Serena and her dad have to build a fence to keep their cows in their pasture and out of the road. Together they walk the line of the new fence, measuring out and marking every three feet where the fence posts will go. But then her dad gets called in for a phone call. He leaves her with the tape measure, the poles, and a shovel and tells her to go ahead and start digging the holes. To figure out the depth he tells her that about half what's left above ground should go below. She measures the poles and finds that they're each 60 inches long. How can Serena figure out how deep the holes need to be?

In this concept, you will learn how to find the opposites of integers.

### Finding Opposites of Integers

Finding the opposite of an integer is useful in many different contexts. If you spend $20, the person who takes your money gains it. If a boat floats half below water and half above it, then the distance below the water is the opposite of the distance above it. All of this can be described using opposite integers.

A **positive number **is a number greater than zero. It can be written with or without a + symbol in front of it. A gain in something is written with a positive number. Profit, increase, and income are all examples of words that mean positive.

A **negative number** is a number that is less than zero. It is always written with a - symbol in front of it. A loss is written with a negative number. Decrease, spend, and decline are examples of words that mean negative.

An **integer **is any positive whole number or its opposite. Here, opposite means sign. So a positive integer has a negative opposite and vice versa.

Let's look at an example.

Write the opposite of -15.

First, determine whether the given integer is positive or negative.

This example has a "-" out front, so it is negative.

Next, figure out what the opposite sign is.

The opposite of negative is positive.

Then, re-write the original integer with the opposite sign.

The answer is +15 or 15.

Let's look at another example.

Find the opposite of 500 feet above sea level.

First, determine whether the original number is positive or negative.

"Above sea level" is positive, since sea level is zero.

Next, write the original example as a positive integer.

It is +500.

Then, write it with the opposite sign.

The answer is -500.

In this example, -500 means 500 feet below sea level.

### Examples

#### Example 1

Earlier, you were given a problem about Serena and her fence making.

The posts are each 60 inches long and her dad tells her that half as much of the post should go in the ground as stays above. Serena realizes that the post can be divided into three even pieces. One that needs to fit in the hole. And two that are above ground.

In order to figure out how much of the post goes underground, she divides the length 60 by 3.

The piece that needs to fit in the hole is 20inches.

Next, she re-writes that as a positive integer.

It is +20.

Then, she figures out that a hole dug into the ground is a negative integer because it is below ground-level.

So, in order for the +20 of the pole to fit in the hole, the hole needs to be -20.

In other words, she needs to dig the whole 20 inches deep.

#### Example 2

Alec spends $15 buying lemonade to sell at his stand. In order to break even, he needs to sell the opposite. Write the integer that represents how much Alec needs to sell.

First, write the original number as an integer.

Alec "spends" the money. This means it's negative. The original integer is -15.

Next, find the opposite sign.

In this case, it is positive.

Then, write the opposite integer.

The answer is +15. This means that Alec needs to make, or gain, $15 in order to break even.

**Write the opposite of each example.**

#### Example 3

A loss of 34.

First, determine whether the given integer is positive or negative.

"Loss" means negative.

Then, write the original integer.

It is -34

Finally, write the opposite.

The answer is **\begin{align*}+34\end{align*}.**

#### Example 4

998

First, determine whether the given integer is positive or negative.

Since there is no sign, you can assume it is positive.

It can also be written as +998.

Then, write the opposite.

The answer is** \begin{align*}-998\end{align*}.**

#### Example 5

67 feet above sea level

First, determine whether the given integer is positive or negative.

"Above sea level" means positive.

Then, write the original integer.

It is +67.

Finally, write the opposite.

The answer is** \begin{align*}-67\end{align*}.**

### Review

Write the opposite of each integer.

- A loss of 18
- A gain of 22
- -78
- 999
- -87
- 30 feet below the surface of the ocean
- A rise of 15 feet
- 25 feet below sea level
- 80 feet above ground
- 100 feet deep
- A profit of ten dollars
- The hill had a grade of 15%
- -987
- 65
- -1,298

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.2.