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# Additive Inverses and Absolute Values

## Opposite numbers and distance from zero

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What if you had a number like 34\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.

### 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: 3+3=0\begin{align*}3 + -3 = 0\end{align*}

The numbers 4.2 and -4.2 are opposites because: 4.2+4.2=0\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 4×1\begin{align*}4 \times -1\end{align*} or -4, and the opposite of -2.3 is 2.3×1\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) 49\begin{align*}- \frac{4}{9}\end{align*}

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

d) xy2\begin{align*}xy^2\end{align*}

e) (x3)\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 49\begin{align*}-\frac{4}{9}\end{align*} is 49\begin{align*}\frac{4}{9}\end{align*}.

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

d) The opposite of xy2\begin{align*}xy^2\end{align*} is xy2\begin{align*}-xy^2\end{align*}.

e) The opposite of (x3)\begin{align*}(x - 3)\end{align*} is (x3)\begin{align*}-(x - 3)\end{align*}, or (3x)\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 (x3)\begin{align*}(x-3)\end{align*} is (x+3)\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 |7|\begin{align*}| -7 |\end{align*}. We read the expression |x|\begin{align*}| x |\end{align*} as “the absolute value of x\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) |5+4|\begin{align*}|5 + 4|\end{align*}

b) |722|\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) |5+4|=|9|=9\begin{align*}| 5 + 4| = | 9 | = 9\end{align*}

b) |722|=|15|=(15)=15\begin{align*}-| 7 - 22 | = - | -15 | = -(15) = -15\end{align*}

Watch this video for help with the Examples above.

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

b) xy2\begin{align*}xy^2\end{align*}

2. Evaluate the following absolute value expressions.

a) 3|49|\begin{align*}3 - |4 - 9|\end{align*}

b) |511|\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 x\begin{align*}x\end{align*} is x\begin{align*}-x\end{align*}.

b) The opposite of xy2\begin{align*}xy^2\end{align*} is xy2\begin{align*}-xy^2\end{align*}.

2. a) 3|49|=3|5|=35=2\begin{align*} 3 - | 4 - 9 | = 3 - | -5 | = 3 - 5 = -2\end{align*}

b) |511|=|16|=16\begin{align*}| -5 - 11 | = | -16 | = 16\end{align*}

### Practice

Find the opposite of each of the following.

1. 1.001
2. \begin{align*} (5 - 11)\end{align*}
3. \begin{align*}( x + y )\end{align*}
4. \begin{align*}(x - y)\end{align*}
5. \begin{align*}(x + y - 4)\end{align*}
6. \begin{align*}(-x + 2y)\end{align*}

Simplify the following absolute value expressions.

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