Alyson and her son, Gage, are tired of Vermont winters, so they decide to get a small winter home somewhere warm. After looking around, they come up with a short list of places that are within their price range. But there's a catch. Alyson gets altitude sickness, so they want to pick the place that has the lowest altitude. Which place should they pick?

Brawley, CA: altitude -112ft

Eldorado, NM: altitude 6,677ft

Punta Gorda, FL: altitude 10 ft

Holtville, CA: altitude -10 ft

In this concept, you will learn how to compare and order integers using a number line.

### Comparing Integers on a Number Line

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.

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.

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.

A **number line **is a line on which numbers are marked at intervals, used to illustrate simple numerical operations. Using a number line allows you to see where a number is in relationship to other numbers and from zero.

This number line shows the integers from -5 to 5.

Number lines are useful for comparing integers. Just plot the integers on the line and then check to see which one is furthest to the right. That one is the greatest. The one furthest to the left is the least. Note, this means that a positive number is always greater than (>) a negative number. (The sideways V opens in the direction of the bigger number.)

Here is an example.

-6 ____ 2

First draw a number line.

Next, plot both integers on it.

Then, decide which one is further to the right.

In this case, the 2 is further to the right. That means it is greater than -6.

The answer is -6 < 2.

Here is another example.

17 ____10

First draw a number line.

Next, plot both integers on it.

Then, decide which one is further to the right.

In this case, the 17 is further to the right. That means it is greater than 10.

The answer is 17 > 10.

Let's look at another example.

-5 ____ -6

First draw a number line.

Next, plot both integers on it.

Then, decide which one is further to the right.

In this case, the -5 is further to the right. That means it is greater than -6.

The answer is -5 > -6.

### Examples

#### Example 1

Earlier, you were given a problem about Alyson and Gage's house-hunt dilemma.

They want to get a winter home that is at the lowest elevation possible. Their four options are:

Brawley, CA: altitude -112ft

Eldorado, NM: altitude 6,677ft

Punta Gorda, FL: altitude 10 ft

Holtville, CA: altitude -10 ft

In order to decide which one to buy, Gage makes a number line.

It is easy to see from this number line that Eldorado is the highest altitude, then Punta Gorda, then Holtville, then Brawley.

Brawley, CA has the lowest altitude, so it is the place that they pick.

#### Example 2

Jessie loses fifteen dollars playing video games. Carmen loses twenty-five dollars. Who has the bigger loss?

First, write both losses as integers.

Since losses are represented by negative integers, Jessie=\begin{align*}-15\end{align*} and Carmen=\begin{align*}-25\end{align*}

Next, draw a number line and plot both integers on it.

Then, compare the two points. The one further to the left is the smaller number, or the bigger loss.

In this case, -25 is further to the left so it is the smaller integer.

This can also be written:

\begin{align*}-25<-15\end{align*}

#### Example 3

Decide which is greater using < or > symbols: -6 ____ 8.

First, draw a number line and plot both points on it.

Next, decide which point is further to the left.

In this case, -6 is further to the left, so it is smaller than 8.

That means -6<8.

The answer is <.

#### Example 4

Decide which is greater using < or > symbols: -99 ____ -9.

First, draw a number line and plot both points on it.

Next, decide which point is further to the left.

In this case, -99 is much further to the left than -9, so it is smaller.

That means, -99<-9.

The answer is <.

Example 5

Decide which is greater using < or > symbols: 12 ____ 6.

First, draw a number line and plot both points on it.

Next, decide which point is further to the left.

In this case, 6 is further to the left so it is smaller.

That means 6<12. Or, re-written: 12>6.

The answer is >.

### Review

Draw a number line, and compare each pair of integers using the symbols for greater than and less than.

- 18 ____ 22
- -12 ____ 12
- -14 ____ -16
- -20 ____ -33
- 19 ____ -1
- 0 ____ -3
- -27 ____ -28
- -233 ____ -300
- -765 ____ -745
- -234 _____ -300

Write integers and then use inequality symbols to compare each pair of values.

- A loss of 16 and a loss of 29
- A loss of 18 and a gain of 28
- A loss of 27 and a loss of 10
- A loss of 16 and a gain of 12
- A loss of 90 and a loss of 99

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.3.