Absolute Value

Algebra 1 Tutorial

Absolute Value

Intro

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. (If you go as far as calculus, the technical definition might come up.) For now, you should view the absolute value of a number as being the distance, on the number line, of that number from zero.

Since we’ll be using the intuitive, number-line-based definition, let’s get started by looking at the number line:

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. You can see this on the following number line:

abs(-3) = abs(3) = 3
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 need to handle the absolute-value part, taking the positive of the insides (the “argument of” the absolute value) and then 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 (that is, if you have a “minus” sign in front of the absolute-value bars), you will get a negative number for your answer.

Sample Problem

Simplify | –8 |.
| –8 | = 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

In the next three examples, pay particular attention to the difference that the location of the square makes, with respect to the “minus” signs.

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

Solution

Simplify | –8 |.
| –8 | = 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

In the next three examples, pay particular attention to the difference that the location of the square makes, with respect to the “minus” signs.

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



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