Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 25 May 2017, 14:29

### 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

# Absolute Value: Tips and hints

 post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 38880
Followers: 7733

Kudos [?]: 106124 [2] , given: 11607

Absolute Value: Tips and hints [#permalink]

### Show Tags

09 Jul 2014, 07:04
2
KUDOS
Expert's post
36
This post was
BOOKMARKED

Absolute Value: Tips and hints

 ! This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

DEFINITION

The absolute value of a number is the value of a number without regard to its sign.

For example, $$|3| = 3$$; $$|-12| = 12$$; $$|-1.3|=1.3$$...

Another way to understand absolute value is as the distance from zero. For example, $$|x|$$ is the distance between x and 0 on a number line.

From that comes the most important property of an absolute value: since the distance cannot be negative, an absolute value expression is ALWAYS more than or equal to zero.

GRAPH OF y=|x|

As you can see for any value of x, the value of y, which is |x|, is ALWAYS more than or equal to zero.

IMPORTANT PROPERTY

When $$x\leq{0}$$, then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$. Notice that in the negative scenario, we don't simply remove the absolute value bars. We remove the absolute value bars and negate the entire expression contained within, thus making it positive again;

When $$x\geq{0}$$, then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$.

OTHER IMPORTANT PROPERTIES

1. $$|x|\geq0$$

2. $$\sqrt{x^2}=|x|$$

3. $$|0|=0$$

4. $$|-x|=|x|$$

5. $$|x-y|=|y-x|$$. |x - y| represents the distance between x and y, so naturally it equals to |y - x|, which is the distance between y and x.

6. $$|x|+|y|\geq|x+y|$$. Note that "=" sign holds for $$xy\geq{0}$$ (or simply when $$x$$ and $$y$$ have the same sign). So, the strict inequality (>) holds when $$xy<0$$;

7. $$|x|-|y|\leq{|x-y|}$$. Note that "=" sign holds for $$xy>{0}$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|\geq{|y|}$$ (simultaneously).

Please share your Absolute Value properties tips below and get kudos point. Thank you.
_________________
Current Student
Joined: 22 Jan 2014
Posts: 46
GPA: 4
Followers: 0

Kudos [?]: 12 [1] , given: 3

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

19 Jul 2014, 09:29
1
KUDOS
7. $$|x|-|y|\leq{|x-y|}$$. Note that "=" sign holds for $$xy>{0}$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|>|y|$$(simultaneously).

For the 7th rule - shouldn't the "=" sign hold when xy>0 and |x|>=|y| (as opposed to just |x|>|y|)? What am I missing?
Math Expert
Joined: 02 Sep 2009
Posts: 38880
Followers: 7733

Kudos [?]: 106124 [0], given: 11607

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

19 Jul 2014, 09:34
rog45 wrote:
7. $$|x|-|y|\leq{|x-y|}$$. Note that "=" sign holds for $$xy>{0}$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|>|y|$$(simultaneously).

For the 7th rule - shouldn't the "=" sign hold when xy>0 and |x|>=|y| (as opposed to just |x|>|y|)? What am I missing?

Yes. Edited. Thank you.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15444
Followers: 649

Kudos [?]: 209 [0], given: 0

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

11 Oct 2015, 11:59
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Manager
Status: Just redeemed Kudos for GMAT Club Test !!
Joined: 14 Sep 2013
Posts: 95
GMAT 1: 530 Q40 V23
GPA: 3.56
WE: Analyst (Commercial Banking)
Followers: 2

Kudos [?]: 14 [0], given: 34

Absolute Value: Tips and hints [#permalink]

### Show Tags

04 Nov 2015, 17:55
Multiplicative properties

8. Module of the product of two (or more) numbers is equal to the product of their modules: |a . b| = |a| . |b|

9. Constant positive factor can be taken out of the module sign: |c . x| = c . |x|
_________________

______________
KUDOS please, if you like the post or if it helps
"Giving kudos" is a decent way to say "Thanks"

Master with structure - Numerical comparison [source: economist.com] https://gmatclub.com/forum/master-with-structure-numerical-comparison-233657.html#p1801987

Intern
Joined: 01 Nov 2015
Posts: 7
Followers: 0

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

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

05 Nov 2015, 00:51
Hi, I've some problem with the graph y=|x| [figure in original post]

would someone please explain it ?

Math Expert
Joined: 02 Sep 2009
Posts: 38880
Followers: 7733

Kudos [?]: 106124 [1] , given: 11607

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

05 Nov 2015, 01:25
1
KUDOS
Expert's post
suravi wrote:
Hi, I've some problem with the graph y=|x| [figure in original post]

would someone please explain it ?

Plug values for x and you get the values of y (|x|). For example, if x=-1, then y=|x|=1, if x=2, then y=|x|=2, if x=0, then y=|x|=0, ... As you can see no matter whether x is negative, positive or 0, the value of y (|x|) will come up as positive or 0, but never as negative. Hence the position of the graph above the x-axis, where only positive y's lie.

Hope it's clear.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15444
Followers: 649

Kudos [?]: 209 [0], given: 0

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

02 Dec 2016, 00:23
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Director
Joined: 14 Nov 2016
Posts: 876
Location: Malaysia
Followers: 24

Kudos [?]: 456 [0], given: 148

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

12 Feb 2017, 20:41
Bunuel wrote:

Absolute Value: Tips and hints

 ! This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

DEFINITION

The absolute value of a number is the value of a number without regard to its sign.

For example, $$|3| = 3$$; $$|-12| = 12$$; $$|-1.3|=1.3$$...

Another way to understand absolute value is as the distance from zero. For example, $$|x|$$ is the distance between x and 0 on a number line.

From that comes the most important property of an absolute value: since the distance cannot be negative, an absolute value expression is ALWAYS more than or equal to zero.

GRAPH OF y=|x|

As you can see for any value of x, the value of y, which is |x|, is ALWAYS more than or equal to zero.

IMPORTANT PROPERTY

When $$x\leq{0}$$, then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$. Notice that in the negative scenario, we don't simply remove the absolute value bars. We remove the absolute value bars and negate the entire expression contained within, thus making it positive again;

When $$x\geq{0}$$, then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$.

OTHER IMPORTANT PROPERTIES

1. $$|x|\geq0$$

2. $$\sqrt{x^2}=|x|$$

3. $$|0|=0$$

4. $$|-x|=|x|$$

5. $$|x-y|=|y-x|$$. |x - y| represents the distance between x and y, so naturally it equals to |y - x|, which is the distance between y and x.

6. $$|x|+|y|\geq|x+y|$$. Note that "=" sign holds for $$xy\geq{0}$$ (or simply when $$x$$ and $$y$$ have the same sign). So, the strict inequality (>) holds when $$xy<0$$;

7. $$|x|-|y|\leq{|x-y|}$$. Note that "=" sign holds for $$xy>{0}$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|\geq{|y|}$$ (simultaneously).

Please share your Absolute Value properties tips below and get kudos point. Thank you.

Dear Bunuel, How to prove the highlighted inequalities property in yellow?
Attachments

Untitled.jpg [ 25.43 KiB | Viewed 885 times ]

Math Expert
Joined: 02 Sep 2009
Posts: 38880
Followers: 7733

Kudos [?]: 106124 [0], given: 11607

Re: Absolute Value: Tips and hints [#permalink]

### Show Tags

13 Feb 2017, 02:10
ziyuenlau wrote:
Bunuel wrote:

Absolute Value: Tips and hints

 ! This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

DEFINITION

The absolute value of a number is the value of a number without regard to its sign.

For example, $$|3| = 3$$; $$|-12| = 12$$; $$|-1.3|=1.3$$...

Another way to understand absolute value is as the distance from zero. For example, $$|x|$$ is the distance between x and 0 on a number line.

From that comes the most important property of an absolute value: since the distance cannot be negative, an absolute value expression is ALWAYS more than or equal to zero.

GRAPH OF y=|x|

As you can see for any value of x, the value of y, which is |x|, is ALWAYS more than or equal to zero.

IMPORTANT PROPERTY

When $$x\leq{0}$$, then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$. Notice that in the negative scenario, we don't simply remove the absolute value bars. We remove the absolute value bars and negate the entire expression contained within, thus making it positive again;

When $$x\geq{0}$$, then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$.

OTHER IMPORTANT PROPERTIES

1. $$|x|\geq0$$

2. $$\sqrt{x^2}=|x|$$

3. $$|0|=0$$

4. $$|-x|=|x|$$

5. $$|x-y|=|y-x|$$. |x - y| represents the distance between x and y, so naturally it equals to |y - x|, which is the distance between y and x.

6. $$|x|+|y|\geq|x+y|$$. Note that "=" sign holds for $$xy\geq{0}$$ (or simply when $$x$$ and $$y$$ have the same sign). So, the strict inequality (>) holds when $$xy<0$$;

7. $$|x|-|y|\leq{|x-y|}$$. Note that "=" sign holds for $$xy>{0}$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|\geq{|y|}$$ (simultaneously).

Please share your Absolute Value properties tips below and get kudos point. Thank you.

Dear Bunuel, How to prove the highlighted inequalities property in yellow?

Try number plugging.
_________________
Re: Absolute Value: Tips and hints   [#permalink] 13 Feb 2017, 02:10
Similar topics Replies Last post
Similar
Topics:
21 Algebra: Tips and hints 3 05 Nov 2016, 09:53
84 Inequalities: Tips and hints 2 06 Aug 2016, 00:46
16 Remainders: Tips and hints 2 22 Jan 2017, 13:29
65 Number Properties: Tips and hints 13 19 Jan 2017, 09:05
1 Absolute Value tips - a call for material 4 27 Aug 2014, 07:03
Display posts from previous: Sort by

# Absolute Value: Tips and hints

 post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.