=−4<0−3−1=−4<0. It satisfies. Otherwise, we would have to reject x=-3.

So...just to clarify...the answer can either be 5 or -3?

Just need the clarification on what this is saying

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In the following example of 3-steps approach for complex problems

Example #1
Q.: |x+3|−|4−x|=|8+x||x+3|−|4−x|=|8+x|. How many solutions does the equation have?
Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) x<−8x<−8. −(x+3)−(4−x)=−(8+x)−(x+3)−(4−x)=−(8+x) --> x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not less than -8) Can any one please explain me how x<-8 because whenever I am doing it I am getting x>=-8 for both |8+x| and -|8+x|

b) −8≤x<−3−8≤x<−3. −(x+3)−(4−x)=(8+x)−(x+3)−(4−x)=(8+x) --> x=−15x=−15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) −3≤x<4−3≤x<4. (x+3)−(4−x)=(8+x)(x+3)−(4−x)=(8+x) --> x=9x=9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x≥4x≥4. (x+3)+(4−x)=(8+x)(x+3)+(4−x)=(8+x) --> x=−1x=−1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

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IN Example #1
Q.: |x+3|−|4−x|=|8+x||x+3|−|4−x|=|8+x|. How many solutions does the equation have?
C condition has typo error
c) −3≤x<4−3≤x<4. (x+3)−(4−x)=(8+x)(x+3)−(4−x)=(8+x) --> x=9x=9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

its not -15 but 9

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chetan2u Hi, sorry I keep bugging you but your explanations are easiest to understand

Can you explain to me the "trick" mentioned in the post? I don't get it :/ What are we doing after finding the mid-point?

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hibobotamuss

chetan2u Hi, sorry I keep bugging you but your explanations are easiest to understand

Can you explain to me the "trick" mentioned in the post? I don't get it :/ What are we doing after finding the mid-point?

Please mention the portion you are talking about..
No problems. Pl keep asking questions and I would reply to each whenever I get time

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"Problem: 1<x<9. What inequality represents this condition?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5
Solution: 10sec. Traditional 3-steps method is too time-consume technique. First of all we find length (9-1)=8 and center (1+8/2=5) of the segment represented by 1<x<9. Now, let’s look at our options. Only B and D has 8/2=4 on the right side and D had left site 0 at x=5. Therefore, answer is D."

This is the "trick" mentioned in the post, don't know what it means chetan2u

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gettinit

Let’s consider following examples,

Example #1 I am not understanding this example and really struggling with modulus? Can someone please elaborate and explain in further detail? From this post I can't see how I would use this on every modulus problem?
Q.: $|x+3| - |4-x| = |8+x|$. How many solutions does the equation have?
Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) $x < -8$. $-(x+3) - (4-x) how did we get -(x+3) here?= -(8+x)$ --> $x = -1$. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) $-8 \leq x < -3$. $-(x+3)-(4-x) =+ (8+x)$ --> $x = -15$. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) $-3 \leq x < 4$. $+ (x+3)-(4-x) =+ (8+x)$ --> $x = 9$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) $x \geq 4$. $+(x+3) + (4-x) = + (8+x)$ --> $x = -1$. We reject the solution because our condition is not satisfied (-1 is not more than 4)

I am totally lost with this post and also with other modulus problems I looked up in Gmat club thank you very much for your help in advance!!!!!

I didn't understand how the sign for each expression was determined.
I'd appreciate any help.
Thanks!!

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Bunuel

10. Absolute Value

Theory
Math: Absolute value (Modulus)
Absolute Value: Tips and hints
Properties of Absolute Values on the GMAT
Absolute modulus : A better understanding
GMAT Question Patterns: Abolute Value
The Reason Behind Absolute Value Questions on the GMAT

Questions
Inequality and absolute value questions from my collection
The E-GMAT Question Series on ABSOLUTE VALUE
DS Questions
PS Questions

For more check Ultimate GMAT Quantitative Megathread

When are we supposed to use the 3-step method? I mean, on which kind of problems?

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OP or Bunuel

There is a typo in the first post.

It says "(-15 is not within (-3,4) interval." for the range in (c) when it should actually say "9 is not within the -3<x<4 range"

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OP or Bunuel

There is a typo in the first post.

It says "(-15 is not within (-3,4) interval." for the range in (c) when it should actually say "9 is not within the -3<x<4 range"

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Edited. Thank you.

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VeritasKarishma

rrsnathan

hi,

i have a doubt in the example

Q.: |x^2-4| = 1. What is x?
Solution: There are 2 conditions:

a) (x^2-4)\geq0

how the range for this example is (X^2-4)>=0 and not (x^2-4)>0?

regards,
RRSNATHAN

You can ignore "=0" part. (x^2 - 4) cannot be 0 since its mod is 1. So you can certainly write the range as (x^2 - 4) > 0. In the overall scheme, it doesn't change anything. People usually write the ranges are ">=0" and "<0" to cover everything. Here "=0" is not needed.

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I have a doubt for this question

Does not this expression mean that x^2 is at a distance of 1 from 4. Then should not the correct answers be only sq root 5 and sq root 3 instead of -sq root 3 and -sq root 5 as these don't have a distance of 1 from 4 ?

Regards

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You said in each range signs of the terms will be different . I am unable to assign proper signs to terms in ranges VeritasKarishma please help

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Vini07

VeritasKarishma

rrsnathan

hi,

i have a doubt in the example

Q.: |x^2-4| = 1. What is x?
Solution: There are 2 conditions:

a) (x^2-4)\geq0

how the range for this example is (X^2-4)>=0 and not (x^2-4)>0?

regards,
RRSNATHAN

Vini07,

It means "x^2" is at a distance 1 from 4, not x.
So x^2 is 5 or 3. Then x can be -sqrt(5) and also sqrt(5) and -sqrt(3) and also sqrt(3)

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Updated on: 26 Nov 2023, 03:01

Originally posted by KarishmaB on 08 Jul 2020, 22:53.
Last edited by KarishmaB on 26 Nov 2023, 03:01, edited 1 time in total.

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Syeedarimshah

You said in each range signs of the terms will be different . I am unable to assign proper signs to terms in ranges VeritasKarishma please help

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The rightmost region is positive and you alternate from there (assuming the factors are in (ax + b) or (ax - b) form). Check out the logic explained in this video:

https://youtu.be/PWsUOe77__E

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walker

ABSOLUTE VALUE

$|x|\geq0$

$|0|=0$

$|-x|=|x|$

$|x|+|y|\geq|x+y|$

$|x|\geq0$

Little typo walker Bunuel. Hope it helps

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Fixed. Thank you.

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worldogvictor

Hi,
In the original post, in the section GMAC books, there are some prob numbers stated. Can you please explain how they relate to the post? How one will use those numbers is what I am asking.

The given problem numbers are official questions on the concept of Modulus.
You need to have those books to be able to access the given questions. The numbers give you the question numbers e.g. in OG12 in sample problem solving questions, question no 22 (on page 155) tests you on Mods.

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yangsta8

Hi Walker, Thanks for posting this.

You've written a property that:
|X + Y| >= |X| + |Y|
Is the same true for negative?
|X - Y| <= |X| - |Y|

No, the comparison sign gets reversed in |X-Y|

Ex,
|X+Y| <= |X|+|Y| - Less than equals

|X-Y| >= |X|-|Y| - Greater than equals

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