# 4.3: Comparison of Integers with Absolute Value

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

**Practice**Comparison of Integers with Absolute Value

This year, Cameron and his family are going to Alaska, but last year they were in a warmer place. Take a look.

Cameron and his parents are all scuba divers. Cameron learned to scuba dive two years ago when he was eleven. Kids between the ages of 11 and 14 can become junior divers and become certified junior divers through an organization called PADI. Since Cameron learned to dive, he has looked forward to his family’s diving vacation each year when they all take off to someplace warm and scuba dive.

One week before the big trip to the Caribbean, Cameron began looking through his dive book. A dive book is a book where divers keep track of their dives. They chart the depth that they went, the time they were underwater, and anything cool that they saw.

As a junior diver, Cameron is only allowed to travel to a maximum depth of 40 feet.

Here are Cameron’s dive depths from his last trip when he went diving in Jamaica.

15 feet deep

40 feet deep

25 feet deep

36 feet deep

30 feet deep

Integers can help a scuba divers in a real-world situation like this. Since Cameron traveled below the surface, we can use integers to write each of his depths. Then we can write them in order from least to greatest.

**To do this, you will need to know about integers. Pay attention to this Concept, and at the end of it, you will know how to help Cameron figure out his diving depths.**

### Guidance

**Now that you know about absolute value, opposites, zero and integers, we can work on learning how to compare and order integers. Often, the easiest way to do this is by using a number line.**

Remember, a number line shows numbers ordered from least to greatest. **So, on a number line, the further a number is to the right, the greater its value.**

This can be a little tricky because when you look at a number line -222, you might think that it is larger number, but it isn’t.

**How can this be?**

Well, think about how far away that number is from zero in the negatives. If you owed someone $222.00 it would be far worse than owing someone $22.00. **The best way to think about it is the farther that a negative number is on the left side of the number line, the smaller that number is.**

Order these numbers on a number line. Then determine which one is the greatest and which is the least: -12, 5, -6, -9, 10, 7

**First, let’s draw a number line and plot the numbers on the number line**

**Now look at the values on this number line and where they are located.**

**You can see that the number furthest to the right is 10. That is the largest number. The number furthest to the left is -12. That is the smallest number.**

Order these integers from least to greatest: -6, 0, 5, -1.

To help you order these integers, draw a number line from -6 to 6. Then plot points for -6, 0, 5 and -1.

**The numbers, ordered from least to greatest, are -6, -1, 0, 5.**

Once you understand how to determine which integers are greater, you can compare the integers using symbols. We can use these inequality symbols to compare and order integers.

> means *is greater than*.

< means *is less than*.

= means *is equal to*.

\begin{align*}\neq\end{align*} means *is not equal to*.

**Here are some statements to think about.**

Choose the inequality symbol that goes in the blank to make each statement true.

a. \begin{align*}-2 \ \underline{\;\;\;\;\;\;} \ -4\end{align*}

b. \begin{align*}-4 \ \underline{\;\;\;\;\;\;} \ 4\end{align*}

c. \begin{align*}|-4| \ \underline{\;\;\;\;\;\;} \ 4\end{align*}

To help you compare these values, draw a number line from -5 to 5, like this.

**Consider statement a.**

-2 is to the right of -4 on the number line. So, -2 is greater than -4.

The symbol > goes in the blank, because \begin{align*}-2 > -4\end{align*}.

**Consider statement b.**

-4 is to the left of 4 on the number line. So, -4 is less than 4.

The symbol < goes in the blank, because \begin{align*}-4 < 4\end{align*}.

**Consider statement c.**

\begin{align*}|-4|\end{align*} is 4, because -4 is 4 units from zero on the number line.

Since \begin{align*}|-4| = 4\end{align*}, the symbol = goes in the blank.

Use a number line and write these numbers in order from least to greatest.

#### Example A

\begin{align*}-4, 2, 8, 9, -11, -5\end{align*}

**Solution: \begin{align*}-11,-5,-4,2,8,9\end{align*}**

#### Example B

\begin{align*}6, -16, 7, -22, 1, 4\end{align*}

**Solution: \begin{align*}-22,-16,1,4,6,7\end{align*}**

#### Example C

\begin{align*}-3, -2, -7, -12, -1\end{align*}

**Solution: \begin{align*}-12,-7,-3,-2,-1\end{align*}**

Here is the original problem once again.

Cameron and his parents are all scuba divers. Cameron learned to scuba dive two years ago when he was eleven. Kids between the ages of 11 and 14 can become junior divers and become certified junior divers through an organization called PADI. Since Cameron learned to dive, he has looked forward to his family’s diving vacation each year when they all take off to someplace warm and scuba dive.

One week before the big trip to the Caribbean, Cameron began looking through his dive book. A dive book is a book where divers keep track of their dives. They chart the depth that they went, the time they were underwater, and anything cool that they saw.

As a junior diver, Cameron is only allowed to travel to a maximum depth of 40 feet.

Here are Cameron’s dive depths from his last trip when he went diving in Jamaica.

15 feet deep

40 feet deep

25 feet deep

36 feet deep

30 feet deep

Integers can help scuba divers in a real-world situation like this. Since Cameron traveled below the surface, we can use integers to write each of his depths. Then we can write them in order from least to greatest.

**First, let’s write each of Cameron’s depths as an integer. Depth is a word that tells us that we are going below the surface of the water. If the surface is zero, then anything below the surface would be represented by a negative number. Cameron’s depths are all negative numbers.**

**-25**

**-30**

**-15**

**-36**

**-40**

**We can order these integers from least to greatest by thinking of the deepest dive as the least and the dive closest to the surface, zero, as the greatest.**

**-40, -36, -30, -25, -15**

### Vocabulary

Here are the vocabulary words in this Concept.

- Whole Numbers
- the positive counting numbers including 0.

- Fractions
- parts of a whole written with a numerator and denominator.

- Decimal
- parts of a whole written with a decimal point using place value.

- Integers
- positive whole numbers and their opposites. Positive and negative numbers

### Guided Practice

Here is one for you to try on your own.

Compare using <, > or =.

\begin{align*}-22\end{align*} and \begin{align*}-32\end{align*}

**Answer**

To compare these two values, let's think about the distance that they are from zero. The farther a value is from zero, the smaller that value.

In this case, \begin{align*}-32\end{align*} is farther away from zero. It is the smaller value.

\begin{align*}-22 > -32\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

- This is a James Sousa video on comparing integers.

### Practice

Directions: Compare the following integers using inequality symbols.

1. \begin{align*}-17 \ \underline{\;\;\;\;\;\;} \ -19\end{align*}

2. \begin{align*}-9 \ \underline{\;\;\;\;\;\;} \ -11\end{align*}

3. \begin{align*}4 \ \underline{\;\;\;\;\;\;} \ -3\end{align*}

4. \begin{align*}5 \ \underline{\;\;\;\;\;\;} \ 7\end{align*}

5. \begin{align*}9 \ \underline{\;\;\;\;\;\;} \ -9\end{align*}

6. \begin{align*}-12 \ \underline{\;\;\;\;\;\;} \ -23\end{align*}

7. \begin{align*}|-9| \ \underline{\;\;\;\;\;\;} \ |8|\end{align*}

8. \begin{align*}|9| \ \underline{\;\;\;\;\;\;} \ |-9|\end{align*}

9. \begin{align*}|-2| \ \underline{\;\;\;\;\;\;} \ |-7|\end{align*}

10. \begin{align*}|-12| \ \underline{\;\;\;\;\;\;} \ 12\end{align*}

11. \begin{align*}|-22| \ \underline{\;\;\;\;\;\;} \ |22|\end{align*}

12. \begin{align*}|14| \ \underline{\;\;\;\;\;\;} \ |-7|\end{align*}

13. \begin{align*}|88| \ \underline{\;\;\;\;\;\;} \ |90|\end{align*}

14. \begin{align*}|-88| \ \underline{\;\;\;\;\;\;} \ |-92|\end{align*}

15. \begin{align*}|-27| \ \underline{\;\;\;\;\;\;} \ |27|\end{align*}

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Term | Definition |
---|---|

Decimal |
In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths). |

fraction |
A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number. |

Integer |
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3... |

### Image Attributions

Here you'll learn to compare and order integers on a number line and by using inequality symbols.