1. OPERATIONS WITH INTEGERS, ADDING AND SUBTRACTING RATIONAL NUMBERS Objective: To add, subtract, multiply, and divide integers, to compare and order rational numbers, and to add and subtract rational numbers.

2. ADDING INTEGERS  The sum of two positive numbers is a positive number.  The sum of two negative numbers is a negative number.  The sum of a positive number and a negative number will either be positive or negative depending on the sign of the number with the larger absolute value.

3. ADDING INTEGERS  4 + 5 = 9 (Positive + Positive = Positive)  -4 + -5 = -9 (Negative + Negative = Negative)  -4 + 5 = 1 (Negative + Positive = Positive because the Positive is the larger absolute value)  4 + -5 = -1 (Positive + Negative = Negative because the Negative is the larger absolute value)

4. ADDING INTEGERS  You may add integers using a number line.  Always start at zero.  Positive numbers move right and negative numbers move left.

5. EXAMPLE 1  Use a number line to find -2 + 3. -2 3 = 1

6. ADDING INTEGERS  You may also add integers by using absolute values.  If the two numbers have the same sign (both positive or both negative), add the absolute value of the two numbers and match the sign to the two numbers.  If the two numbers have different signs (one positive and one negative), subtract the absolute value of the two numbers and match the sign of the larger absolute value number.

7. EXAMPLE 2  Add the following. a.-15 + (-4) b.-15 + 4 c.15 + -4 = -19 = -11 = 11

8. SUBTRACTING INTEGERS  Every positive integer can be paired with a negative integer. These pairs are called opposites.  A number and its opposite are additive inverses. Additive inverses can be used when you subtract integers.  Subtracting integers is the same as adding the inverse.  a – b = a + (-b)

9. EXAMPLE 3  Subtract the following. a.11 – 15 b.-11 – 15 c.11 – (-15) = 11 + (-15)= -4 = -11 + (-15)= -26 = 11 + 15= 26

10. MULTIPLYING AND DIVIDING INTEGERS  If you multiply or divide integers with the same sign, your answer will be positive. Positive x Positive = Positive Negative x Negative = Positive Positive ÷ Positive = Positive Negative ÷ Negative = Positive

11. MULTIPLYING AND DIVIDING INTEGERS  If you multiply or divide integers with different signs, your answer will be negative. Positive x Negative = Negative Negative x Positive = Negative Positive ÷ Negative = Negative Negative ÷ Positive = Negative

12. EXAMPLE 4  Find each product or quotient. a.-9(4) b.-112 ÷ (-8) c.-11(-11) d.96 ÷ (-6) = -36 = 14 = 121 = -16

13. COMPARING RATIONAL NUMBERS  You can compare rational numbers in two different ways.  When comparing fractions, always make them have common denominators or turn them into decimals and compare the decimals.

14. EXAMPLE 5  Replace  with >, <, or = to make  a true statement. 32 25 >

15. EXAMPLE 6  Order from least to greatest. 15 14 100N = 45.4545... -N = 0.4545... 99N = 45 40 33

16. ADDING OR SUBTRACTING FRACTIONS WITH LIKE DENOMINATORS  To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference over the common denominator.  Follow the addition and subtraction rules for integers to determine the sign of your answer.

17. EXAMPLE 7  Find each sum or difference. Write in simplest form. a.. b.. c..

18. ADDING OR SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS  To add or subtract fractions with unlike denominators, first find the least common denominator (LCD).  Rename each fraction with the LCD, and then add or subtract.  Simplify if possible.

19. EXAMPLE 8  Find each sum or difference. Write in simplest form. a.. b.. c.. 87 15 5 6

20. EXAMPLE 9  Use a number line to find -1.25 + 1.5. -1.25 1.5 = 0.25

21. EXAMPLE 10  Find each sum. a.-12.6 + (-3.9) b.. = -16.5 6 5

22. EXAMPLE 11  Find -17.55 – (-24.5). = -17.55 + 24.5 = 6.95