Absolute Value in Algebra

This symbol--> |x| represents the absolute value of x, which is the number without its sign. |+3| = 3. |−3| = 3.

Geometrically, |x| is the distance of x from 0.

Both 3 and −3 are a distance of 3 units from 0. |3| = |−3| = 3. Distance, in mathematics, is never negative.


The absolute value leaves a positive unchanged, and makes a negative positive
An absolute value is written like this: |x|, and is read as "the absolute value of x." Note: In certain places, such as calculator and computer programs, you may see it written as abs(x), which naturally means "the absolute value of x," but |x| is the accepted way to write it on your homework and tests.

To force a number to be negative, you can write -|x|. This takes the number, makes it positive, and then negates it. Remember -- just putting a negative sign in front of a number doesn't make it negative. If the number was already negative then you just made it positive! Using the absolute value guarantees we have a positive value so that the negative sign will definitely make it negative.



EXAMPLE:

Rewrite this expression without using absolute value notation:

1) |x+1|+4|x+3| given that x<-3

If x < –3 then x+ 1< 0 and x+ 3 <0. In general, when a<0 |a| = –a, thus

|x+1|+ 4|x+3| =

–(x+1) – 4(x +3) provided x < –3 or simplifying ...

–x – 1 – 4x – 12 =

–5x – 13 when x < –3

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