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Bunuel
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Absolute Value: Tips and hints 09 Jul 2014, 07:04
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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|



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 when \(x \leq y \leq 0\) or when \(0 \leq y \leq x\). So, essentially when \(xy \geq 0\) AND \(|x| \geq |y|\)


Please share your Absolute Value properties tips below and get kudos point. Thank you.
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Absolute Value: Tips and hints 19 Jul 2014, 09:29
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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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Absolute Value: Tips and hints 19 Jul 2014, 09:34
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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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Yes. Edited. Thank you.
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Absolute Value: Tips and hints 04 Nov 2015, 17:55
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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|
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Absolute Value: Tips and hints 05 Nov 2015, 00:51
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Hi, I've some problem with the graph y=|x| [figure in original post]

would someone please explain it ?

Thanks in advance.
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Absolute Value: Tips and hints 05 Nov 2015, 01:25
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suravi
Hi, I've some problem with the graph y=|x| [figure in original post]

would someone please explain it ?

Thanks in advance.
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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.
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Absolute Value: Tips and hints 12 Feb 2017, 20:41
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Bunuel

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.
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Dear Bunuel, How to prove the highlighted inequalities property in yellow?
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Absolute Value: Tips and hints 13 Feb 2017, 02:10
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ziyuenlau
Bunuel

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?
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Try number plugging.
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