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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).

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.
_________________

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

would someone please explain it ?

Thanks in advance.

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.

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.
_________________

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).

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).

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