# 2.2: Additive Inverses and Absolute Values

**At Grade**Created by: CK-12

**Practice**Additive Inverses and Absolute Values

### Additive Inverses and Absolute Values

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.

The sum of a number and its opposite is always zero, as shown in the following example.

#### Proving the Sum of a Number and its Opposite Equals Zero

The numbers 3 and -3 are opposites because: \begin{align*}3 + -3 = 0\end{align*}

The numbers 4.2 and -4.2 are opposites because: \begin{align*}4.2 + -4.2 = 0\end{align*}

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.

#### Additive Inverses

The opposite of 4 is \begin{align*}4 \times -1\end{align*} or -4, and the opposite of -2.3 is \begin{align*}-2.3 \times -1\end{align*} or just 2.3.

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

#### Finding the Opposite

Find the opposite of each of the following:

Since we know that opposite numbers are 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) 19.6

The opposite of 19.6 is -19.6.

b) \begin{align*}- \frac{4}{9}\end{align*}

The opposite of \begin{align*}- \frac{4}{9}\end{align*} is \begin{align*}\frac{4}{9}\end{align*}.

c) \begin{align*}x\end{align*}

The opposite of \begin{align*}x\end{align*} is \begin{align*}-x\end{align*}.

d) \begin{align*}xy^2\end{align*}

The opposite of \begin{align*}xy^2\end{align*} is \begin{align*}-xy^2\end{align*}.

e) \begin{align*}(x - 3)\end{align*}

The opposite of \begin{align*}(x - 3)\end{align*} is \begin{align*}-(x - 3)\end{align*}, or \begin{align*}(3 - x)\end{align*}.

**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 \begin{align*}(x-3)\end{align*} is \begin{align*}(x + 3)\end{align*}. 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 \begin{align*}| -7 |\end{align*}. We **read** the expression \begin{align*}| x |\end{align*} as “the absolute value of \begin{align*}x\end{align*}.”

- 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.

#### Evaluating Absolute Value Expressions

Evaluate the following absolute value expressions.

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

a) \begin{align*}|5 + 4|\end{align*}

\begin{align*}| 5 + 4| = | 9 | = 9\end{align*}

b) \begin{align*}- |7 - 22|\end{align*}

\begin{align*}-| 7 - 22 | = - | -15 | = -(15) = -15\end{align*}

### Examples

#### Example 1

Find the opposite of each of the following:

Since we know that opposite numbers are on opposite sides of zero, we can simply multiply each expression by -1.

a) \begin{align*}\frac{2}{x}\end{align*}

The opposite of \begin{align*}\frac{2}{x}\end{align*}is \begin{align*}\frac{-2}{x}\end{align*}.

b) \begin{align*}-2y\end{align*}

The opposite of \begin{align*}-2y\end{align*} is \begin{align*}2y\end{align*}.

#### Example 2

Evaluate the following absolute value expressions.

a) \begin{align*}3 - |4 - 9|\end{align*}

\begin{align*} 3 - | 4 - 9 | = 3 - | -5 | = 3 - 5 = -2\end{align*}

b) \begin{align*}|-5 - 11|\end{align*}

\begin{align*}| -5 - 11 | = | -16 | = 16\end{align*}

### Review

Find the opposite of each of the following.

- 1.001
- \begin{align*} (5 - 11)\end{align*}
- \begin{align*}( x + y )\end{align*}
- \begin{align*}(x - y)\end{align*}
- \begin{align*}(x + y - 4)\end{align*}
- \begin{align*}(-x + 2y)\end{align*}

Simplify the following absolute value expressions.

- \begin{align*}11 - | -4 | \end{align*}
- \begin{align*}| 4 - 9 | - | -5 | \end{align*}
- \begin{align*}| -5 - 11 |\end{align*}
- \begin{align*}7 - | 22 - 15 - 19 |\end{align*}
- \begin{align*}- | -7 |\end{align*}
- \begin{align*}| -2 - 88 | - | 88 + 2 |\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.2.

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### Image Attributions

Here you'll learn the property that makes two numbers opposites of each other. You'll also learn how to evaluate absolute value expressions.

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