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# 4.3: Comparison of Integers with Absolute Value

Difficulty Level: At Grade Created by: CK-12
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Practice Comparison of Integers with Absolute Value

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Josh and his four friends are comparing how much money they each have. Josh has $23. His friend Krista has$25. His friend Mark has $8. His friends Luis and Tessa each owe their parents money so they basically have negative money. Luis has -$5 and Tessa has -10. How could Josh use a number line to compare and order the amount of money he and his friends have from least to greatest? In this concept, you will learn how to order integers on a number line and how to compare integers using inequality symbols. ### Comparing Integers with Absolute Value Recall that integers are a set of numbers that include the positive whole numbers (1, 2, 3, 4, 5, ...), their opposites (-1, -2, -3, -4, -5, ...) and zero. Sometimes you will want to compare different integers or put them in order from smallest to largest. A number line can help you to do this. On a number line, the further a number is to the right, the larger its value. Conversely, the further a number is to the left, the smaller its value. Let's look at an example. Plot these numbers on a number line. Then determine which one is the largest and which is the smallest: -12, 5, -6, -9, 10, 7 First, 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. Let's look at another example. Order these integers from least to greatest: -6, 0, 5, -1. Again, to help you to order these integers, draw a number line. Then, plot points for -6, 0, 5 and -1. You can see that -6 is furthest to the left. This means -6 is the smallest number in your set. Moving to the right, the next plotted point is -1. This means -1 is the next smallest number in your set. Continuing in this way you see that the next smallest number is 0 and then the largest number is 6. Ordered from least to greatest, the numbers are -6, -1, 0, 5. Once you know how to order integers from smallest to largest, you can compare integers using symbols. You can use the following inequality symbols to compare integers: • > means is greater than. • < means is less than. • = means is equal to. • ≠ means is not equal to. Let's look at an example. Choose the inequality symbol that goes in the blank to make the statement true: \begin{align*}-2 \ \underline{\;\;\;\;\;\;\;\;\;} \ -4\end{align*} First, draw a number line. Next, notice that -2 is to the right of -4 on the number line. This means that -2 is greater than -4 so the symbol \begin{align*}>\end{align*} should go in the blank. The answer is \begin{align*}\text{-}2 > \text{-}4\end{align*}. Let's look at another example. Choose the inequality symbol that goes in the blank to make the statement true: \begin{align*}|\text{-}4| \ \underline{\;\;\;\;\;\;\;\;\;} \ 4\end{align*} First, evaluate the absolute value. \begin{align*}|\text{-}4|\end{align*} is 4, because -4 is 4 units from zero on the number line. This means \begin{align*}|\text{-}4|\end{align*} is equal to \begin{align*}|4|\end{align*} so the symbol = goes in the blank. The answer is \begin{align*}|\text{-}4| = 4\end{align*}. ### Examples #### Example 1 Earlier, you were given a problem about Josh and his four friends. Josh has23, his friend Krista has $25, his friend Mark has$8, his friend Luis has -$5 (he owes his parents), and his friend Tessa has -$10 (she also owes her parents). Josh wanted to compare and order the amount of money he and his friends have.

First, draw a number line. Make sure it is big enough to include all the numbers you need to plot. Plot all the money values on the number line.



Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -10 is the smallest number which means Tessa has the least amount of money.

Then, look to see which plotted number is to the right of -10. -5 comes next so it is the second smallest number. Luis has the next smallest amount of money.

Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.

The answer is that the money amounts in order from least to greatest are -$10, -$5, $8,$23, \$25. Tessa has the smallest amount of money, then Luis, then Mark, then Josh. Krista has the most amount of money.

#### Example 2

Choose the inequality symbol that goes in the blank to make the statement true: \begin{align*}-22 \ \underline{\;\;\;\;\;\;\;\;\;} \ -32\end{align*}

First, visualize a number line and imagine plotting the points -22 and -32.

Next, determine where -22 is on the number line compared to -32. -22 will be further to the right. This means -22 is greater than -32.

The answer is \begin{align*}-22 > -32\end{align*}.

#### Example 3

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

−4, 2, 8, 9, −11, −5

First, draw a number line. Make sure it is big enough to include all the numbers you need to plot.

Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -11 is the smallest number.

Then, look to see which plotted number is to the right of -11. -5 comes next so it is the second smallest number.

Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.

The answer is -11, -5, -4, 2, 8, 9.

#### Example 4

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

5, −16, 6, -12, 1, 4

First, draw a number line. Make sure it is big enough to include all the numbers you need to plot.

Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -16 is the smallest number.

Then, look to see which plotted number is to the right of -16. -12 comes next so it is the second smallest number.

Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.

The answer is -16, -12, 1, 4, 5, 6.

#### Example 5

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

−3, −2, −7, −12, −1

First, draw a number line. Make sure it is big enough to include all the numbers you need to plot.

Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -12 is the smallest number.

Then, look to see which plotted number is to the right of -12. -7 comes next so it is the second smallest number.

Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.

The answer is -12, -7, -3, -2, -1.

### Review

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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