1-3 Multiplying and Dividing Real Numbers Warm Up Lesson Presentation Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz

Warm Up Multiply or divide. 1. 6(7) 42 3. 42 7 ÷ 6 2. 4. 4 42 7 ÷ 6 2. 4. 4 5. Write as a mixed number. 6. Write as an improper fraction.

Objectives Multiply real numbers. Divide real numbers.

Vocabulary reciprocal multiplicative inverse

Factors Product When you multiply two numbers, the signs of the numbers you are multiplying determine whether the product is positive or negative. Factors Product 3(5) Both positive 15 Positive 3(–5) One negative –15 Negative –3(–5) Both negative 15 Positive This is true for division also.

Multiplying and Dividing Signed Numbers WORDS Multiplying and Dividing Numbers with the Same Sign If two numbers have the same sign, their product or quotient is positive. NUMBERS 4  5 = 20 –15 ÷ (–3) = 5

Multiplying and Dividing Signed Numbers WORDS Multiplying and Dividing Numbers with Different Signs If two numbers have different signs, their product or quotient is negative. NUMBERS 6(–3) = –18 –18 ÷ 2 = –9

Example 1: Multiplying and Dividing Signed Numbers Find the value of each expression. A. The product of two numbers with different signs is negative. –5 B. The quotient of two numbers with the same sign is positive. 12

Example 1C: Multiplying and Dividing Signed Numbers Find the value of the expression. First substitute for x. The quotient of two numbers with different signs is negative.

Check It Out! Example 1a and 1b Find the value of each expression. 1a. 35  (–5) The quotient of two numbers with different signs is negative. –7 1b. –11(–4) The product of two numbers with the same sign is positive. 44

Check It Out! Example 1c Find the value of each expression. 1c. –6x for x = 7 –6x = –6(7) First substitute 7 for x. The product of two numbers with different signs is negative. = –42

Two numbers are reciprocals if their product is 1. A number and its reciprocal are called multiplicative inverses. To divide by a number, you can multiply by its multiplicative inverse. Dividing by a nonzero number is the same as Multiplying by the reciprocal of the number.

Multiplicative inverses 1 10 10 ÷ 5 = 2 10 ∙ = = 2 5 5 Dividing by 5 is the same as multiplying by the reciprocal of 5, .

You can write the reciprocal of a number by switching the numerator and denominator. A whole number has a denominator of 1. Helpful Hint

Example 2A: Dividing by Fractions Example 2 Dividing by Fractions Divide. To divide by , multiply by . Multiply the numerators and multiply the denominators. and have the same sign, so the quotient is positive.

Example 2B: Dividing by Fractions Divide. Write as an improper fraction. To divide by , multiply by . and have different signs, so the quotient is negative.

Check It Out! Example 2a Divide. Write as an improper fraction. To divide by , multiply by . and have the same signs, so the quotient is positive.

Check It Out! Example 2b Divide. Write the reciprocal of . To divide by , multiply by . and have different signs, so the quotient is negative.

Check It Out! Example 2c Divide. Write as an improper fraction. To divide by multiply by . The signs are different so the quotient is negative.

No number can be multiplied by 0 to give a product of 1, so 0 has no reciprocal. Because 0 has no reciprocal, division by 0 is not possible. We say that division by zero is undefined.

Properties of Zero Multiplication by Zero The product of any number and 0 is 0. WORDS 1 3 · 0 = 0 0(–17) = 0 NUMBERS ALGEBRA a · 0 = 0 0 · a = 0

Properties of Zero Zero Divided by a Number The quotient of 0 and any nonzero number is 0. WORDS 6 = 0 0 ÷ 2 3 = 0 NUMBERS a = 0 ALGEBRA 0 ÷ a = 0

Properties of Zero Division by Zero WORDS Division by 0 is undefined. –5 12 ÷ 0 NUMBERS Undefined a ALGEBRA a ÷ 0 Undefined

Example 3: Multiplying and Dividing with Zero Multiply or divide. 15 A. Zero is divided by a nonzero number. The quotient of zero and any nonzero number is 0. B. –22  0 A number is divided by zero. undefined Division by zero is undefined. C. –8.45(0) A number is multiplied by zero. The product of an number and 0 is 0.

Check It Out! Example 3a Divide: Turn into an improper fraction. Multiply by the reciprocal. Any number multiplied by 0 equals 0.

Check It Out! Example 3b and 3c Divide: 3b. 0 ÷ 0 Any number divided by 0 is undefined. 0 ÷ 0 = undefined 3c. (–12.350)(0) The product of any number and 0 is 0. (–12.350)(0) = 0

Example 4: Recreation Application The speed of a hot-air balloon is 3 mi/h. It travels in a straight line for 1 hour before landing. How many miles away from the liftoff site does the balloon land? 1 3 4 Find the distance traveled at a rate of 3 mi/h for 1 hour. To find distance, multiply rate by time. 3 4 1 rate times time 3 4 1 3  1 3

Example 4: Recreation Application 3 4 • 1 = 15 Write and as improper fractions. 3 4 1 15(4) 4(3) = 60 12 Multiply the numerators and multiply the denominators. 3 4 and have the same sign, so the quotient is positive. 1 = 5 The hot-air balloon lands 5 miles from the liftoff site.

Check It Out! Example 4 What if…? On another hot-air balloon trip, the wind speed is 5.25 mi/h. The trip is planned for 1.5 hours. The balloon travels in a straight line. How many miles away from the liftoff site will the balloon land? 5.25(1.5) Speed times time equals distance. = 7.875 mi Distance traveled.

Lesson Quiz Find the value of each expression. 1. 35 –7 –5 2. 2x for x = –6 – 12 Multiply or divide if possible. undefined 3. –3 ÷ 1 3 4 (0) 4. –2 1 3 – 12 7 5. –  0 3 4 6. A cyclist traveled on a straight road for 1 hours at a speed of 12 mi/h. How many miles did the cyclist travel? 1 4 15 miles