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Absolute Value: Tips and hints

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Absolute Value: Tips and hints [#permalink] New post 09 Jul 2014, 06:04
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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|

Image

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.
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Re: Absolute Value: Tips and hints [#permalink] New post 19 Jul 2014, 08:29
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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?
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Re: Absolute Value: Tips and hints [#permalink] New post 19 Jul 2014, 08:34
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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.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Re: Absolute Value: Tips and hints   [#permalink] 19 Jul 2014, 08:34
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