Warm Up Add or subtract. 1. 6 + 104 2. 12(9) 3. 23 – 8 4. 1. 6 + 104 2. 12(9) 3. 23 – 8 4. Multiply or divide. 5. 324 ÷ 18 6. 7. 13.5(10) 8. 18.2 ÷ 2 110 108 15 18 6 135 9.1

Objectives Translate between words and algebra. Evaluate algebraic expressions.

A variable is a letter or a symbol used to represent a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic expression may contain variables, constants, and operations.

These expressions all mean “2 times y”: 2y 2(y) 2•y (2)(y) 2 x y (2)y Writing Math

Example 1: Translating from Algebra to Words Give two ways to write each algebra expression in words. A. 9 + r B. q – 3 the sum of 9 and r the difference of q and 3 9 increased by r 3 less than q C. 7m D. j  6 the product of m and 7 the quotient of j and 6 m times 7 j divided by 6

Check It Out! Example 1 Give two ways to write each algebra expression in words. 1a. 4 - n 1b. 4 decreased by n the quotient of t and 5 n less than 4 t divided by 5 1c. 9 + q 1d. 3(h) the sum of 9 and q the product of 3 and h 3 times h q added to 9

Multiply Divide Add Subtract Put together, combine To translate words into algebraic expressions, look for words that indicate the action that is taking place. Add Subtract Multiply Divide Put together, combine Find how much more or less Put together equal groups Separate into equal groups

Example 2A: Translating from Words to Algebra John types 62 words per minute. Write an expression for the number of words he types in m minutes. m represents the number of minutes that John types. 62 · m or 62m Think: m groups of 62 words

Example 2B: Translating from Words to Algebra Roberto is 4 years older than Emily, who is y years old. Write an expression for Roberto’s age y represents Emily’s age. y + 4 Think: “older than” means “greater than.”

Example 2C: Translating from Words to Algebra Joey earns $5 for each car he washes. Write an expression for the number of cars Joey must wash to earn d dollars. d represents the total amount that Joey will earn. Think: How many groups of $5 are in d?

To evaluate an expression is to find its value. To evaluate an algebraic expression, substitute numbers for the variables in the expression and then simplify the expression.

Check It Out! Example 3 Evaluate each expression for m = 3, n = 2, and p = 9. a. mn mn = 3 · 2 Substitute 3 for m and 2 for n. = 6 Simplify. b. p – n p – n = 9 – 2 Substitute 9 for p and 2 for n. = 7 Simplify. c. p ÷ m p ÷ m = 9 ÷ 3 Substitute 9 for p and 3 for m. = 3 Simplify.

Example 4a: Recycling Application Approximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag. Write an expression for the number of bottles needed to make s sleeping bags. The expression 85s models the number of bottles to make s sleeping bags.

Example 4b: Recycling Application Continued Approximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag. Find the number of bottles needed to make 20, 50, and 325 sleeping bags. Evaluate 85s for s = 20, 50, and 325. s 85s 20 50 325 To make 20 sleeping bags 1700 bottles are needed. 85(20) = 1700 To make 50 sleeping bags 4250 bottles are needed. 85(50) = 4250 To make 325 sleeping bags 27,625 bottles are needed. 85(325) = 27,625

A replacement set is a set of numbers that can be substituted for a variable. The replacement set in Example 4 is {20, 50, and 325}. Helpful Hint

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

-3 -2 -1 1 2 3 4 5 6 7 8 9 Check It Out! Example 1a Add or subtract using a number line. –3 + 7 Start at 0. Move left to –3. To add 7, move right 7 units. +7 –3 -3 -2 -1 1 2 3 4 5 6 7 8 9 –3 + 7 = 4

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

Example 2A: Adding Real Numbers When the signs of numbers are different, find the difference of the absolute values: Use the sign of the number with the greater absolute value. The sum is negative.

Example 2B: Adding Real Numbers y + (–2) for y = –6 y + (–2) = (–6) + (–2) First substitute –6 for y. When the signs are the same, find the sum of the absolute values: 6 + 2 = 8. (–6) + (–2) –8 Both numbers are negative, so the sum is negative.

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.

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.

Example 3A: Subtracting Real Numbers –6.7 – 4.1 –6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1. When the signs of the numbers are the same, find the sum of the absolute values: 6.7 + 4.1 = 10.8. = –10.8 Both numbers are negative, so the sum is negative.

Example 3B: Subtracting Real Numbers 5 – (–4) 5 − (–4) = 5 + 4 To subtract –4 add 4. 9 Find the sum of the absolute values.

Example 3C: Subtracting Real Numbers First substitute for z. To subtract , add . Rewrite with a denominator of 10.

Example 3C Continued When the signs of the numbers are the same, find the sum of the absolute values: . Write the answer in the simplest form. Both numbers are negative, so the sum is negative.

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.

elevation at top of iceberg Check It Out! Example 4 What if…? The tallest known iceberg in the North Atlantic rose 550 feet above the oceans surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet? elevation at top of iceberg 550 Minus elevation of the Titanic –12,468 – 550 – (–12,468) To subtract –12,468, add 12,468. 550 – (–12,468) = 550 + 12,468 Find the sum of the absolute values. = 13,018 Distance from the iceberg to the Titanic is 13,018 feet.

Warm-Up Add or subtract. 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.

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

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 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.

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.

Warm-Up 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

5 Minute Warm-Up Directions: Solve the following problems. 1. 3 (15) 2. (4) 5 5 12 3. 4 – (15) 4. 5 + |-12| 5. (-121) ÷ 11 6. 44 ÷ 4 11

Objective Evaluate expressions containing exponents.

A power is an expression written with an exponent and a base or the value of such an expression. 3² is an example of a power. The base is the number that is used as a factor. 3 2 The exponent, 2 tells how many times the base, 3, is used as a factor.

When a number is raised to the second power, we usually say it is “squared.” The area of a square is s  s = s2, is the side length. S When a number is raised to the third power, we usually say its “cubed.” The of volume of a cube is s  s  s = s3 is the side length. S

Check It Out! Example 1 Write the power represented by each geometric model. a. The figure is 2 units long and 2 units wide. 2  2 22 The factor 2 is used 2 times. x b. The figure is x units long, x units wide, and x units tall. x  x  x The factor x is used 3 times. x3

There are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them using repeated multiplication or a base and exponent. Reading Exponents Words Multiplication Power Value 3 to the first power 3 31 3 3 to the second power, or 3 squared 3  3 32 9 3 to the third power, or 3 cubed 3  3  3 33 27 3 to the fourth power 34 3  3  3  3 81 3 to the fifth power 3  3  3  3  3 35 243

Caution! In the expression –52, 5 is the base because the negative sign is not in parentheses. In the expression (–2), –2 is the base because of the parentheses.

Example 2: Evaluating Powers Evaluate each expression. A. (–6)3 (–6)(–6)(–6) Use –6 as a factor 3 times. –216 B. –102 Think of a negative sign in front of a power as multiplying by a –1. –1 • 10 • 10 Find the product of –1 and two 10’s. –100

Example 2: Evaluating Powers Evaluate the expression. C. 2 9  Use as a factor 2 times. 2 9 = 4 81 2 9 

Example 3: Writing Powers Write each number as a power of the given base. A. 64; base 8 8  8 The product of two 8’s is 64. 82 B. 81; base –3 (–3)(–3)(–3)(–3) The product of four –3’s is 81. (–3)4

Example 4: Problem-Solving Application In case of a school closing, the PTA president calls 3 families. Each of these families calls 3 other families and so on. How many families will have been called in the 4th round of calls? Understand the problem 1 The answer will be the number of families contacted in the 4th round of calls. List the important information: • The PTA president calls 3 families. • Each family then calls 3 more families.

Example 4 Continued 2 Make a Plan Draw a diagram to show the number of Families called in each round of calls. PTA President 1st round of calls 2nd round of calls

Notice that after each round of calls the Example 4 Continued Solve 3 Notice that after each round of calls the number of families contacted is a power of 3. 1st round of calls: 1  3 = 3 or 31 families contacted 2nd round of calls: 3  3 = 9 or 32 families contacted 3rd round of calls: 9  3 = 27 or 33 families contacted So, in the 4th round of calls, 34 families will have been contacted. 34 = 3  3  3  3 = 81 Multiply four 3’s. In the fourth round of calls, 81 families will have been contacted.

n n Warm-Up 1. Write the power represented by the geometric model. n2 Simplify each expression. 3. –63 −216 2. 4. 6 216 5. (–2)6 64 Write each number as a power of the given base. 6. 343; base 7 73 7. 10,000; base 10 104

Objectives Evaluate expressions containing square roots. Classify numbers within the real number system.

A number that is multiplied by itself to form a product is called a square root of that product. The operations of squaring and finding a square root are inverse operations. The radical symbol , is used to represent square roots. Positive real numbers have two square roots. = 4 Positive square root of 16 4  4 = 42 = 16 (–4)(–4) = (–4)2 = 16 – = –4 Negative square root of 16

The nonnegative square root is represented by The nonnegative square root is represented by . The negative square root is represented by – . A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table. 1 4 9 16 25 36 49 64 81 100 02 12 22 32 42 52 62 72 82 92 102

The expression does not represent a real number because there is no real number that can be multiplied by itself to form a product of –36. Reading Math

Example 1: Finding Square Roots of Perfect Squares Find each square root. A. Think: What number squared equals 16? 42 = 16 Positive square root positive 4. = 4 B. Think: What is the opposite of the square root of 9? 32 = 9 = –3 Negative square root negative 3.

Example 1C: Finding Square Roots of Perfect Squares Find the square root. Think: What number squared equals ? 25 81 Positive square root positive . 5 9

Check It Out! Example 1 Find the square root. 1a. 22 = 4 Think: What number squared equals 4? = 2 Positive square root positive 2. 1b. 52 = 25 Think: What is the opposite of the square root of 25? Negative square root negative 5.

The square roots of many numbers like , are not whole numbers The square roots of many numbers like , are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics. Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, …

Rational numbers can be expressed in the form , where a and b are both integers and b ≠ 0: , , . a b 1 2 7 9 10

Irrational numbers cannot be expressed in the form Irrational numbers cannot be expressed in the form . They include square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat: , ,  a b Terminating decimals are rational numbers in decimal form that have a finite number of digits: 1.5, 2.75, 4.0 Repeating decimals are rational numbers in decimal form that have a block of one or more digits that repeat continuously: 1.3, 0.6, 2.14

Example 3: Classifying Real Numbers Write all classifications that apply to each Real number. A. –32 32 can be written as a fraction and a decimal. 32 1 –32 = – = –32.0 rational number, integer, terminating decimal B. 5 5 can be written as a fraction and a decimal. 5 1 5 = = 5.0 rational number, integer, whole number, natural number, terminating decimal

Write all classifications that apply to each real number. Check It Out! Example 3 Write all classifications that apply to each real number. 7 can be written as a repeating decimal. 49 3a. 7 4 9 67  9 = 7.444… = 7.4 rational number, repeating decimal 3b. –12 32 can be written as a fraction and a decimal. –12 = – = –12.0 12 1 rational number, terminating decimal, integer 3c. The digits continue with no pattern. = 3.16227766… irrational number

Warm-Up Find each square root. 3 7 1 2 -8 1. 12 2. 3. 4. – 5. The area of a square piece of cloth is 68 in2. How long is each side of the piece of cloth? Round your answer to the nearest tenth of an inch. 8.2 in. Write all classifications that apply to each real number. 6. 1 rational, integer, whole number, natural number, terminating decimal 7. –3.89 rational, repeating decimal 8. irrational

Objective Use the order of operations to simplify expressions.

When a numerical or algebraic expression contains more than one operation symbol, the order of operations tells which operation to perform first. Order of Operations Perform operations inside grouping symbols. First: Second: Evaluate powers. Third: Perform multiplication and division from left to right. Perform addition and subtraction from left to right. Fourth:

Grouping symbols include parentheses ( ), brackets [ ], and braces { } Grouping symbols include parentheses ( ), brackets [ ], and braces { }. If an expression contains more than one set of grouping symbols, evaluate the expression from the innermost set first.

Helpful Hint The first letter of these words can help you remember the order of operations. Please Excuse My Dear Aunt Sally Parentheses Exponents Multiply Divide Add Subtract

Check It Out! Example 1b Simplify the expression. 5.4 – 32 + 6.2 There are no grouping symbols. 5.4 – 32 + 6.2 5.4 – 9 + 6.2 Simplify powers. –3.6 + 6.2 Subtract 2.6 Add.

Check It Out! Example 1c Simplify the expression. –20 ÷ [–2(4 + 1)] There are two sets of grouping symbols. –20 ÷ [–2(4 + 1)] Perform the operations in the innermost set. –20 ÷ [–2(5)] Perform the operation inside the brackets. –20 ÷ –10 2 Divide.

Check It Out! Example 2b Evaluate the expression for the given value of x. (x · 22) ÷ (2 + 6) for x = 6 (x · 22) ÷ (2 + 6) (6 · 22) ÷ (2 + 6) First substitute 6 for x. (6 · 4) ÷ (2 + 6) Square two. (24) ÷ (8) Perform the operations inside the parentheses. 3 Divide.

Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.

Example 3A: Simplifying Expressions with Other Grouping Symbols 2(–4) + 22 42 – 9 The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing. 2(–4) + 22 42 – 9 –8 + 22 42 – 9 Multiply to simplify the numerator. –8 + 22 16 – 9 Evaluate the power in the denominator. Add to simplify the numerator. Subtract to simplify the denominator. 14 7 2 Divide.

Example 3B: Simplifying Expressions with Other Grouping Symbols 3|42 + 8 ÷ 2| The absolute-value symbols act as grouping symbols. 3|42 + 8 ÷ 2| Evaluate the power. 3|16 + 8 ÷ 2| Divide within the absolute-value symbols. 3|16 + 4| 3|20| Add within the absolute-symbols. 3 · 20 Write the absolute value of 20. 60 Multiply.

Example 4: Translating from Words to Math Translate each word phrase into a numerical or algebraic expression. A. the sum of the quotient of 12 and –3 and the square root of 25 Show the quotient being added to . B. the difference of y and the product of 4 and Use parentheses so that the product is evaluated first.

Check It Out! Example 4 Translate the word phrase into a numerical or algebraic expression: the product of 6.2 and the sum of 9.4 and 8. Use parentheses to show that the sum of 9.4 and 8 is evaluated first. 6.2(9.4 + 8)

Warm-Up Simply each expression. 2. 52 – (5 + 4) |4 – 8| 1. 2[5 ÷ (–6 – 4)] –1 4 3. 5  8 – 4 + 16 ÷ 22 40 Translate each word phrase into a numerical or algebraic expression. 4. 3 three times the sum of –5 and n 3(–5 + n) 5. the quotient of the difference of 34 and 9 and the square root of 25 6. the volume of a storage box can be found using the expression l · w(w + 2). Find the volume of the box if l = 3 feet and w = 2 feet. 24 cubic feet

Objectives Use the Commutative, Associative, and Distributive Properties to simplify expressions. Combine like terms.

Compatible numbers help you do math Helpful Hint Compatible numbers help you do math mentally. Try to make multiples of 5 or 10. They are simpler to use when multiplying.

Check It Out! Example 1a Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 21

Check It Out! Example 1b Simplify. 410 + 58 + 90 + 2 410 + 90 + 58 + 2 Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. (410 + 90) + (58 + 2) (500) + (60) 560

The Distributive Property is used with Addition to Simplify Expressions. The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

Example 2A: Using the Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 5(59) 5(50 + 9) Rewrite 59 as 50 + 9. 5(50) + 5(9) Use the Distributive Property. 250 + 45 Multiply. 295 Add.

Example 2B: Using the Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 8(33) 8(30 + 3) Rewrite 33 as 30 + 3. 8(30) + 8(3) Use the Distributive Property. 240 + 24 Multiply. 264 Add.

The terms of an expression are the parts to be added or subtracted The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms. Like terms Constant 4x – 3x + 2

A coefficient is a number multiplied by a variable A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1. Coefficients 1x2 + 3x

Using the Distributive Property can help you combine like terms Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression. 7x2 – 4x2 = (7 – 4)x2 Factor out x2 from both terms. = (3)x2 Perform operations in parenthesis. = 3x2 Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

Example 3A: Combining Like Terms Simplify the expression by combining like terms. 72p – 25p 72p – 25p 72p and 25p are like terms. 47p Subtract the coefficients.

Example 3B: Combining Like Terms Simplify the expression by combining like terms. A variable without a coefficient has a coefficient of 1. and are like terms. Write 1 as . Add the coefficients.

Example 4: Simplifying Algebraic Expressions Simplify 14x + 4(2 + x). Justify each step. Procedure Justification 1. 14x + 4(2 + x) 2. 14x + 4(2) + 4(x) Distributive Property 3. 14x + 8 + 4x Multiply. Commutative Property 4. 14x + 4x + 8 5. (14x + 4x) + 8 Associative Property 6. 18x + 8 Combine like terms.

Check It Out! Example 4b Simplify −12x – 5x + 3a + x. Justify each step. Procedure Justification 1. –12x – 5x + 3a + x 2. –12x – 5x + x + 3a Commutative Property 3. –16x + 3a Combine like terms.

Warm-Up Simplify each expression. 1. 165 +27 + 3 + 5 200 2. 8 Write each product using the Distributive Property. Then simplify. 3. 5($1.99) 5($2) – 5($0.01) = $9.95 4. 6(13) 6(10) + 6(3) = 78

Objectives Graph ordered pairs in the coordinate plane. Graph functions from ordered pairs.

The coordinate plane is formed by the intersection of two perpendicular number lines called axes. The point of intersection, called the origin, is at 0 on each number line. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.

Points on the coordinate plane are described using ordered pairs Points on the coordinate plane are described using ordered pairs. An ordered pair consists of an x-coordinate and a y-coordinate and is written (x, y). Points are often named by a capital letter. The x-coordinate tells how many units to move left or right from the origin. The y-coordinate tells how many units to move up or down. Reading Math

Example 1: Graphing Points in the Coordinate Plane Graph each point. A. T(–4, 4) Start at the origin. T(–4, 4) • Move 4 units left and 4 units up. B. U(0, –5) Start at the origin. Move 5 units down. C. V (–2, –3) • V(–2, −3) Start at the origin. • U(0, –5) Move 2 units left and 3 units down.

Example 2: Locating Points in the Coordinate Plane Name the quadrant in which each point lies. •E •F •H •G y A. E Quadrant ll B. F no quadrant (y-axis) x C. G Quadrant l D. H Quadrant lll

An equation that contains two variables can be used as a rule to generate ordered pairs. When you substitute a value for x, you generate a value for y. The value substituted for x is called the input, and the value generated for y is called the output. Output Input y = 10x + 5 In a function, the value of y (the output) is determined by the value of x (the input). All of the equations in this lesson represent functions.

Example 3: Art Application An engraver charges a setup fee of $10 plus $2 for every word engraved. Write a rule for the engraver’s fee. Write ordered pairs for the engraver’s fee when there are 5, 10, 15, and 20 words engraved. Let y represent the engraver’s fee and x represent the number of words engraved. Engraver’s fee is $10 plus $2 for each word y = 10 + 2 · x y = 10 + 2x

The engraver’s fee is determined by the number of words in the engraving. So the number of words is the input and the engraver’s fee is the output. Writing Math

Example 3 Continued Number of Words Engraved Rule Charges Ordered Pair x (input) y = 10 + 2x y (output) (x, y) 5 y = 10 + 2(5) 20 (5, 20) 10 y = 10 + 2(10) 30 (10, 30) 15 y = 10 + 2(15) 40 (15, 40) 20 y = 10 + 2(20) 50 (20, 50)

When you graph ordered pairs generated by a function, they may create a pattern.

Example 4A: Generating and Graphing Ordered Pairs Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern. y = 2x + 1; x = –2, –1, 0, 1, 2 • Input Output Ordered Pair x y (x, y) –2 2(–2) + 1 = –3 (–2, –3) –1 2(–1) + 1 = –1 (–1, –1) 2(0) + 1 = 1 (0, 1) 1 2(1) + 1 = 3 (1, 3) 2(2) + 1 = 5 2 (2, 5) The points form a line.

Example 4B: Generating and Graphing Ordered Pairs y = x2 – 3; x = –2, –1, 0, 1, 2 Input Output Ordered Pair x y (x, y) –2 (–2)2 – 3 = 1 (–2, 1) –1 (–1)2 – 3 = –2 (–1, –2) (0)2 – 3 = –3 (0, –3) 1 (1)2 – 3 = –2 (1, –2) (2)2 – 3 = 1 2 (2, 1) The points form a U shape.

Example 4C: Generating and Graphing Ordered Pairs y = |x – 2|; x = 0, 1, 2, 3, 4 Input Output Ordered Pair x y (x, y) |0 – 2| = 2 (0, 2) 1 |1 – 2| = 1 (1, 1) 2 |2 – 2| = 0 (2, 0) 3 |3 – 2| = 1 (3, 1) 4 |4 – 2| = 2 (4, 2) The points form a V shape.

A(2,2) G(3,2) M(4,2) S(5,2) Y(6,2) B(2,3) H(3,3) N(4,3) T(5,3) Z(6,3) C(2,4) I(3,4) O(4,4) U(5,4) D(2,5) J(3,5) P(4,5) V(5,5) E(2,6) K(3,6) Q(4,6) W(5,6) F(2,7) L(3,7) R(4,7) X(5,7)