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 \(|x| \geq |y|\)


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