Absolute Value

The concept of absolute value has many uses, but you probably won't see anything interesting for a few more classes yet.

There is a technical definition for absolute value, but you could easily never need it. For now, you should view the absolute value of a number as its distance from zero.

Let's look at the number line:

The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?", not "in which direction?" This means not only that | 3 | = 3, because 3 is three units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.

Warning: The absolute-value notation is bars, not parentheses or brackets. Use the proper notation; the other notations do not mean the same thing.

It is important to note that the absolute value bars do NOT work in the same way as do parentheses. Whereas –(–3) = +3, this is NOT how it works for absolute value:

• Simplify –| –3 |.

Given –| –3 |, I first handle the absolute value part, taking the positive and converting the absolute value bars to parentheses:

–| –3 | = –(+3)

Now I can take the negative through the parentheses:

–| –3 | = –(3) = –3

As this illustrates, if you take the negative of an absolute value, you will get a negative number for your answer.

When typing math as text, such as in an e-mail, the "pipe" character is usually used to indicate absolute values. The "pipe" is probably a shift-key somewhere north of the "Enter" key on your keyboard. While the "pipe" denoted on the physical keyboard key may look like a "broken" line, the typed character should display on your screen as a solid vertical bar. If you cannot locate a "pipe" character, you can use the "abs()" notation instead, so that "the absolute value of negative 3" would be typed as "abs(–3)".

Here are some more sample simplifications:

• Simplify | –8 |.

• Simplify | 0 – 6 |.

| 0 – 6 | = | –6 | = 6

• Simplify | 5 – 2 |.
• | 5 – 2 | = | 3 | = 3

• Simplify | 2 – 5 |.
• | 2 – 5 | = | –3 | = 3

• Simplify | 0(–4) |.
• | 0(–4) | = | 0 | = 0

Why is the absolute value of zero equal to "0"? Ask yourself: How far is zero from 0? Zero units, right? So | 0 | = 0.

• Simplify | 2 + 3(–4) |.
• | 2 + 3(–4) | = | 2 – 12 | = | –10 | = 10

• Simplify –| –4 |.
• –| –4| = –(4) = –4

• Simplify –| (–2)2 |.
• –| (–2)2 | = –| 4 | = –4

• Simplify –| –2 |2
• –| –2 |2 = –(2)2 = –(4) = –4

• Simplify (–| –2 |)2.
• (–| –2 |)2 = (–(2))2 = (–2)2 = 4

Sometimes you will be asked to insert an inequality sign between two absolute values, such as:

• Insert the correct inequality:  | –4 | _____ | –7 |

Whereas –4 > –7 (because it is further to the right on the number line than is –7), I am dealing here with the absolute values. Since:

| –4 | = 4

| –7 | = 7,

...and since 4 < 7, then the solution is:

| –4 |    <   | –7 |.

When the number inside the absolute value (the "argument" of the absolute value) was positive anyway, we didn't change the sign when we took the absolute value. But when the argument was negative, we did change the sign; namely, we changed the "understood" "plus" into a "minus". This leads to one fiddly point which may not come up in your homework now, but will probably show up on tests later:

When you are dealing with variables, you cannot tell the sign of the number or the value that is contained in the variable. For instance, given the variable x, you cannot tell by looking whether there is, say, a "2" or a "–4" contained inside. If I ask you for the absolute value of x, what would you do? Since you cannot tell, just by looking at the letter, whether or not the variable contains a positive or negative value, you would have to consider these two different cases.

If x > 0 (that is, if x is positive), then the value won't change when you take the absolute value. For instance, if x = 2, then you have | x | = | 2 | = 2 = x. In fact, for any positive value of x (or if x equals zero), the sign would be unchanged, so:

For x > 0, | x | = x

On the other hand, if x < 0 (that is, if x is negative), then it will change its sign when you take the absolute value. For instance, if x = –4, then | x | = | –4 | = + 4 = –(–4) = –x. In fact, for any negative value of x, the sign would have to be changed, so:

For x < 0, | x | = –x

This is a case in which the "minus" sign on the variable does not indicate "a number to the left of zero", but "a change of the sign from whatever the sign originally was". This "–" does not mean "the number is negative" but instead means that "I've changed the sign on the original value".

• Must x be negative? Why or why not?
• No, it does not have to be negative:
If the original value of
x was negative, then x, the opposite-signed version of x, would have to be positive. For instance, if I start with x = –3, then x = –(–3) = +3, which is positive.

You can use the Mathway widget below to practice simplifying an absolute-value expression. Try the entered exercise, or type in your own exercise. Then click the "paper-airplane" button to compare your answer to Mathway's.

(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)