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

 It is currently 27 May 2015, 13:08

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Math: Absolute value (Modulus)

Author Message
TAGS:
CEO
Joined: 17 Nov 2007
Posts: 3577
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 408

Kudos [?]: 2175 [70] , given: 359

Math: Absolute value (Modulus) [#permalink]  06 Nov 2009, 18:49
70
KUDOS
Expert's post
73
This post was
BOOKMARKED
ABSOLUTE VALUE
(Modulus)

This post is a part of [GMAT MATH BOOK]

created by: walker
edited by: bb, Bunuel

--------------------------------------------------------
Get The Official GMAT Club's App - GMAT TOOLKIT 2.
The only app you need to get 700+ score!

[iOS App] [Android App]

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

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:

$$|x|\geq0$$

$$|0|=0$$

$$|-x|=|x|$$

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

$$|x|\geq0$$

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 $$(x-1)<0$$, 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.

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?

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]

A lot of questions as well as a separate topic of PrepGame on absolute value is included in GMAT ToolKit 2

--------------------------------------------------------
Get The Official GMAT Club's App - GMAT TOOLKIT 2.
The only app you need to get 700+ score!

[iOS App] [Android App]

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

[Reveal] Spoiler: Images
Attachment:

lineAXYZ.png [ 8.44 KiB | Viewed 103414 times ]
Attachment:

line1x9.png [ 4.07 KiB | Viewed 85969 times ]
Attachment:

graph_modulus.png [ 7.61 KiB | Viewed 87809 times ]
Attachment:

Math_icon_absolute_value.png [ 1.78 KiB | Viewed 84482 times ]
Attachment:

Math_abs_example1.png [ 5.37 KiB | Viewed 84673 times ]
Attachment:

Math_abs_example2.png [ 3.67 KiB | Viewed 84600 times ]
Attachment:

Math_abs_example0.png [ 3.19 KiB | Viewed 84680 times ]

_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | PrepGame

Last edited by walker on 02 Jun 2013, 14:01, edited 27 times in total.
 Kaplan GMAT Prep Discount Codes Knewton GMAT Discount Codes GMAT Pill GMAT Discount Codes
Retired Moderator
Status: The last round
Joined: 18 Jun 2009
Posts: 1317
Concentration: Strategy, General Management
GMAT 1: 680 Q48 V34
Followers: 64

Kudos [?]: 600 [5] , given: 157

Re: Math: Absolute value (Modulus) [#permalink]  06 Nov 2009, 20:02
5
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: 330
Followers: 8

Kudos [?]: 200 [2] , given: 13

Re: Math: Absolute value (Modulus) [#permalink]  06 Nov 2009, 20:06
2
KUDOS
Great post walker!!! + 1 to you.

PS. I have just made 2 corrections above.
CEO
Joined: 17 Nov 2007
Posts: 3577
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 408

Kudos [?]: 2175 [1] , given: 359

Re: Math: Absolute value (Modulus) [#permalink]  06 Nov 2009, 20:26
1
KUDOS
Expert's post
hgp2k wrote:
PS. I have just made 2 corrections above.

+1
Thanks
_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | PrepGame

CEO
Joined: 17 Nov 2007
Posts: 3577
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 408

Kudos [?]: 2175 [1] , given: 359

Re: Math: Absolute value (Modulus) [#permalink]  06 Nov 2009, 20:34
1
KUDOS
Expert's post
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!

_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | PrepGame

CEO
Joined: 15 Aug 2003
Posts: 3467
Followers: 61

Kudos [?]: 705 [0], given: 781

Re: Math: Absolute value (Modulus) [#permalink]  07 Nov 2009, 16:06
kudos!
Senior Manager
Joined: 31 Aug 2009
Posts: 420
Location: Sydney, Australia
Followers: 6

Kudos [?]: 147 [0], given: 20

Re: Math: Absolute value (Modulus) [#permalink]  13 Nov 2009, 22:07
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: 3577
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 408

Kudos [?]: 2175 [2] , given: 359

Re: Math: Absolute value (Modulus) [#permalink]  14 Nov 2009, 03:53
2
KUDOS
Expert's post
1
This post was
BOOKMARKED
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|
_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | PrepGame

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

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

Re: Math: Absolute value (Modulus) [#permalink]  13 Dec 2009, 04:40
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: 13 Dec 2009
Posts: 79
Followers: 1

Kudos [?]: 33 [1] , given: 20

Re: Math: Absolute value (Modulus) [#permalink]  15 Dec 2009, 22:45
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: 3577
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 408

Kudos [?]: 2175 [0], given: 359

Re: Math: Absolute value (Modulus) [#permalink]  15 Dec 2009, 23:08
Expert's post
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
_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | PrepGame

Manager
Joined: 24 Jul 2009
Posts: 193
Location: Anchorage, AK
Schools: Mellon, USC, MIT, UCLA, NSCU
Followers: 4

Kudos [?]: 24 [0], given: 10

Re: Math: Absolute value (Modulus) [#permalink]  21 Dec 2009, 20:49
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 [?]: 12 [0], given: 3

Re: Math: Absolute value (Modulus) [#permalink]  23 Dec 2009, 19:42
This is exactly what I was looking for past few days. Great Post.
Intern
Joined: 21 Nov 2009
Posts: 30
Location: London
Followers: 0

Kudos [?]: 4 [0], given: 9

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 03:26
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: 28
Followers: 0

Kudos [?]: 11 [0], given: 6

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 06:55
wonderful post walker
kudos..
_________________

Deserve before you Desire

Intern
Joined: 22 Dec 2009
Posts: 28
Followers: 0

Kudos [?]: 11 [0], given: 6

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 07:05
1
This post was
BOOKMARKED
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: 3577
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 408

Kudos [?]: 2175 [0], given: 359

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 09:22
Expert's post

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: $$-\sqrt{5}$$, $$-\sqrt{3}$$, $$\sqrt{3}$$, $$\sqrt{5}$$
_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | PrepGame

CEO
Joined: 17 Nov 2007
Posts: 3577
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 408

Kudos [?]: 2175 [1] , given: 359

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 10:32
1
KUDOS
Expert's post
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
_________________

HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | PrepGame

Intern
Joined: 21 Nov 2009
Posts: 30
Location: London
Followers: 0

Kudos [?]: 4 [0], given: 9

Re: Math: Absolute value (Modulus) [#permalink]  27 Dec 2009, 11:16
Many thanks for the illustration. This is now made clear. Kudos!
Manager
Joined: 05 Mar 2010
Posts: 218
Followers: 1

Kudos [?]: 25 [0], given: 8

Re: Math: Absolute value (Modulus) [#permalink]  26 Apr 2010, 04:56
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, 04:56

Go to page    1   2   3   4   5   6   7    Next  [ 135 posts ]

Similar topics Replies Last post
Similar
Topics:
absolute value modulus from math book 4 22 Jun 2010, 14:33
Absolute values 2 03 Sep 2007, 00:40
absolute value 4 11 Oct 2006, 22:10
Absolute Value? 4 29 May 2006, 22:54
Absolut Valu 5 23 Oct 2005, 06:34
Display posts from previous: Sort by