1-2&1-3 Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz

Warm Up Simplify. 1. |–3| 3 2. –|4| –4 Write an improper fraction to represent each mixed number. 2 14 6 55 3. 4 4. 7 3 3 7 7 Write a mixed number to represent each improper fraction. 2 3 12 2 5 24 5. 6. 5 9

Objectives Add real numbers. Subtract real numbers.

All the numbers on a number line are called real numbers. You can use a number line to model addition and subtraction of real numbers. Addition To model addition of a positive number, move right. To model addition of a negative number move left. Subtraction To model subtraction of a positive number, move left. To model subtraction of a negative number move right.

Example 1A: Adding and Subtracting Numbers on a Number line Add or subtract using a number line. –4 + (–7) Start at 0. Move left to –4. To add –7, move left 7 units. + (–7) –4 11 10 9 8 7 6 5 4 3 2 1 –4+ (–7) = –11

Check It Out! Example 1c Add or subtract using a number line. –5 – (–6.5) Start at 0. Move left to –5. To subtract negative 6.5 move right 6.5 units. –5 – (–6.5) 8 7 6 5 4 3 2 1 1 2 –5 – (–6.5) = 1.5

The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|. 5 units 5 units - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 |–5| = 5 |5| = 5

Check It Out! Example 2a Add. –5 + (–7) –5 + (–7) = 5 + 7 When the signs are the same, find the sum of the absolute values. 5 + 7 = 12 Both numbers are negative, so the sum is negative. –12

Check It Out! Example 2b Add. –13.5 + (–22.3) –13.5 + (–22.3) When the signs are the same, find the sum of the absolute values. 13.5 + 22.3 –35.8 Both numbers are negative so, the sum is negative.

Check It Out! Example 2c Add. x + (–68) for x = 52 First substitute 52 for x. x + (–68) = 52 + (–68) When the signs of the numbers are different, find the difference of the absolute values. 68 – 52 Use the sign of the number with the greater absolute value. –16 The sum is negative.

Two numbers are opposites if their sum is 0 Two numbers are opposites if their sum is 0. A number and its opposite are on opposite sides of zero on a number line, but are the same distance from zero. They have the same absolute value.

Additive inverses 11 – 6 = 5 11 + (–6) = 5 A number and its opposite are additive inverses. To subtract signed numbers, you can use additive inverses. Subtracting 6 is the same as adding the inverse of 6. Additive inverses 11 – 6 = 5 11 + (–6) = 5 Subtracting a number is the same as adding the opposite of the number.

Check It Out! Example 3a Subtract. 13 – 21 13 – 21 = 13 + (–21) To subtract 21 add –21. When the signs of the numbers are different, find the difference of the absolute values: 21 – 13 = 8. –8 Use the sign of the number with the greater absolute value.

Check It Out! Example 3b Subtract. To subtract add . –3 1 2 3 When the signs of the numbers are the same, find the sum of the absolute values: = 4. 3 1 2 + 4 Both numbers are positive so, the sum is positive.

Check It Out! Example 3c Subtract. x – (–12) for x = –14 x – (–12) = –14 – (–12) First substitute –14 for x. –14 + (12) To subtract –12, add 12. When the signs of the numbers are different, find the difference of the absolute values: 14 – 12 = 2. Use the sign of the number with the greater absolute value. –2

Lesson Quiz Add or subtract using a number line. 2. –5 – (–3) –2 1. –2 + 9 7 Add or subtract. 3. –23 + 42 19 4. 4.5 – (–3.7) 8.2 5. 6. The temperature at 6:00 A.M. was –23°F. At 3:00 P.M. it was 18°F. Find the difference in the temperatures. 41°F

Objectives Multiply real numbers. Divide real numbers.

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

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

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

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

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

Homework: Vocabulary for Sections 1-2 and 1-3 p Homework: Vocabulary for Sections 1-2 and 1-3 p. 17-18, #’s 17-43 Odd, 53, 55 p. 23-24, 13-61 Odd This is a lot of problems!!! Don’t put it off!!!