What if you had a number like . How could you find its opposite and its absolute value? After completing this Concept, you'll be able to find both values for any number.

### Watch This

CK-12 Foundation: 0202S Opposites and Absolute Values

### Guidance

Every number has an opposite. On the number line, a number and its opposite are, predictably, *opposite* each other. In other words, they are the same distance from zero, but on opposite sides of the number line. The opposite of zero is defined to be simply zero.

#### Example A

The sum of a number and its opposite is always zero, as shown in Example B.

#### Example B

The numbers 3 and -3 are opposites because:

The numbers 4.2 and -4.2 are opposites because:

This is because adding 3 and -3 is like moving 3 steps to the right along the number line, and then 3 steps back to the left. The number and its opposite cancel each other out, leaving zero.

Another way to think of the opposite of a number is that it is simply the original number multiplied by -1.

#### Example C

The opposite of 4 is or -4, and the opposite of -2.3 is or just 2.3.

Another term for the opposite of a number is the **additive inverse**.

#### Example D

*Find the opposite of each of the following:*

a) 19.6

b)

c)

d)

e)

**Solution**

Since we know that opposite numbers are on opposite sides of zero, we can simply multiply each expression by -1. This changes the sign of the number to its opposite—if it’s negative, it becomes positive, and vice versa.

a) The opposite of 19.6 is -19.6.

b) The opposite of is is .

c) The opposite of is .

d) The opposite of is .

e) The opposite of is , or .

**Note:** With the last example you must multiply the **entire expression** by -1. A common mistake in this example is to assume that the opposite of is . Avoid this mistake!

**Find Absolute Values**

When we talk about absolute value, we are talking about distances on the number line. For example, the number 7 is 7 units away from zero—and so is the number -7. The absolute value of a number is the distance it is from zero, so the absolute value of 7 and the absolute value of -7 are both 7.

We **write** the absolute value of -7 as . We **read** the expression as “the absolute value of .”

- Treat absolute value expressions like parentheses. If there is an operation inside the absolute value symbols, evaluate that operation first.
- The absolute value of a number or an expression is
**always**positive or zero. It cannot be negative. With absolute value, we are only interested in how far a number is from zero, and not in which direction.

#### Example E

*Evaluate the following absolute value expressions.*

a)

b)

(Remember to treat any expressions inside the absolute value sign as if they were inside parentheses, and evaluate them first.)

**Solution**

a)

b)

Watch this video for help with the Examples above.

CK-12 Foundation: Opposites and Absolute Values

### Vocabulary

- The
**absolute value**of a number is the distance it is from zero on the number line. The absolute value of any expression will always be positive or zero. - Two numbers are
**opposites**if they are the same distance from zero on the number line and on opposite sides of zero. The opposite of an expression can be found by multiplying**the entire expression**by -1.

### Guided Practice

1. *Find the opposite of each of the following:*

a)

b)

2. *Evaluate the following absolute value expressions.*

a)

b)

**Solution**

1. Since we know that opposite numbers are on opposite sides of zero, we can simply multiply each expression by -1. This changes the sign of the number to its opposite—if it’s negative, it becomes positive, and vice versa.

a) The opposite of is .

b) The opposite of is .

2. a)

b)

### Practice

Find the opposite of each of the following.

- 1.001

Simplify the following absolute value expressions.