1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 1.1 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 Identify types of numbers. o Graph sets of numbers on a real number line. o Determine if given numbers are greater than, less than, or equal to other given numbers. o Determine absolute values.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Types of Numbers Notes The negative sign (−) indicates the opposite of a number which we call a negative number. It is also used, as we will see in Section 1.3, to indicate subtraction. To avoid confusion, you must learn (by practice) just how the − sign is used in each particular situation.

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

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 b.State the opposite of −3. Solution −(−3) or +3 In words, the opposite of −3 is +3.

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

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Types of Numbers Variable A variable is a symbol (generally a letter of the alphabet) that is used to represent an unknown number (or numbers).

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rational Numbers 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 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.”)

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

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

11. 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

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Inequality Symbols Special Note About the Inequality Symbols 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 3.4.

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

14. 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.”

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Inequalities (cont.) Solution

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

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Inequalities (cont.) d. Graph all integers less than 0. Solution Remember, the three dots above the number line indicate that the pattern in the graph continues without end.

18. 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. |a|  aif a is a positive number or 0. |a|  aif a is a negative number.

19. 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.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Absolute Value Notes (cont.) Similarly, If x = −1, then −x = −(−1) = 1. If y = −10, then −y = −(−10) = 10. Remember that −a (the opposite of a) represents a positive number whenever a represents a negative number.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Absolute Value a.|6.3| 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. b.|−5.1| The number −5.1 is 5.1 units from 0. = 6.3 = −(−5.1) = 5.1

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Absolute Value (cont.) c.−|−2.9| The opposite of the absolute value of −2.9. d.If |x| = 7, what are the possible values for x? Solution x = 7 or x = −7 since |7|= 7 and |−7| = 7. e.True or false: |−4|≥ 4 Solution True, since |−4| = 4 and 4 ≥ 4. = −(2.9)= −2.9

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Absolute Value (cont.) Solution

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Absolute Value (cont.) g.If |x| = −3, what are the possible values for x? Solution There are no values of x for which |x| = −3. The absolute value can never be negative. There is no solution.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Absolute Value (cont.) h.If |x| < 3, what are the possible integer values for x? Graph these numbers on a number line. Solution The integers are within 3 units of 0: −2, −1, 0, 1, 2.

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Absolute Value (cont.) i.If |x| ≥ 4, what are the possible integer values for x? Graph these numbers on a number line. Solution The integers must be 4 or more units from 0:...,  7,  6,  5,  4, 4, 5, 6, 7,...

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Fill in the blank with the appropriate symbol:, or =. 1.−2 ____ 1 2. ____ 1.63. −(−4.1) ____ −7.2 4.Graph the set of all negative integers on a number line. 5.True or false: 3.6  |−3.6| 6.If |x| = 8, what are the possible values for x? 7.If |x| = −6, what are the possible values for x?

28. 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,  8 7. None