Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 24 May 2013, 04:56

# Math: Absolute value (Modulus)

Author Message
TAGS:
CEO
Joined: 17 Nov 2007
Posts: 3596
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 231

Kudos [?]: 1300 [39] , given: 346

Math: Absolute value (Modulus) [#permalink]  06 Nov 2009, 19:49
39
KUDOS
ABSOLUTE VALUE
(Modulus)

This post is a part of [GMAT MATH BOOK]

created by: walker
edited by: bb, Bunuel

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

This topic is included in GMAT ToolKit App (iPhone/iPod Touch)

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

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:

Important properties:

|x|\geq0

|x| = \sqrt{x^2}

|0|=0

|-x|=|x|

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

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)<0, we can rewrite the equation as: -(x-1)=4
We can also think about conditions like graphics. x=1 is a key point in which the expression under modulus equals zero. All points right are the first condition (x>1) and all points left are second condition (x<1).

2. Solve new equations:
a) x-1=4 --> 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<0. -3-1=-4<0. It satisfies. Otherwise, we would have to reject x=-3.

3-steps approach for complex problems

Let’s consider following examples,

Example #1
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) = -(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)

(Optional) The following illustration may help you understand how to open modulus at different conditions.

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)<0 --> -2 < x < 2. -(x^2-4) = 1 --> 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.

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?

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.

II. Converting inequalities with modulus into a range expression.
In many cases, especially in DS problems, it helps avoid silly mistakes.

For example,
|x|<5 is equal to x e (-5,5).
|x+3|>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:

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| < |Y-A| and at the same time the distance between X and B will be larger than that between Y and B (|B-Y|<|B-X|). Therefore, statement 1 is sufficient.

Pitfalls

The most typical pitfall is ignoring the third step in opening modulus - always check whether your solution satisfies conditions.

Official GMAC Books:

The Official Guide, 12th Edition: PS #22; PS #50; PS #130; DS #1; DS #153;
The Official Guide, Quantitative 2th Edition: PS #152; PS #156; DS #96; DS #120;
The Official Guide, 11th Edition: DT #9; PS #20; PS #130; DS #3; DS #105; DS #128;

Generated from [GMAT ToolKit]

Resources

Absolute value DS problems: [search]
Absolute value PS problems: [search]

Fig's post with absolute value problems: [Absolute Value Problems]

------------------------------------------------
[Reveal] Spoiler: Images
Attachment:

lineAXYZ.png [ 8.44 KiB | Viewed 54738 times ]
Attachment:

line1x9.png [ 4.07 KiB | Viewed 45505 times ]
Attachment:

graph_modulus.png [ 7.61 KiB | Viewed 45636 times ]
Attachment:

Math_icon_absolute_value.png [ 1.78 KiB | Viewed 44172 times ]
Attachment:

Math_abs_example1.png [ 5.37 KiB | Viewed 44203 times ]
Attachment:

Math_abs_example2.png [ 3.67 KiB | Viewed 44119 times ]
Attachment:

Math_abs_example0.png [ 3.19 KiB | Viewed 44194 times ]

_________________

iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit
PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for only $0.99 (read more) Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do. Last edited by walker on 17 May 2012, 07:01, edited 26 times in total. added a new example + illustrations  Kaplan Promo Code Knewton GMAT Discount Codes Veritas Prep GMAT Discount Codes VP Status: The last round Joined: 18 Jun 2009 Posts: 1327 Concentration: Strategy, General Management GMAT 1: 680 Q48 V34 Followers: 43 Kudos [?]: 383 [3] , given: 156 Re: Math: Absolute value (Modulus) [#permalink] 06 Nov 2009, 21:02 3 This post received KUDOS Great post!! Kudos!! If some problems links with categorization (like 700 level, 600-700 level) are posted, it will be great. I was in search of range approach for solving modules problems, I got some stuff here. Thanks again! _________________ Senior Manager Joined: 18 Aug 2009 Posts: 340 Followers: 5 Kudos [?]: 120 [2] , given: 13 Re: Math: Absolute value (Modulus) [#permalink] 06 Nov 2009, 21:06 2 This post received KUDOS Great post walker!!! + 1 to you. PS. I have just made 2 corrections above. CEO Joined: 17 Nov 2007 Posts: 3596 Concentration: Entrepreneurship, Other Schools: Chicago (Booth) - Class of 2011 GMAT 1: 750 Q50 V40 Followers: 231 Kudos [?]: 1300 [1] , given: 346 Re: Math: Absolute value (Modulus) [#permalink] 06 Nov 2009, 21:26 1 This post received KUDOS hgp2k wrote: PS. I have just made 2 corrections above. +1 Thanks _________________ iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for only$0.99 (read more)
Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation
The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do.

CEO
Joined: 17 Nov 2007
Posts: 3596
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 231

Kudos [?]: 1300 [1] , given: 346

Re: Math: Absolute value (Modulus) [#permalink]  06 Nov 2009, 21:34
1
KUDOS
Hussain15 wrote:
If some problems links with categorization (like 700 level, 600-700 level) are posted, it will be great. I was in search of range approach for solving modules problems, I got some stuff here. Thanks again!

_________________

iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit
PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for only $0.99 (read more) Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do. CEO Joined: 15 Aug 2003 Posts: 3550 Followers: 55 Kudos [?]: 626 [0], given: 781 Re: Math: Absolute value (Modulus) [#permalink] 07 Nov 2009, 17:06 kudos! Senior Manager Joined: 31 Aug 2009 Posts: 426 Location: Sydney, Australia Followers: 4 Kudos [?]: 76 [0], given: 20 Re: Math: Absolute value (Modulus) [#permalink] 13 Nov 2009, 23:07 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| CEO Joined: 17 Nov 2007 Posts: 3596 Concentration: Entrepreneurship, Other Schools: Chicago (Booth) - Class of 2011 GMAT 1: 750 Q50 V40 Followers: 231 Kudos [?]: 1300 [1] , given: 346 Re: Math: Absolute value (Modulus) [#permalink] 14 Nov 2009, 04:53 1 This post received KUDOS yangsta8 wrote: 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| |X + Y| <= |X| + |Y| |X - Y| >=|X| - |Y| _________________ iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for only$0.99 (read more)
Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation
The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do.

Intern
Joined: 30 Aug 2009
Posts: 26
Followers: 0

Kudos [?]: 2 [1] , given: 3

Re: Math: Absolute value (Modulus) [#permalink]  13 Dec 2009, 05:40
1
KUDOS
Not that it matters in any way, but can you correct the "witch" under 1b in the initial post. Other than that: awesome post!
+1
Manager
Joined: 14 Dec 2009
Posts: 84
Followers: 1

Kudos [?]: 27 [1] , given: 20

Re: Math: Absolute value (Modulus) [#permalink]  15 Dec 2009, 23:45
1
KUDOS
Hi Walker!
I found a mistake in your expl.
In 3-step approach for more than one module: (d) is not -9, it's -1
Correct me if I'm wrong.
Thank you.
CEO
Joined: 17 Nov 2007
Posts: 3596
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 231

Kudos [?]: 1300 [0], given: 346

Re: Math: Absolute value (Modulus) [#permalink]  16 Dec 2009, 00:08
Igor010 wrote:
Hi Walker!
I found a mistake in your expl.
In 3-step approach for more than one module: (d) is not -9, it's -1
Correct me if I'm wrong.
Thank you.

Thanks! You are right
_________________

iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit
PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for only $0.99 (read more) Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do. Manager Joined: 24 Jul 2009 Posts: 197 Location: Anchorage, AK Schools: Mellon, USC, MIT, UCLA, NSCU Followers: 4 Kudos [?]: 17 [0], given: 10 Re: Math: Absolute value (Modulus) [#permalink] 21 Dec 2009, 21:49 Thank you! Thank you! Thank you! This is the clearest explanation of absolute value that I've come across thus far. _________________ Reward wisdom with kudos. Manager Joined: 21 Jul 2003 Posts: 68 Followers: 2 Kudos [?]: 5 [0], given: 3 Re: Math: Absolute value (Modulus) [#permalink] 23 Dec 2009, 20:42 This is exactly what I was looking for past few days. Great Post. Intern Joined: 21 Nov 2009 Posts: 33 Location: London Followers: 0 Kudos [?]: 2 [0], given: 9 Re: Math: Absolute value (Modulus) [#permalink] 27 Dec 2009, 04:26 Sorry, but please clarify why is it not... a) x < -8. -(x+3) + (4-x) = -(8+x) The -ve sign is given in the question stem, once we take the solution -(4-x) with the -ve sign in the equation, then two -ves should be positive, shouldn't it? Intern Joined: 22 Dec 2009 Posts: 29 Followers: 0 Kudos [?]: 8 [0], given: 6 Re: Math: Absolute value (Modulus) [#permalink] 27 Dec 2009, 07:55 wonderful post walker kudos.. _________________ Deserve before you Desire Intern Joined: 22 Dec 2009 Posts: 29 Followers: 0 Kudos [?]: 8 [0], given: 6 Re: Math: Absolute value (Modulus) [#permalink] 27 Dec 2009, 08:05 arjunrampal wrote: Sorry, but please clarify why is it not... a) x < -8. -(x+3) + (4-x) = -(8+x) The -ve sign is given in the question stem, once we take the solution -(4-x) with the -ve sign in the equation, then two -ves should be positive, shouldn't it? hai arjunrampal.. when x< -8, (x+3) and (8+x) are both negative, while (4-x) is positive. (if you want to confirm this, u can plug in values for x and try) when x<0 |x| = -x ( i.e. |x| = - (-ve x), ultimately modulus x is positive. i hope this point is clear) when x>0 |x| = x so for x< -8, |(4-x)| remains positive while modulus of the other two expressions become negative. hope it is clear... _________________ Deserve before you Desire CEO Joined: 17 Nov 2007 Posts: 3596 Concentration: Entrepreneurship, Other Schools: Chicago (Booth) - Class of 2011 GMAT 1: 750 Q50 V40 Followers: 231 Kudos [?]: 1300 [0], given: 346 Re: Math: Absolute value (Modulus) [#permalink] 27 Dec 2009, 10:22 I've added a new example: 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)<0 --> -2 < x < 2. -(x^2-4) = 1 --> x^2 = 3. x e {-\sqrt{3}, \sqrt{3}} and both solutions satisfy the condition. Answer: -\sqrt{5}, -\sqrt{3}, \sqrt{3}, \sqrt{5} _________________ iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for only$0.99 (read more)
Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation
The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do.

CEO
Joined: 17 Nov 2007
Posts: 3596
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 231

Kudos [?]: 1300 [1] , given: 346

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 11:32
1
KUDOS
arjunrampal wrote:
Sorry, but please clarify why is it not...
a) x < -8. -(x+3) + (4-x) = -(8+x)

The -ve sign is given in the question stem, once we take the solution -(4-x) with the -ve sign in the equation, then two -ves should be positive, shouldn't it?

Thanks for the question. I've added illustration to the example as well as one more example and hope they help
_________________

iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit
PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for only \$0.99 (read more)
Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation
The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do.

Intern
Joined: 21 Nov 2009
Posts: 33
Location: London
Followers: 0

Kudos [?]: 2 [0], given: 9

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 12:16
Many thanks for the illustration. This is now made clear. Kudos!
Manager
Joined: 05 Mar 2010
Posts: 220
Followers: 1

Kudos [?]: 15 [0], given: 8

Re: Math: Absolute value (Modulus) [#permalink]  26 Apr 2010, 05:56
Thank you walker. This post helped me understand the concepts of absolute value
+1
_________________

Success is my Destiny

Re: Math: Absolute value (Modulus)   [#permalink] 26 Apr 2010, 05:56
Similar topics Replies Last post
Similar
Topics:
Absolut Valu 5 23 Oct 2005, 07:34
Absolute Value? 4 29 May 2006, 23:54
absolute value 4 11 Oct 2006, 23:10
Absolute values 2 03 Sep 2007, 01:40
absolute value modulus from math book 4 22 Jun 2010, 15:33
Display posts from previous: Sort by