In this lesson you learned how to use inequality symbols to compare integers. The symbols you can use are less than (\( \lt \)), greater than (\( \gt \)), less than or equal to (\( \leq \)), greater than or equal to (\( \geq \)), and not equal to \( ( \neq ) \). You can use a number line to help you make comparisons among integers with these symbols.
You can also use inequalities to compare integers that are inside absolute value symbols. Remember that the absolute value of an integer measures its distance away from 0. The absolute value of an integer is always positive or 0.
For Example
Which inequality symbol makes the statement \( \left| - 20 \right| \text{ __ } \left| 17 \right| \) true?
The numbers in the example above are enclosed within absolute value symbols. Click the Show Me button to see how to work with this kind of inequality.
Step 1: Simplify the absolute value(s). |
\[ \left| - 20 \right| = 20 \]
The integer \( - \)20 is 20 units to the left of 0. Its absolute value is 20. \[ \left| 17 \right| = 17 \]The integer 17 is 17 units to the right of 0. Its absolute value is 17. |
Step 2: Compare the integers. |
Compare the integers 20 and 17. The number 20 is further to the right on the number line than 17. 
A horizontal number line from 13 to 23 in increments of 1. There are two points plotted. One is at 17 and the other is at 20. |
Step 3: Write the inequality. |
Since \( \left| - 20 \right| = 20, \) and \( 20 \) is larger than \( \left| 17 \right| = 17 \): \[ \left| - 20 \right| \gt \left| 17 \right| \] |
Level Up
How well can you compare integers using inequality symbols? Practice your skills with the activity below. Respond to the question on each tab. You can click the Answer button to check your work. If you need help, refer to the tutorial video on the previous page or to the example above.
Which inequality symbol, \( \lt \) or \( \gt \), completes the following statement?
\[ | - 3| \text{ __ } |7| \]If you need help arriving at this answer, click the Solution button.
Step 1: Simplify the absolute value(s). |
\[ \left| - 3 \right| = 3 \]
The integer −3 is 3 units to the left of 0. Its absolute value is 3. \[ \left| 7 \right| = 7 \]The integer 7 is 7 units to the right of 0. Its absolute value is 7. |
Step 2: Compare the integers. |
Compare the integers 3 and 7. The integer 3 is further to the left on the number line than 7. 
A horizontal number line from −4 to 9 in increments of 1. There are two points plotted. One is at \( 3 \) and the other is at \( 7 \). |
Step 3: Write the inequality. |
Since \( \left| - 3 \right| = 3, \) and \( 3 \) is smaller than \( \left| 7 \right| = 7 \): \[ \left| - 3 \right| \lt \left| 7 \right| \] |
Last year 86 fewer students than average attended the dance. This year 45 fewer students than average attended the dance.
Write an inequality that compares the integers \( - 45 \) and \( - 86 \). Use either \( \leq \) or \( \geq . \)
\( - 45 \geq - 86 \) or \( - 86 \leq - 45 \)
If you need help arriving at this answer, click the Solution button.
Step 1: Simplify the absolute value(s). |
This question does not involve absolute value symbols. You can skip this step. |
Step 2: Compare the integers. |
Compare the integers \( - 45 \) and \( - 86 \). The integer \( - 45 \) is further to the right on the number line than \( - 86 \). Notice that this number line does not show an equal number of values to the left and right of 0, but it still can be used for comparing. 
A horizontal number line from \( - 100 \) to \( - 40 \) in increments of 5. There are two points plotted. One is at \( - 86 \) and the other is at \( - 45 \). |
Step 3: Write the inequality. |
Since \( - 86 \) is further to the left than \( - 45 \), you could write either \( - 45 \geq - 86 \) or \( - 86 \leq - 45 \). Both of these inequalities are true. |
Chandler and Rebecca are standing in the middle of the football field. Chandler is facing left and throws a football 25 feet. Rebecca is facing right and throws her football 40 feet.
You can express these integers as \( - 25 \) and \( 40 \).
Compare the integers using the greater than symbol.
\( 40 \gt - 25 \)
If you need help arriving at this answer, click the Solution button.
Step 1: Simplify the absolute value(s). |
This question does not involve absolute value symbols. You can skip this step. |
Step 2: Compare the integers. |
Compare the integers \( - 25 \) and \( 40 \). The integer \( - 25 \) is further to the left on the number line than \( 40 \). 
A horizontal number line from \( - 30 \) to 40 in increments of 5. There are two points plotted. One is at \( - 25 \) and the other is at \( 40 \). |
Step 3: Write the inequality. |
Since \( - 25 \) is further to the left than \( 40 \), and you must use the inequality symbol \( \gt \), the inequality is: \[ 40 \gt - 25 \] |
Which inequality symbol, \( \lt \) or \( \gt \), completes the following statement?
\[ \left| - 8 \right| \text{ __ } \left| - 12 \right| \]\( \lt \)
If you need help arriving at this answer, click the Solution button.
Step 1: Simplify the absolute value(s). |
\[ \left| - 8 \right| = 8 \]
The integer −8 is 8 units to the left of 0. Its absolute value is 8. \[ \left| - 12 \right| = 12 \]The integer −12 is 12 units to the left of 0. Its absolute value is 12. |
Step 2: Compare the integers. |
Compare the integers 8 and 12. The number 12 is further to the right on the number line than 8. 
A horizontal number line from \( - 10 \) to 15 in increments of 1. There are two points plotted. One is at \( 8 \) and the other is at \( 12 \). |
Step 3: Write the inequality. |
Since \( \left| - 12 \right| = 12, \) and \( 12 \) is larger than \( \left| - 8 \right| = 8 \): \[ \left| - 8 \right| \lt \left| - 12 \right| \] |
Jesse and Wriston went hiking into the Grand Canyon. Jesse descended 936 feet before stopping for a drink. Wriston descended 785 feet before stopping for a drink. You can express these values as \( - 936 \) and \( - 785 \).
Write an inequality using the absolute values of the depths of where each hiker was when they took a water break. Use one of the \( \geq \) or \( \leq \) symbols.
Compare \( | - 936| \) and \( | - 785| \).
\( \left| - 936 \right| \geq \left| - 785 \right| \) or \( \left| - 785 \right| \leq \left| - 936 \right| \)
If you need help arriving at this answer, click the Solution button.
Step 1: Simplify the absolute value(s). |
\[ \left| - 936 \right| = 936 \]
The integer \( - \)936 is 936 units to the left of 0. Its absolute value is 936. \[ \left| - 785 \right| = 785 \]The integer \( - \)785 is 785 units to the left of 0. Its absolute value is 785. |
Step 2: Compare the integers. |
Compare the integers 936 and 785. The number 936 is further to the right on the number line than 785. 
A horizontal number line from \( 750 \) to 950 in increments of 10. There are two points plotted. One is at \( 785 \) and the other is at \( 936 \). |
Step 3: Write the inequality. |
Since \( \left| - 936 \right| = 936, \) and \( 936 \) is larger than \( \left| - 785 \right| = 785 \), you could write either \( \left| - 936 \right| \geq \left| - 785 \right| \) or \( \left| - 785 \right| \leq \left| - 936 \right|. \) |
