Simplify Expressions with Absolute Values

There are several ways in which we can characterise the absolute value function. 

Absolute value is used when we are concerned with the magnitude of a quantity and not with its sign. Thus, we can define the absolute value of a number to be the positive number that has the same magnitude as the number. The treatment of absolute values with numbers is described here, check that out to refresh.  

The notation $|x|$|x| is used for the absolute value of $x$x.

For example, $|3|=3$|3|=3 and $|-5|=5$|−5|=5.

We can define the function piecewise, as follows.

Alternatively, we can write $|f(x)|=\sqrt{\left(f(x)\right)^2}$|f(x)|=√(f(x))2. This works because the square root sign is always understood to give the positive square root.

 

Let's look at some simple cases, 

Case 1

Case 2

The expression $\left|2x\right|$|2x| can be simplified for values of $x$x less than $0$0, or greater than $0$0. 

So, the expression is equivalent to $2x$2x for $x\ge0$x≥0 and $-2x$−2x for $x<0$x<0

Case 3

 

Case 4

 

Properties

Combining absolute values by multiplication presents no difficulty. It is easy to check that $|a|\times|b|=|ab|$|a|×|b|=|ab| for any values of $a$a and $b$b.

However, absolute values that are to be added cannot be combined so simply. It is not generally true that $|a|+|b|=|a+b|$|a|+|b|=|a+b|. In fact, there is a different statement that can be made concerning addition, called the triangle inequality, which is: $|a+b|\le|a|+|b|$|a+b|≤|a|+|b|.

(Think of $a$a and $b$b as forming two sides of a triangle. The sum of any two sides is less in magnitude than the sum of the separate magnitudes.)

We can use the absolute value function to express the situation that a quantity is between two others. The statement $|x||x|<y is equivalent to the pair of inequalities $-y−y<x<y. Thus, if $|x|\le10$|x|≤10, then $-10\le x\le10$−10≤x≤10.

For example, to express the fact that $x$x is strictly between $-2$−2 and $6$6, we could begin by writing $-2−2<x<6. An equivalent but symmetrical statement would be $-4−4<x−2<4 and this can be written $|x-2|<4$|x−2|<4.