1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 7.3 Subtraction with Real Numbers

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Subtract real numbers. o Subtract real numbers in a vertical format. o Find the change in value between two real numbers. o Determine if a given real number is a solution to an equation.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Additive Inverse The opposite of a real number is called its additive inverse. The sum of a number and its additive inverse is zero. Symbolically, for any real number a, Subtraction with Real Numbers

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. a.Find the additive inverse (opposite) of 3. Solution b.Find the additive inverse (opposite) of  7.3. Solution Example 1: Additive Inverse The additive inverse of 3 is  3. The additive inverse of  7.3 is

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. c.Find the additive inverse (opposite) of 0. Solution Example 1: Additive Inverse (cont.) The additive inverse of 0 is −0 = 0. That is, 0 is its own opposite.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Subtraction For any real numbers a and b, In words, to subtract b from a, add the opposite of b to a. Subtraction with Real Numbers

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the following differences. Example 2: Subtraction

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Subtraction (cont.)

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Subtraction a. b. sign change sign change

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Subtraction (cont.) c. d. sign change sign change

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Change in Value a.At noon on Tuesday, the temperature was 34  F. By noon on Thursday, the temperature had changed to −5  F. How much did the temperature change between Tuesday and Thursday?

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Change in Value (cont.) Solution For change in value: Between Tuesday and Thursday, the temperature changed  39  F (or dropped 39  F).

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Change in Value (cont.) b. A jet pilot flew her plane from an altitude of 30,000 ft to an altitude of 12,000 ft. What was the change in altitude? Solution (This means that the plane descended 18,000 ft.)

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Net Change a.Susan is a salesperson for a shoe store. Last week her sales of pairs of shoes were as follows. DaySalesReturnsDaily Net Sales Monday716 Tuesday303 Wednesday24 22 Thursday615 Friday835

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. What were Susan’s net sales for last week? Solution Susan’s net sales for the week were 17 pairs of shoes. Example 5: Net Change (cont.)

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Net Change (cont.) b.Robert weighed 230 lb when he started to diet. The first month he lost 7 lb, the second month he gained 2 lb, and the third month he lost 5 lb. What was his weight after 3 months of dieting? Solution

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Evaluating Possible Solutions Determine whether or not the given real number is a solution to the given equation by substituting for the variable and then evaluating.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solution −14 is not a solution. Example 6: Evaluating Possible Solutions (cont.)

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. b. 7  y   1 given that y  8 Solution 8 is a solution. Example 6: Evaluating Possible Solutions (cont.)

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. c. given that Solution Example 6: Evaluating Possible Solutions (cont.)

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. 1.What is the additive inverse of 85? 2.Find the difference: 3.Simplify: 4.Perform the indicated subtraction. Practice Problems

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. 5.Perform the operations on each side of the inequality symbol to determine whether the expression is true or false: 6.Determine whether or not x = −4 is a solution to the equation Practice Problems (cont.)

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.  85 2.  1 3.  8 4.90 5.true 6.  4 is a solution