What if you had a number like \begin{align*}- \frac{3}{4}\end{align*}. 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.

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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: \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.

#### Example C

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**.

#### Example D

*Find the opposite of each of the following:*

a) 19.6

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

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

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

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

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

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

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

e) 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.

#### Example E

*Evaluate the following absolute value expressions.*

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

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

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

**Solution**

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

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

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

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

2. *Evaluate the following absolute value expressions.*

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

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

**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 \begin{align*}x\end{align*} is \begin{align*}-x\end{align*}.

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

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

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

### Practice

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*}