1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 7.1 The Real Number Line and Absolute Value

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Graph numbers on number lines. o Identify different types of numbers. o Understand inequality symbols such as. o Know the meaning of absolute value. o Graph absolute value inequalities.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Notes The negative sign indicates the opposite of a number as well as a negative number. It is also used, as we will see in Section 7.3, to indicate subtraction. (Note on your calculator, the subtraction key on the right side and the negative key at the bottom of the key pad.) Number Lines

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Integers The set of numbers consisting of the whole numbers and their opposites is called the set of integers: Number Lines

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Opposites a.State the opposite of 7. Solution 77 b.State the opposite of −4. Solution In words, the opposite of −4 is +4.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Number Line a.Graph the set of integers Solution b.Graph the set of integers Solution The three dots above the number line indicate that the pattern in the graph continues without end.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rational Numbers A rational number is a number that can be written in the form of where a and b are integers and b ≠ 0. (≠ is read “is not equal to”). OR A rational number is a number that can be written in decimal form as a terminating decimal or as an infinite repeating decimal. Types of Numbers

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. List the numbers in the set that are: a. Whole numbers Solution 0 and 17 are whole numbers b.Integers Solution  5, 0, and 17 are integers Example 3: Identifying Types of Numbers

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. c.Rational numbers Solution d.Real numbers Solution All numbers in S are real numbers. Example 3: Identifying Types of Numbers (cont.)

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Inequality Symbols Symbols of Equality and Inequality =is equal to≠ is not equal to is greater than ≤is less than or equal to≥ is greater than or equal to

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Inequality Symbols Notes Each symbol can be read from left to right as was just indicated in “Symbols of Equality and Inequality.” However, each symbol can also be read from right to left. Thus any inequality can be read in two ways. For example, 6 < 10 can be read from left to right as “6 is less than 10,” but also from right to left as “10 is greater than 6.” We will see that this flexibility is particularly useful when reading expressions with variables in Section 8.7.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Inequalities Determine whether each of the following statements is true or false.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Inequalities (cont.) Note: 7 < 15 can be read as “7 is less than 15” or as “15 is greater than 7.” 3 > −1 can be read as “3 is greater than −1” or as “−1 is less than 3.” 4 ≥ −4 can be read as “4 is greater than or equal to −4” or as “−4 is less than or equal to 4.”

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. a.Graph the set of real numbers Solution Example 5: Graphing Sets of Numbers

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Graphing Sets of Numbers (cont.) b. Graph all natural numbers less than or equal to 3. Solution Remember that the natural numbers are 1, 2, 3, 4, …

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Graphing Sets of Numbers (cont.) c.Graph all integers that satisfy the following condition: a < 0. Solution Remember, the three dots above the number line indicate that the pattern in the graph continues without end.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Absolute Value The absolute value of a real number is its distance from 0. Note that the absolute value of a real number is never negative. Absolute Value

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Absolute Value Notes The symbol  a should be thought of as the “opposite of a.” Since a is a variable, a might represent a positive number, a negative number, or 0. This use of symbols can make the definition of absolute value difficult to understand at first. As an aid to understanding the use of the negative sign, consider the following examples. If a = – 6, then –a = –(– 6) = 6.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Notes (cont.) Similarly, Remember that −a (the opposite of a) represents a positive number whenever a represents a negative number. Absolute Value

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The number 6.3 is 6.3 units from 0. Also, 6.3 is positive so its absolute value is the same as the number itself. Example 6: Absolute Value The opposite of the absolute value of −2.9.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. True, since and 4 ≥ 4. True or false: ? Solution Example 7: Absolute Value

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. a. If what are the possible values for x? Solution x = 7 or x = −7 since and b.If what are the possible values for x? Solution There are no values of x for which The absolute value can never be negative. There is no solution. Example 8: Absolute Value

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. If what are the possible integer values for x? Graph these integers on a number line. Solution There are an infinite number of integers 4 or more units from 0, both negative and positive. These integers are Example 9: Graphing Absolute Value Inequalities

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. If what are the possible integer values for x? Graph these integers on a number line. Solution The integers that are less than 4 units from 0 have absolute values less than 4. These integers are _______________________. Completion Example 10: Graphing Absolute Value Inequalities

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Fill in the blank with the appropriate symbol:, or =. 1.  2 ____ 12. ____ 1.6 3. 4.Graph the set of all negative integers on a number line. Practice Problems

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. 5.True or false: 6.If what are the possible values for x? 7.If what are the possible values for x? 8.If what are the possible integer values for x? Graph these integers on a number line. Practice Problems (cont.)

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1. 4. 5. True 6. –8, 87. no solution 8.