Adding and Subtracting Integers

1. Adding and Subtracting Integers

2. -5 0 5 -10 10 Absolute Value -7 +7 A number line has many functions. Previously, we learned that numbers to the right of zero are positive and numbers to the left of zero are negative. By putting points on the number line, we can graph values. If one were to start at zero and move seven places to the right, this would represent a value of positive seven. If one were to start at zero and move seven places to the left, this would represent a value of negative seven.

3. -5 0 5 -10 10 Absolute Value -7 +7 Both of these numbers, positive seven (+7) and negative seven (-7), represent a point that is seven units away from the origin. The absolute value of a number is the distance between that number and zero on a number line. Absolute value is shown by placing two vertical bars around the number as follows: │ 5 │ The absolute value of five is five. │ -3 │ The absolute value of negative three is three.

4. Adding Integers - Same Sign What is 5 + 7? We can show how to do this by using algebra chips. + = 5 + 7 = 12 What is –5 + -7? = + -12 = + -7 -5

5. 9 -5 0 5 -10 10 Adding Integers - Same Sign We can show this same idea using a number line. What is 5 + 4? Move five (5) units to the right from zero. Now move four more units to the right. The final point is at 9 on the number line. Therefore, 5 + 4 = 9.

6. -9 -5 0 5 -10 10 Adding Integers - Same Sign What is -5 + (-4)? Move five (5) units to the left from zero. Now move four more units to the left. The final point is at -9 on the number line. Therefore, -5 + (-4) = -9.

7. Adding Integers - Same Sign To add integers with the same sign, add the absolute values of the integers. Give the answer the same sign as the integers. Solution Examples

8. -5 0 5 -10 10 Additive Inverse What is (-7) + 7? To show this, we can begin at zero and move seven units to the left. Now, move seven units to the right. Notice, we are back at zero (0). For every positive integer on the number line, there is a corresponding negative integer. These integer pairs are opposites or additive inverses. Additive Inverse Property – For every number a, a + (-a) = 0

9. Additive Inverse When using algebra chips, the additive inverses make what is called a zero pair. For example, the following is a zero pair. 1 + (-1) = 0.

10. -5 0 5 -10 10 Adding Integers - Different Signs Add the following integers: (-4) + 7. Start at zero and move four units to the left. Now move seven units to the right. The final position is at 3. Therefore, (-4) + 7 = 3.

11. -5 0 5 -10 10 Adding Integers - Different Signs Add the following integers: (-4) + 7. Notice that seven minus four also equals three. In our example, the number with the larger absolute value was positive and our solution was positive. Let’s try another one.

12. -5 0 5 -10 10 Adding Integers - Different Signs Add (-9) + 3 Start at zero and move nine places to the left. Now move three places to the right. The final position is at negative six, (-6). Therefore, (-9) + 3 = -6.

13. -5 0 5 -10 10 Adding Integers - Different Signs Add (-9) + 3 In this example, the number with the larger absolute value is negative. The number with the smaller absolute value is positive. We know that 9-3 = 6. However, (-9) + 3 = -6. 6 and –6 are opposites. Comparing these two examples shows us that the answer will have the same sign as the number with the larger absolute value.

14. Adding Integers - Different Signs To add integers with different signs determine the absolute value of the two numbers. Take the difference or subtract the smaller value from the larger value. Keep the sign of the number with the larger absolute value. Solution Example Subtract

15. Adding Integers We can show the addition of numbers with opposite signs by using algebra chips. For example, 3 + (-5) would look like this: + Group the zero pairs. Remove the zero pairs The remainder is the solution. Therefore, 3 + (-5) = -2

16. Subtracting Integers Let’s see how subtraction works using algebra chips for 5 - 2 Begin with five positive chips. There are enough positive chips to remove 2 from the board Now remove 2 positive ones. There are 3 positive ones remaining.

17. Subtracting Integers Let’s see how subtraction works using algebra chips for 3 - 5. Begin with three algebra chips. There are not enough positive chips to remove so put 2 zero pairs on the board Now remove the five positive ones.

18. Subtracting Integers The remainder is negative two (-2). Therefore, 3 – 5 = -2. This is the same as 3 + (-5). To subtract a number, add its additive inverse. This means KEEP the sign of first number CHANGE the operation to addition and CHANGE the sign of the second number. Then use the addition rules.

20. You Try It Find each sum or difference. • -24 – 11 2. 18 + (-40) 3. -9 + 9 • 4. -16 – (-14) 5. 13 – 35 6. -29 + 65 Simplify each expression. 7. 18 – 27 8. 9 – (-12) 9. -7 + 45 Evaluate each expression. • │ 89 │ 11. │ -14 │ • 12. - │ -7 │ 13. │ c + 5 │ if c = -19

21. Solutions • -16 – (-14) = 16 + 14 • = 30

22. Solutions 5. 13 – 35 =13 + (-35) =35 - 13 = -22 {keeping -35’s sign} • 7. 18 – 27= 18 + (-27) 8. 9 – (-12) = 9 + 12 • = 27 - 18 = 17 • = -9 {keeping -27’s sign} • 9. -7 + 45 = 45 – 7 • = 38

23. Solutions 13. │ c + 5 │ if c= -19