Integers: Learn It 3

Absolute Value

We saw that numbers such as [latex]5[/latex] and [latex]-5[/latex] are opposites because they are the same distance from [latex]0[/latex] on the number line. They are both five units from [latex]0[/latex]. The distance between [latex]0[/latex] and any number on the number line is called the absolute value of that number.

Because distance is never negative, the absolute value of any number is never negative.

The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[/latex] is written as [latex]|5|[/latex], and the absolute value of [latex]-5[/latex] is written as [latex]|-5|[/latex] as shown below.

A number line, with the points negative 5 and 5 labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units and the distance from 0 to 5 is labeled as 5 units.

 

Absolute Value

The absolute value of a number is its distance from [latex]0[/latex] on the number line.

 

The absolute value of a number [latex]n[/latex] is written as [latex]|n|[/latex].

 

Absolute value can be never be negative so [latex]|n|\ge 0[/latex] for all numbers.

Comparing Absolute Values

Just as we did with positive and negative numbers, we can use inequality symbols to show the ordering of absolute values. It is important to know, we treat absolute value bars just like we treat parentheses in the order of operations – we simplify the expression inside first.

Order of operations is a set of rules in mathematics that dictates the sequence in which operations should be performed to correctly solve an expression. It is commonly remembered by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

For instance, in the expression [latex]8 + (4 * 2)^2 ÷ 4 - 2[/latex] using the order of operations we must:

  1. Parentheses/Brackets: Perform the operation inside the parentheses first.
    • This gives us: [latex]8 + (8)^2 ÷ 4 - 2[/latex]
  2. Exponents/Orders: Next, solve for the exponent.
    • This gives us: [latex]8 + 64 ÷ 4 - 2[/latex]
  3. Multiplication and Division: Perform multiplication and division operations from left to right.
    • This gives us: [latex]8 + 16 - 2[/latex]
  4. Addition and Subtraction: Finally, carry out addition and subtraction from left to right.
    • This gives us: [latex]24 - 2 = 22[/latex]

So, the result of the expression [latex]8 + (4 * 2)^2 ÷ 4 - 2[/latex] following the order of operations is [latex]22[/latex].

Fill in [latex]\text{< },\text{ > },\text {or }=[/latex] for each of the following:

  1. [latex]|-5|[/latex]   ___  [latex]-|-5|[/latex]
  2. [latex]8[/latex]  ___  [latex]-|-8|[/latex]
  3. [latex]-9[/latex]  ___  [latex]- |-9|[/latex]
  4. [latex]-|-7|[/latex]  ___  [latex]- 7[/latex]

Notice that the result is negative only when there is a negative sign outside the absolute value symbol.

Simplify Absolute Values

Absolute value bars act like grouping symbols. First, simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

Simplify:

  1. [latex]|9 - 3|[/latex]
  2. [latex]|8+7|-|5+6|[/latex]
  3. [latex]24-|19 - 3\left(6 - 2\right)|[/latex]