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Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Problem of the Day The only socks in Jerry’s drawer are 245 black socks and 246 green socks. How many socks does he have to pull out in order to be sure that he has a matched pair? 3 Lesson Main Lesson 1-1 Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Check Skills You’ll Need (For help, go to the Skills Handbook page 632) 1. Vocabulary Review Which expression does not use multiplication: 3 • 4, 3 , 3 x 4, or 3(4)? 4 Multiply. 2. 12 • 8 3. 2.5 • 4 4. 7.4 • 6 5. 0.6 • 5 Check Skills You’ll Need Lesson Main Lesson 1-1 Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Check Skills You’ll Need Solutions 1. 3 2. 96 3. 10 4. 44.4 4 5. 3 Lesson Main Lesson 1-1 Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Additional Examples A student earns $5 an hour babysitting. The hours vary from week to week. Define a variable. Write an algebraic expression for how much the student earns in a week. Words $5 an hour times number of hours spent babysitting Let h = number of hours spent babysitting. Equation 5 • h The algebraic expression 5h represents the student’s weekly earnings. Quick Check Lesson Main Lesson 1-1 Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Additional Examples Evaluate p – 23 for p = 10. p – 23 = 10 – 23 = – 13 Substitute 10 for p. Simplify by subtracting 23 from 10. Quick Check Lesson Main Lesson 1-1 Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Additional Examples Evaluate the expression t + (12 – t ) ÷ 2 for t = 6. t + (12 – t ) ÷ 2 = 6 + (12 – 6) ÷ 2 Substitute 6 for t. =6+6÷2 Work within parentheses. =6+3 Divide. =9 Add. Quick Check Lesson Main Lesson 1-1 Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Additional Examples An Internet provider charges $25 for a connection fee and then $16 per month. Write an expression to model the total cost and then evaluate the expression for 1 to 5 months of Internet access. Words connection fee plus 16 times number of months Let m = the number of months. Equation Lesson Main 25 + 16 Lesson 1-1 • m Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Additional Examples (continued) m 25 + 16m Total Cost 1 25 + 16(1) $41 2 25 + 16(2) $57 3 25 + 16(3) $73 4 25 + 16(4) $89 5 25 + 16(5) $105 Evaluate the expression for each value of the variable. Use the values 1–5 for m. Quick Check Lesson Main Lesson 1-1 Feature Algebraic Expressions and the Order of Operations LESSON 1-1 Course 3 Lesson Quiz 1. Write an expression for the number of months in y years. 12y 2. Evaluate your expression in Exercise 1 for 7 years. 84 months 3. Evaluate 5n – n ÷ 2 for n = 10. 45 4. Evaluate 3c – 5 for c = 2, 4, and 6. 1, 7, 13 Lesson Main Lesson 1-1 Feature Integers and Absolute Value LESSON 1-2 Course 3 Problem of the Day CDs take up half of Antancio’s music shelf. Cassette tapes fill up 1 as much 3 space as the CDs. If the shelf is 6 ft long, how much space remains? 2 ft Lesson Main Lesson 1-2 Feature Integers and Absolute Value LESSON 1-2 Course 3 Check Skills You’ll Need (For help, go to Lesson 1-1.) 1. Vocabulary Review What is an algebraic expression? Evaluate each expression for x = 4. 2. 2x + 3 3. 5(x – 1) 4. 7x – 4x Check Skills You’ll Need Lesson Main Lesson 1-2 Feature Integers and Absolute Value LESSON 1-2 Course 3 Check Skills You’ll Need Solutions 1. An algebraic expression is a mathematical phrase that uses numbers, variables, and operation symbols. 2. 2(4) + 3 = 8 + 3 = 11 3. 5(4 – 1) = 5(3) = 15 4. 7(4) – 4(4) = 28 – 16 = 12 Lesson Main Lesson 1-2 Feature Integers and Absolute Value LESSON 1-2 Course 3 Additional Examples Find each absolute value. a. |3| On the number line, 3 is 3 units from 0. This means |3| = 3. b. |–2| On the number line, –2 is 2 units from 0. This means |–2| = 2. Quick Check Lesson Main Lesson 1-2 Feature Integers and Absolute Value LESSON 1-2 Course 3 Additional Examples Order –7, 5, and –4 from least to greatest. Put the integers on the same number line. The numbers from least to greatest are –7, –4, and 5. Quick Check Lesson Main Lesson 1-2 Feature Integers and Absolute Value LESSON 1-2 Course 3 Additional Examples The lowest recorded temperature in Asia is –90ºF; in North America it is –81ºF. Which continent has the lower recorded temperature? | –90| = 90 | –81| = 81 Find the negative integer with the greater absolute value. The number with the greater absolute value is –90. Asia has the lower recorded temperature. Quick Check Lesson Main Lesson 1-2 Feature Integers and Absolute Value LESSON 1-2 Course 3 Additional Examples Evaluate 5 |z| for z = –2.4 5 |z| = 5 | –2.4| Substitute –2.4 for z. = 5(2.4) Find the absolute value of –2.4. = 12 Simplify. Quick Check Lesson Main Lesson 1-2 Feature Integers and Absolute Value LESSON 1-2 Course 3 Lesson Quiz Compare. Write <, >, or =. 1. |7| < |–8| 2. |–1| < |–6| 3. |–5| = |5| 4. |–10| > –2 5. 14 = |14| 6. 9 = |–9| Lesson Main Lesson 1-2 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Problem of the Day How many weeks are in 363 days? 51 6 weeks 7 Lesson Main Lesson 1-3 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Check Skills You’ll Need (For help, go to Lesson 1-1.) 1. Vocabulary Review How is simplifying an expression different from evaluating an expression? Evaluate each expression for x = 7. 2. x + 12 3. x – 5 4. 7x – 11 5. 12 + 9 x Check Skills You’ll Need Lesson Main Lesson 1-3 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Check Skills You’ll Need Solutions 1. Answers may vary. Sample: When you simplify an expression, you write its simplest name. When you evaluate an expression, you replace each variable with a number and then simplify. 2. 7 + 12 = 19 3. 7 – 5 = 2 4. 7(7) – 11 = 49 – 11 = 38 5. 12 + 9 = 21 = 3 7 7 Lesson Main Lesson 1-3 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Additional Examples Simplify each expression. a. –7 + (–7) –7 + (–7) = –14 Both 7s are negative, so the sum is negative. b. 8 + (–32) |–32| – |8| = 24 32 – 8 = 24 8 + (–32) = –24 Find the absolute value of each integer. Subtract. |–32| > |8|, so the sum is negative. Quick Check Lesson Main Lesson 1-3 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Additional Examples Simplify the expression 10 – 17. 10 – 17 = 10 + (–17) = –7 Add the opposite of 17, which is –17. Simplify. Quick Check Lesson Main Lesson 1-3 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Additional Examples The elevation of the Dead Sea is 396 m below sea level. A group of archaeologists leaves the shore of the Dead Sea. They stop after ascending 500 m. What is their elevation? Draw a diagram. The diagram shows that the elevation of the archaeologists is given by the expression –396 + 500. Lesson Main Lesson 1-3 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Additional Examples (continued) –396 + 500 = 104 Simplify by using the rule for adding integers with different signs. The elevation of the archaeologists is 104 m above sea level. Quick Check Lesson Main Lesson 1-3 Feature Adding and Subtracting Integers LESSON 1-3 Course 3 Lesson Quiz Simplify each expression. 1. 11 – 17 –6 4. 15 – 12 3 Lesson Main 2. 8 + (–6) 3. –1 – (–20) 19 2 5. –4 + (–6) –10 6. 9 – (–2) 11 Lesson 1-3 Feature Multiplying and Dividing Integers LESSON 1-4 Course 3 Problem of the Day How many numbers are in this sequence: 3, 6, 9, . . . 144? 48 Lesson Main Lesson 1-4 Feature Multiplying and Dividing Integers LESSON 1-4 Course 3 Check Skills You’ll Need (For help, go to Lesson 1-3) 1. Vocabulary Review Integers are the set of whole numbers, their ? , and 0. Simplify each expression. 2. –3 + (–3) 3. –5 + (–5) + (–5) 4. –2 + (–2) + (–2) Check Skills You’ll Need Lesson Main Lesson 1-4 Feature Multiplying and Dividing Integers LESSON 1-4 Course 3 Check Skills You’ll Need Solutions 1. opposites 2. –6 3. –10 + (–5) = –15 4. –6 Lesson Main Lesson 1-4 Feature Multiplying and Dividing Integers LESSON 1-4 Course 3 Additional Examples Simplify –11 • 2 • (–1). –11 and 2 have different signs, so the product is negative. –11 • 2 • (–1) = (–22)(–1) –22 and –1 have the same sign, so the product is positive. = 22 Quick Check Lesson Main Lesson 1-4 Feature Multiplying and Dividing Integers LESSON 1-4 Course 3 Additional Examples A diver descended 80 ft in 4 min. What was the rate of this descent? Represent a descent of 80 feet with –80. Then divide the descent by the number of minutes to find the change in elevation per minute. –80 = –20 4 The quotient of two integers with different signs is negative. The diver descended an average of 20 feet per minute. Quick Check Lesson Main Lesson 1-4 Feature Multiplying and Dividing Integers LESSON 1-4 Course 3 Additional Examples Evaluate y ÷ (x – y) – zy for x = 6, y = 9, and z = –2. y ÷ (x – y) – zy = 9(6 – 9) – (–2)(9) Substitute 6 for x, 9 for y, and –2 for z. = 9 ÷ (–3) – (–2)(9) Simplify 6 – 9 first. = –3 – (–18) Multiply and divide. = 15 Subtract. Quick Check Lesson Main Lesson 1-4 Feature Multiplying and Dividing Integers LESSON 1-4 Course 3 Lesson Quiz Simplify each expression. 1. 7(–4) 2. 10(2) 3. (–3)(6) –28 20 –18 4. (–5)(–9)(–2) –90 Lesson Main 5. –21 –3 6. 18 –2 –9 7 Lesson 1-4 Feature Properties of Numbers LESSON 1-5 Course 3 Problem of the Day Divide. Round answers to the nearest tenth. a. 75 ÷ 8 b. 740 ÷ 7 9.4 105.7 Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Check Skills You’ll Need (For help, go to Lesson 1-4.) 1. Vocabulary Review What does it mean to simplify an expression? Simplify each expression. 2. 23 + 15 + 73 – 12 3. 5 • 7 + 5 • 13 4. 6 • 7 + (–9) • 7 5. 8(–1) – 8(–5) Check Skills You’ll Need Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Check Skills You’ll Need Solutions 1. To simplify an expression means to replace it with its simplest name. 2. 38 + 73 – 12 = 111 – 12 = 99 3. 35 + 65 = 100 4. 42 + (–63) = –21 5. (–8) – (–40) = 32 Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Additional Examples Use mental math to simplify the expression. 3.8 + 17 + 6.2 What you think Look for numbers that are easy to add. The sum of 3.8 and 6.2 is 10. The sum of 10 and 17 is 27. So, 3.8 + 17 + 6.2 = 27. Why it works 3.8 + 17 + 6.2 = 3.8 + 6.2 + 17 Commutative Property = 10 + 17 Order of operations = 27 Simplify. Quick Check Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Additional Examples Use mental math to simplify the expression. 89 – 67 What you think Make the problem easier. Split 89 into parts. First, do 87 – 67 which is 20. Then add the rest of 89: 2 + 20 = 22. Why it works 89 – 67 = (2 + 87) – 67 Write 89 as 2 + 87. = 2 + (87 – 67) Associative Property = 2 + 20 Order of operations = 22 Simplify. Quick Check Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Additional Examples Use mental math to simplify –2 • 13 • 5. What you think It is easy to multiply with multiples of 10. –2 times 5 is –10, a multiple of 10. –10 times 13 is –130. So, –2 • 13 • 5 = –130. Why it works –2 • 13 • 5 = –2 • 5 • 13 Commutative Property = –10 • 13 Order of operations = –130 Simplify. Quick Check Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Additional Examples Find each product. a. 7(t – 5) 7(t – 5) = 7t – 7 • 5 Distributive Property = 7t – 35 Simplify. b. (d + 23)(–4) Distributive Property (d + 23)(–4) = d(–4) + 23(–4) = –4d + (–92) Simplify. = –4d – 92 Rewrite as a subtraction expression. Quick Check Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Additional Examples A student buys 11 CDs that cost $6.10 each. How much will the CDs cost? 11(6.1) = 11 (6 + 0.1) Replace 6.1 with 6 + 0.1. = 11(6) + 11(0.1) Distributive Property = 66 + 1.1 Multiply. = 67.1 Add. The CDs will cost $67.10 Quick Check Lesson Main Lesson 1-5 Feature Properties of Numbers LESSON 1-5 Course 3 Lesson Quiz Use the Distributive Property to rewrite each expression. 1. 8(a + 2) 8a + 16 2. –2(3x + 1) –6x – 2 3. 4 • 3 + 6 • 3 (4 + 6)3 4. (6 – r)(–4) –24 + 4r 5. 5 • 1 + 5 • 9 5(1 + 9) Lesson Main Lesson 1-5 Feature Solving Equations by Adding and Subtracting LESSON 1-6 Course 3 Problem of the Day How many different ways can the first five letters of the alphabet be arranged? 120 Lesson Main Lesson 1-6 Feature Solving Equations by Adding and Subtracting LESSON 1-6 Course 3 Check Skills You’ll Need (For help, go to Lesson 1-3.) 1. Vocabulary Review 4 and –4 are additive ? . Simplify each expression. 2. –4 + (–7) 3. 12 + (–12) 4. 3 – 10 5. –5 – 1 Check Skills You’ll Need Lesson Main Lesson 1-6 Feature Solving Equations by Adding and Subtracting LESSON 1-6 Course 3 Check Skills You’ll Need Solutions 1. inverses 2. –11 4. 3 + (–10) = –7 5. –5 + (–1) = –6 Lesson Main Lesson 1-6 3. | 12 | – | –12 | = 0; 0 Feature Solving Equations by Adding and Subtracting LESSON 1-6 Course 3 Additional Examples Solve –3 = m – 16. –3 = m – 16 –3 + 16 = m – 16 + 16 13 = m Check –3 = m – 16 –3 13 – 16 –3 = –3 Lesson Main Since 16 is subtracted from m, you must add 16 to each side to isolate m. Add 16 to each side. Simplify. Check the solution in the original equation. Substitute 13 for m. Subtract. Lesson 1-6 Quick Check Feature Solving Equations by Adding and Subtracting LESSON 1-6 Course 3 Additional Examples After Vida adds 17 new shells to her collection, she has a total of 52 shells. Write and solve an equation to find how many shells she had before adding the new ones. Words original number of shells plus new shells = total Let s = the original number of shells. Equation s + 17 = 52 Lesson Main s + 17 = 52 Since 17 is added to s, you must subtract 17 from each side to isolate s. Lesson 1-6 Feature Solving Equations by Adding and Subtracting LESSON 1-6 Course 3 Additional Examples (continued) s + 17 – 17 = 52 – 17 s = 35 Subtract 17 from each side. Simplify. Vida had 35 shells before adding the new ones. Check s + 17 = 52 35 + 17 52 52 = 52 Lesson Main Check the solution in the original equation. Substitute 35 for s. Add. Lesson 1-6 Quick Check Feature Solving Equations by Adding and Subtracting LESSON 1-6 Course 3 Lesson Quiz Solve each equation. 1. 8 = –7 + m 15 2. – 61 + t = 23 84 3. n – 15 = – 12 4. – 82 = r – 36 3 Lesson Main –46 Lesson 1-6 Feature Solving Equations by Multiplying and Dividing LESSON 1-7 Course 3 Problem of the Day Find the GCF for each pair of numbers. a. 12 and 18 6 Lesson Main b. 15 and 60 15 c. 54 and 60 6 Lesson 1-7 Feature Solving Equations by Multiplying and Dividing LESSON 1-7 Course 3 Check Skills You’ll Need (For help, go to Lesson 1-4.) 1. Vocabulary Review Inverse operations are operations that ? each other. Simplify each expression. 2. 6 • 4 3. –7 • 3 4. 10 5. –27 –5 –9 Check Skills You’ll Need Lesson Main Lesson 1-7 Feature Solving Equations by Multiplying and Dividing LESSON 1-7 Course 3 Check Skills You’ll Need Solutions 1. undo 2. 24 3. –21 4. –2 5. 3 Lesson Main Lesson 1-7 Feature Solving Equations by Multiplying and Dividing LESSON 1-7 Course 3 Additional Examples Solve y = 14. –3 y = 14 –3 Since y is divided by –3, you must multiply each side by –3 to isolate y. y Multiply each side by –3. –3 • –3 = –3 • 14 Check y = –42 Simplify. y = 14 –3 Check the solution in the original equation. –42 –3 14 14 = 14 Lesson Main Substitute –42 for y. Quick Check Divide. Lesson 1-7 Feature Solving Equations by Multiplying and Dividing LESSON 1-7 Course 3 Additional Examples Solve 265 = –5x. 265 = –5x Since x is multiplied by –5, you must divide each side by –5 to isolate x. 265 –5x = –5 –5 Divide each side by –5. –53 = x Simplify. Check 265 = –5x 265 –5(–53) 265 = 265 Lesson Main Check the solution in the original equation. Substitute –53 for x. Multiply. Lesson 1-7 Quick Check Feature Solving Equations by Multiplying and Dividing LESSON 1-7 Course 3 Lesson Quiz Solve each equation. 1. –12w = –60 5 3. 5x = –20 –4 Lesson Main n =9 –3 2. –27 4. –2 = r –16 32 Lesson 1-7 Feature