1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: Prealgebra Section 2.1: Introduction to Integers

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o Know that integers are the whole numbers and their opposites. o Know that 0 is neither positive nor negative. o Be able to graph a set of integers on a number line. o Understand and be able to read inequality symbols such as. o Know the meaning of the absolute value of an integer. o Be aware that an expression of the form  a may represent a positive or a negative number.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. A Special Note about Calculators Note that all of the operations with integers can be done with or without a calculator. However, at this beginning stage, to develop a thorough understanding of operating with negative numbers, you should use a calculator as little as possible.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Number Lines The concepts of positive and negative numbers occur frequently in our daily lives: NegativeZeroPositive Temperatures are recorded as:below zerozeroabove zero The stock market will show:a lossno changea gain Altitude can be measured as:below sea level sea levelabove sea level Business will report:lossesno gainprofits

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Number Lines (cont.) A number line is used to represent the positive and negative numbers. We start with a horizontal line and label it with the number 0. Now we chose a point to the right of zero and label it with the number 1. The graph of a number is indicated by marking the corresponding with a large dot. The number 6 is graphed on this number line. We continue numbering the whole numbers. All the numbers are the same distance apart. 0 1 2 3 4 5 6 7 8 9 10 0 1 0 The point one unit to the left of 0 is the opposite of 1. It is called “negative 1” and is symbolized as – 1. Similarly the opposite of 2 is called “negative 2” and is symbolized as – 2, and so on.  10  9  8  7  6  5  4  3  2  1

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 1: Integers Integers The set of integers is the set of whole numbers and their opposites (or additive inverses). Integers: …, – 3, – 2, – 1, 0, 1, 2, 3,…

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The counting numbers (all whole numbers except 0) are called positive integers. The opposites of the counting numbers are called negative integers. 0 is neither positive nor negative. Obj 2: Integers  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10 Zero Negative integersPositive integers

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 2: Integers 1. The opposite of a positive integer is a negative integer. For example, 2. The opposite of a negative integer is a positive integer. For example, 3. The opposite of 0 is 0. [That is, –0 = +0 = 0. This shows that –0 should be thought of as the opposite of 0 and not as “negative 0.” Remember the number 0 is neither positive nor negative, and 0 is its own additive inverse.] opposite of +1 opposite of +6 opposite of  2 opposite of  8

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Integers Find the opposite of: a. –5 b. –11 c. +14 – ( – 5)= 5 – ( – 11) = 11 – (+14) = – 14

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 1: Integers Find the opposite of: a. –10 b. –8 c. +17 – ( – 10) = 10 – ( – 8) = 8 – (+17) = – 17

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Graph Integers Graph the set of integers  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 2: Graph Integers Graph the set of integers  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 3: Graph Integers A set of numbers is infinite if it is so large that it cannot be counted. The set of whole numbers and the set of integers are both infinite. To graph an infinite set of integers, three dots are marked above the number line to indicate that the pattern shown continues without ending.

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 3: Graph Integers Graph the set of integers The three dots above the number line indicate that the pattern in the graph continues without end.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 3: Graph Integers Graph the set of integers on a number line.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 4: Inequality Symbols On a horizontal number line, smaller numbers are always to the left of larger numbers. Each number is smaller than any number to its right and larger than any number to its left.

17. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Symbols for Order < is read “is less than” is read “is less than or equal to” > is read “ is greater than” is read “is greater than or equal to” Obj 4: Inequality Symbols (cont.)

18. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 4: Inequality Symbols (cont.) The numbers 3 and 5 have been graphed on the number line below. To show the relationship on the number line, we could write: 3 < 5 3 is less than 5 OR 5 > 35 is greater than 3  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

19. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 4: Inequality Symbols (cont.) The numbers –4 and –1 have been graphed on the number line below. To show the relationship on the number line, we could write: –4 < –1 –4 is less than –1 OR –1 > –4 –1 is greater than –4  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

20. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 4: Inequality Symbols (cont.) We can read the inequality symbols from right to left or from left to right. 2 < 8 2 is less than 8 OR 8 is greater than 2

21. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 4: Inequality Symbols (cont.) The inequality signs allow for both equality and inequality. For example,

22. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. a.b. c.d. a. true, since 4 is less than 12 b. true, since 4 is equal to 4 c. false, we can write 4 > 0 or 0 < 4 d. false, we can write Determine whether each of the following statements is true or false. Rewrite any false statement so that it is true. Example 4: Inequality Symbols

23. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. a.b. c.d. a. falseb. true c. trued. false Determine whether each of the following statements is true or false. Rewrite any false statement so that it is true. Self-Check 4: Inequality Symbols

24. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 5: Absolute Value Absolute value, symbolized with two vertical bars, is closely related to our discussion of signed numbers. As we have seen, any integer and it opposite lie the same number of units from 0. For example +5 and – 5 are both 5 units from 0 on the number line. The + and – signs indicates the direction, and the 5 indicates distance from 0. Thus  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10 5 units

25. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Absolute Value The absolute value of an integer is it distance from 0. Symbolically, for any integer a, If a, is a positive integer or 0, If a, is a negative integer, The absolute value of an integer is never negative. Obj 5: Absolute Value (cont.)

26. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. When a represents a negative number, the symbol –a represents a positive number. That is, the opposite of a negative number is a positive number. For example, Also, Obj 6: “The Opposite Of”

27. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Obj 6: “The Opposite Of” (cont.) Remember to read –x as “the opposite of x” and not negative x, because –x may not be a negative number. In summary, 1. If x represents a positive number, then –x represents a negative number. 2. If x represents a negative number, then –x represents a positive number.

28. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 5: Absolute Value Find the absolute value of The absolute value for both is 2 because they are both 2 units from 0 on the number line.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10 2 units

29. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 5: Absolute Value Find the absolute value The absolute value of both is 3 because they are both 3 units from 0 on the number line.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10 3 units

30. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 6: Absolute Value Find the absolute value of The absolute value is 0.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10 No units

31. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 6: Absolute Value Find the absolute value The absolute value is 4.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10 4 units

32. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 7: Absolute Value True or False: True since (Remember that since 9 = 9 the statement is true.)

33. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 7: Absolute Value True or False: True

34. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 8: Absolute Value If

35. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 8: Absolute Value If

36. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 9: Absolute Value If There are no values of

37. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 9: Absolute Value If There are no possible values.

38. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Graphing Inequalities To graph an inequality, it is easiest to begin by determining the numbers that make the statement true (the replacement set). For example: The solution set can be represented as {3, 4, 5, 6, 7,….}. The solution set can be represented as {…, –4, –3, –2, –1, 0}. We don’t include 1 because y is not equal to 1.

39. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Graphing Inequalities (cont.) Next, the graphs of the solutions are indicated by making large dots over a few numbers in the solution set and then putting three dots above the number line to show the pattern is to continue without end.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

40. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Graphing Absolute Value Inequalities When graphing absolute value inequalities we need to remember that absolute value means the number of spaces from zero. If we graph we need to find the solution set. We think of all the numbers that are more than or equal to three units from zero.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

41. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 10: Graphing Absolute Value Inequalities There are an infinite number of integers 4 or more units from 0, both negative and positive. These integers are {…, –7, –6, –5, –4, 4, 5, 6, 7, …}.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

42. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 10: Graphing Absolute Value Inequalities There are an infinite number of integers more than 1 unit from 0, both negative and positive. These integers are {…, –5, –4, –3, –2, 2, 3, 4, 5, …}.  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

43. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 11: Graphing Absolute Value Inequalities The integers that are less than 4 units from zero are {–3, –2, –1, 0, 1, 2, 3}  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10

44. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Self-Check 11: Graphing Absolute Value Inequalities The integers that are less than 3 units from zero are { –2, –1, 0, 1, 2, }  10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10