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Math Expert V
Joined: 02 Sep 2009
Posts: 60646
Absolute Value: Tips and hints  [#permalink]

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80

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

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5
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?
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 60646
Re: Absolute Value: Tips and hints  [#permalink]

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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.
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Absolute Value: Tips and hints  [#permalink]

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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|
Intern  Joined: 01 Nov 2015
Posts: 7
Re: Absolute Value: Tips and hints  [#permalink]

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Hi, I've some problem with the graph y=|x| [figure in original post]

would someone please explain it ?

Math Expert V
Joined: 02 Sep 2009
Posts: 60646
Re: Absolute Value: Tips and hints  [#permalink]

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1
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.
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Re: Absolute Value: Tips and hints  [#permalink]

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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 18916 times ]

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Math Expert V
Joined: 02 Sep 2009
Posts: 60646
Re: Absolute Value: Tips and hints  [#permalink]

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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
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