Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Inequalities and Relationships ? MODULE 13 LESSON 13.1 ESSENTIAL QUESTION Writing Inequalities How can you use inequalities and relationships to solve real-world problems? 6.9.A, 6.9.B, 6.10.B LESSON 13.2 Addition and Subtraction Inequalities 6.9.B, 6.9.C, 6.10 LESSON 13.3 Multiplication and Division Inequalities with Positive Numbers 6.9.B, 6.9.C, 6.10 LESSON 13.4 Multiplication and Division Inequalities with Rational Numbers © Houghton Mifflin Harcourt Publishing Company 6.9.B, 6.10.A, 6.10.B Real-World Video my.hrw.com my.hrw.com Some rides at amusement parks indicate a minimum height required for riders. You can model all the heights that are allowed to get on the ride with an inequality. my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 345 Are YOU Ready? Personal Math Trainer Complete these exercises to review skills you will need for this chapter. Understand Integers EXAMPLE my.hrw.com Online Assessment and Intervention Decide whether the integer is positive or negative: into the ground → negative Write the integer. A water well was drilled 735 feet into the ground. -735 Write an integer to represent each situation. 1. a loss of $75 2. a football player’s 3. spending $1,200 4. a climb of 2,400 gain of 9 yards on a flat screen feet TV Integer Operations EXAMPLE 3 × 8 = 24 -30 ÷ (-5) = 6 The product or quotient of two integers is positive if the signs of the integers are the same. 7 × (-4) = -28 -72 ÷ 9 = -8 The product or quotient of two integers is negative if the signs of the integers are different. Find the product or quotient. 9. 3 × (-7) 6. 15 ÷ (-5) 10. -64 ÷ 8 7. -8 × 6 11. -8 × (-2) 8. -100 ÷ (10) 12. 32 ÷ 2 Solve Multiplication Equations EXAMPLE 3 _ h = 15 4 4 _ _ · 3 h = 15 · _43 3 4 ·4 _____ h = 15 3 h = 20 Write the equation. 3 . Multiply both sides by the h is multiplied by __ 4 4 __ reciprocal, 3 , to isolate the variable. Simplify. Solve. 13. 9p = 108 346 Unit 4 14. _35 n = 21 15. _47 k = 84 3 16. __ e = 24 20 © Houghton Mifflin Harcourt Publishing Company 5. 6 × 9 Reading Start-Up Visualize Vocabulary Use the ✔ words to complete the graphic. >, < 3x - 5 4x + 4 = 12; x = 2 Evaluating Expressions 6×4 Vocabulary Review Words ✔ algebraic expression (expresión algebraica) evaluating (evaluar) ✔ greater than (mayor que) ✔ less than (menor que) like terms (términos semejantes) ✔ numerical expression (expresión numérica) properties of operations (propiedades de las operaciones) ✔ solution (solución) term (término, en una expresión) Preview Words Understand Vocabulary © Houghton Mifflin Harcourt Publishing Company Match the term on the left to the correct expression on the right. 1. solution of an inequality A. A value or values that make the inequality true. 2. coefficient B. A specific number whose value does not change. 3. constant C. The number that is multiplied by the variable in an algebraic expression. coefficient (coeficiente) constant (constante) solution of an inequality (solución de una desigualdad) variable (variable) Active Reading Two-Panel Flip Chart Create a two-panel flip chart to help you understand the concepts in this module. Label one flap “Adding and Subtracting Inequalities.” Label the other flap “Multiplying and Dividing Inequalities.” As you study each lesson, write important ideas under the appropriate flap. Module 13 347 MODULE 13 Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module. Represent solutions for onevariable, one-step equations and inequalities on number lines. What It Means to You You will learn to graph the solution of an inequality on a number line. Key Vocabulary UNPACKING EXAMPLE 6.9.B equation (ecuación) A mathematical sentence that shows that two expressions are equivalent. The temperature in a walk-in freezer must stay under 5 °C. Write and graph an inequality to represent this situation. inequality (desigualdad) A mathematical sentence that shows the relationship between quantities that are not equal. solution of an inequality (solución de una desigualdad) A value or values that make the inequality true. Write the inequality. Let t represent the temperature in the freezer. The temperature must be less than 5 °C. 6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. t<5 Graph the inequality. 0 5 10 What It Means to You You can model and solve a one-variable, one-step inequality. UNPACKING EXAMPLE 6.10.A Donny buys 3 binders and spends more than $9. How much did he spend on each binder? Let x represent the cost of one binder. Number of binders · Cost of a binder > Total cost of binders x > Use algebra tiles to model 3x > 9 and solve the inequality. x>3 Donny spent more than $3 on each binder. + + + 3 Visit my.hrw.com to see all the unpacked. my.hrw.com 348 Unit 4 · 9 > + + + + + + + + + © Houghton Mifflin Harcourt Publishing Company • Image Credits: Image Source/Corbis 6.9.B LESSON 13.1 Writing Inequalities ? ESSENTIAL QUESTION Expressions, equations, and relationships— 6.9.A Write … inequalities to represent constraints or conditions within problems. Also 6.9.B, 6.10.B. How can you use inequalities to represent real-world constraints or conditions? 6.9.A EXPLORE ACTIVITY Using Inequalities to Describe Quantities You can use inequality symbols with variables to describe quantities that can have many values. Symbol Meaning Word Phrases < Is less than Fewer than, below > Is greater than More than, above ≤ Is less than or equal to At most, no more than ≥ Is greater than or equal to At least, no less than A The lowest temperature ever recorded in Florida was -2 °F. Graph this temperature on the number line. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 © Houghton Mifflin Harcourt Publishing Company B The temperatures 0 °F, 3 °F, 6 °F, 5 °F, and -1 °F have also been recorded in Florida. Graph these temperatures on the number line. C How do the temperatures in B compare to -2? How can you see this relationship on the number line? D How many other numbers have the same relationship to -2 as the temperatures in B ? Give some examples. E Suppose you could graph all of the possible answers to line. What would the graph look like? F Let x represent all the possible answers to Complete this inequality: x D D on a number . -2 Lesson 13.1 349 Graphing the Solutions of an Inequality Math On the Spot my.hrw.com A solution of an inequality that contains a variable is any value of the variable that makes the inequality true. For example, 7 is a solution of x > -2, since 7 > -2 is a true statement. EXAMPLE 1 6.9.B Graph the solutions of each inequality. Check the solutions. A y ≤ -3 Math Talk STEP 1 Draw a solid circle at -3 to show that -3 is a solution. STEP 2 Shade the number line to the left of -3 to show that numbers less than -3 are solutions. Mathematical Processes Is -4 _14 a solution of y ≤ -3? Is -5.6? -5 -4 -3 -2 -1 STEP 3 Use a solid circle for an inequality that uses ≥ or ≤. 0 1 2 3 4 5 Check your solution. Choose a number that is on the shaded section of the number line, such as -4. Substitute -4 for y. -4 ≤ -3 -4 is less than -3, so -4 is a solution. B 1<m STEP 1 Draw an empty circle at 1 to show that 1 is not a solution. STEP 2 Shade the number line to the right of 1 to show that numbers greater than 1 are solutions. STEP 3 0 1 2 3 4 5 Check your answer. Substitute 2 for m. 1<2 1 is less than 2, so 2 is a solution. Reflect 1. 350 Unit 4 How is x < 5 different from x ≤ 5? © Houghton Mifflin Harcourt Publishing Company -5 -4 -3 -2 -1 Use an open circle for an inequality that uses > or <. YOUR TURN 2. Graph the solution of the inequality t ≤ -4. - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 Personal Math Trainer Online Assessment and Intervention my.hrw.com Writing Inequalities You can write an inequality to model the relationship between an algebraic expression and a number. You can also write inequalities to represent certain real-world situations. Math On the Spot my.hrw.com EXAMPL 2 EXAMPLE 6.9.A, 6.10.B Write an inequality that represents the phrase the sum of y and 2 is greater than 5. Draw a graph to represent the inequality. STEP 1 Write the inequality. Animated Math The sum of y and 2 is greater than 5. y+2 STEP 2 > Graph the solution. For y + 2 to have a value greater than 5, y must be a number greater than 3. -5 -4 -3 -2 -1 © Houghton Mifflin Harcourt Publishing Company STEP 3 my.hrw.com 5 Use an open circle at 3 and shade to the right of 3. 0 1 2 3 4 5 Check your solution by substituting a number greater than 3, such as 4, into the original inequality. 4+2>5 6>5 Substitute 4 for y. 6 is greater than 5, so 4 is a solution. B To test the temperature rating of a coat, a scientist keeps the temperature below 5 °C. Write and graph an inequality to represent this situation. STEP 1 Write the inequality. Let t represent the temperature in the lab. t<5 STEP 2 The temperature must be less than 5 °C. Graph the inequality. 0 1 2 3 4 5 6 7 8 9 10 Lesson 13.1 351 YOUR TURN Personal Math Trainer 3. Write an inequality that represents the phrase the sum of 1 and y is greater than or equal to 3 . Check to see if y = 1 is a solution. Online Assessment and Intervention my.hrw.com Write and graph an inequality to represent each situation. 4. The highest temperature in February was 6 °F. 0 1 2 3 4 5 6 7 8 9 10 11 12 5. Each package must weigh more than 2 ounces. -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 Guided Practice -5 -4 -3 -2 -1 0 1 2 3 4 5 2. Graph -3 > z. Check the graph using substitution. (Example 1) -5 -4 -3 -2 -1 0 1 2 3 4 5 3. Write an inequality that represents the phrase “the sum of 4 and x is less than 6.” Draw a graph that represents the inequality, and check your solution. (Example 2) -5 -4 -3 -2 -1 0 1 2 3 4 5 4. During hibernation, a garter snake’s body temperature never goes below 3 °C. Write and graph an inequality that represents this situation. (Example 2) -5 -4 -3 -2 -1 0 1 2 3 4 5 ? ? ESSENTIAL QUESTION CHECK-IN 5. Write an inequality to represent this situation: Nina wants to take at least $15 to the movies. How did you decide which inequality symbol to use? 352 Unit 4 © Houghton Mifflin Harcourt Publishing Company 1. Graph 1 ≤ x. Use the graph to determine which of these numbers are solutions of the inequality: -1, 3, 0, 1 (Explore Activity and Example 1) Name Class Date 13.1 Independent Practice Personal Math Trainer 6.9.A, 6.9.B, 6.10.B my.hrw.com Online Assessment and Intervention 6. Which of the following numbers are solutions to x ≥ 0? -5, 0.03, -1, 0, 1.5, -6, _12 Graph each inequality. 7. t ≤ 8 8. -7 < h 9. x ≥ -9 10. n > 2.5 11. -4 _12 >x -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 - 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Write an inequality that matches the number line model. 12. © Houghton Mifflin Harcourt Publishing Company 13. 14. 15. - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 16. A child must be at least 48 inches tall to ride a roller coaster. a. Write and graph an inequality to represent this situation. 38 40 42 44 46 48 50 52 54 56 58 b. Can a child who is 46 inches tall ride the roller coaster? Explain. Lesson 13.1 353 Write and graph an inequality to represent each situation. 17. The stock is worth at least $14.50. 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 18. The temperature is less than 3.5 °F. -5 -4 -3 -2 -1 19. The goal of the fundraiser is to make more than $150. 0 50 100 150 200 250 300 Work Area FOCUS ON HIGHER ORDER THINKING 20. Communicate Mathematical Ideas Explain how to graph the inequality 8 ≥ y. 21. Represent Real-World Problems The number line shows an inequality. Describe a real-world situation that the inequality could represent. 1 2 3 4 5 22. Critique Reasoning Natasha is trying to represent the following situation with a number line model: There are fewer than 5 students in the cafeteria. She has come up with two possible representations, shown below. Which is the better representation, and why? 354 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 Unit 4 © Houghton Mifflin Harcourt Publishing Company 0 LESSON 13.2 ? Addition and Subtraction Inequalities Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step… inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B. ESSENTIAL QUESTION How can you solve an inequality involving addition or subtraction? 6.10.A EXPLORE ACTIVITY Modeling One-Step Inequalities You can use algebra tiles to model an inequality involving addition. On a day in January in Watertown, NY, the temperature was 5 °F at dawn. By noon it was at least 8 °F. By how many degrees did the temperature increase? A Let x represent the increase in temperature. Write an inequality. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Janusz Wrobel/ Alamy Temperature at dawn + Increase in temperature + B The model shows 5 + x ≥ 8. How many tiles must you remove from each side to isolate x on one side of the inequality? ≥ 8 ≥ 8 + + + + + 5 Circle these tiles. + + ≥ + + + + + + + + x ≥ 8 C What values of x make this inequality true? Graph the solution of the inequality on the number line. x≥ -5 -4 -3 -2 -1 0 1 2 3 4 5 Reflect 1. Analyze Relationships How is solving the inequality 5 + x ≥ 8 like solving the equation 5 + x = 8? How is it different? Math Talk Mathematical Processes Could the temperature have increased by 2 degrees by noon? Could it have increased by 5 degrees? Explain. Lesson 13.2 355 Using Properties of Inequalities Addition and Subtraction Properties of Inequality Math On the Spot Addition Property of Inequality Subtraction Property of Inequality my.hrw.com You can add the same number to You can subtract the same number both sides of an inequality and the from both sides of an inequality inequality will remain true. and the inequality will remain true. EXAMPLE 1 6.9.B, 6.10.B Solve each inequality. Graph and check the solution. A x + 5 < -12 STEP 1 Solve the inequality. x + 5 < -12 5 ____ -5 ____ x < -17 STEP 2 Use the Subtraction Property of Inequality. Subtract 5 from both sides. Graph the solution. -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 STEP 3 Math Talk Mathematical Processes -13 < -12 B 8≤ y-3 STEP 1 Solve the inequality. 8 ≤y - 3 + 3 + 3 _ _ 11 ≤ y 356 Unit 4 The inequality is true. Use the Addition Property of Inequality. Add 3 to both sides. You can rewrite 11 ≤ y as y ≥ 11. STEP 2 Graph the solution. STEP 3 Check the solution. Substitute a solution from the shaded part of your number line into the original inequality. 5 6 7 8 9 10 11 12 13 14 15 ? 8 ≤ 12 - 3 Substitute 12 for y in 8 ≤ y - 3 8≤ 9 The inequality is true. © Houghton Mifflin Harcourt Publishing Company What would it tell you if the inequality is false when you check the solution? Check the solution. Substitute a solution from the shaded part of your number line into the original inequality. ? -18 + 5 < -12 Substitute -18 for x into x + 5 < -12 YOUR TURN Solve each inequality. Graph and check the solution. 2. y - 5 ≥ -7 Personal Math Trainer 3. 21 > 12 + x Online Assessment and Intervention my.hrw.com -5 -4 -3 -2 -1 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 Interpreting Inequalities as Comparisons You can write a real-world problem for a given inequality. Examine each number and mathematical operation in the inequality. EXAMPL 2 EXAMPLE Math On the Spot 6.9.C my.hrw.com Write a real-world problem for the inequality 60 ≥ w + 5. Then solve the inequality. STEP 1 Examine each part of the inequality. w is the unknown quantity. 5 is added to w. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Lew Robertson/Corbis 60 is greater than or equal to a number added to 5. STEP 2 Write a comparison that the inequality could describe. June’s dog will travel to a dog show in a pet carrier. The pet carrier weighs 5 pounds. The total weight of the pet carrier and the dog must be no more than 60 pounds. What inequality describes the weight of June’s dog? STEP 3 Solve the inequality. 60 ≥ w + 5 -5 ____ 55 ≥ w -5 ____ June’s dog currently weighs ≤ 55 pounds. Reflect 4. If you were to graph the solution, would all points on the graph make sense for the situation? Lesson 13.2 357 YOUR TURN Personal Math Trainer 5. Write a real-world problem that can be modeled by x - 13 > 20. Solve your problem and tell what values make sense for the situation. Online Assessment and Intervention my.hrw.com Guided Practice 1. Write the inequality shown on the model. Circle the tiles you would remove from each side and give the solution. (Explore Activity) Inequality: + + + Solution: + ≤ + + + + + Solve each inequality. Graph and check the solution. (Example 1) 2. x + 4 ≥ 9 0 1 2 3 4 5 6 7 3. 5 > z - 3 8 9 10 4. t + 5 > 12 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 5. y - 4 < 2 0 1 2 3 4 5 6 7 8 9 10 ? ? ESSENTIAL QUESTION CHECK-IN 7. Explain how to solve 7 + x ≥ 12. Tell what property of inequality you would use. 358 Unit 4 © Houghton Mifflin Harcourt Publishing Company 6. Write a real-world problem that can be represented by the inequality y - 4 < 2. Solve the inequality and tell whether all values in the solution make sense for the situation. (Example 2) Name Class Date 13.2 Independent Practice Personal Math Trainer 6.9.B, 6.9.C, 6.10.A, 6.10.B my.hrw.com Online Assessment and Intervention Solve each inequality. Graph and check the solution. 8. x - 35 > 15 0 10 20 30 40 50 60 70 80 90 100 9. 193 + y ≥ 201 0 1 2 3 4 5 6 7 8 9 10 10. y - 5 ≥ -15 - 12 - 11 - 10 - 9 -8 -7 -6 -5 -4 -3 -2 - 15 - 14 - 13 - 12 - 11 - 10 - 9 -8 -7 -6 -5 11. 15 ≥ z + 26 Write an inequality to solve each problem. © Houghton Mifflin Harcourt Publishing Company 12. The water level in the aquarium’s shark tank is always greater than 25 feet. If the water level decreased by 6 feet during cleaning, what was the water level before the cleaners took out any water? 13. Danny has at least $15 more than his big brother. Danny’s big brother has $72. How much money does Danny have? 14. The vet says that Ray’s puppy will grow to be at most 28 inches tall. Ray’s puppy is currently 1 foot tall. How much more will the puppy grow? 15. Pierre’s parents ordered some pizzas for a party. 4.5 pizzas were eaten at the party. There were at least 5_12 whole pizzas left over. How many pizzas did Pierre’s parents order? 16. To get a free meal at his favorite restaurant, Tom needs to spend $50 or more at the restaurant. He has already spent $30.25. How much more does Tom need to spent to get his free meal? Lesson 13.2 359 17. Multistep The table shows Marco’s checking account activity for the first week of June. a. Marco wants his total deposits for the month of June to exceed $1,500. Write and solve an inequality to find how much more he needs to deposit to meet this goal. Deposit – Paycheck $520.45 Purchase – Grocery Store $46.50 Purchase – Movie Theatre $24.00 Purchase – Water bill $22.82 b. Marco wants his total purchases for the month to be less than $450. Write and solve an inequality to find how much more he can spend and still meet this goal. c. There are three weeks left in June. If Marco spends the same amount in each of these weeks that he spent during the first week, will he meet his goal of spending less than $450 for the entire month? Justify your answer. FOCUS ON HIGHER ORDER THINKING Work Area 19. Critical Thinking José solved the inequality 3 > x + 4 and got x < 1. Then, to check his solution, he substituted -2 into the original inequality to check his solution. Since his check worked, he believes that his answer is correct. Describe another check José could perform that will show his solution is not correct. Then explain how to solve the inequality. 20. Look for a Pattern Solve x + 1 > 10, x + 11 > 20, and x + 21 > 30. Describe a pattern. Then use the pattern to predict the solution of x + 9,991 > 10,000. 360 Unit 4 © Houghton Mifflin Harcourt Publishing Company 18. Critique Reasoning Kim solved y - 8 ≤ 10 and got y ≤ 2. What might Kim have done wrong? LESSON 13.3 ? Multiplication and Division Inequalities with Positive Numbers ESSENTIAL QUESTION Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B. How can you solve an inequality involving multiplication or division with positive numbers? 6.10.A EXPLORE ACTIVITY Modeling One-Step Inequalities You can use algebra tiles to solve inequalities that involve multiplying positive numbers. Dominic is buying school supplies. He buys 3 binders and spends more than $9. How much did he spend on each binder? A Let x represent the cost of one binder. Write an inequality. Number of binders • Cost of a binder • B The model shows the inequality from There are 9 > 9 . + + + x-tiles, so draw circles to separate the tiles into © Houghton Mifflin Harcourt Publishing Company A > > equal groups. + + + + + + + + + How many units are in each group? C What values make the inequality you wrote in A -5 -4 - 3 - 2 - 1 0 1 2 3 4 5 true? Graph the solution of the inequality. Reflect 1. Analyze Relationships Is 3.25 a solution of the inequality you wrote in A ? If so, does that solution make sense for the situation? 2. Represent Real-World Problems Rewrite the situation in represent the inequality 3x < 9. A to Lesson 13.3 361 Solving Inequalities Involving Multiplication and Division Math On the Spot You can use properties of inequality to solve inequalities involving multiplication and division with positive integers. my.hrw.com Multiplication and Division Properties of Inequality • You can multiply both sides of an inequality by the same positive number and the inequality will remain true. • You can divide both sides of an inequality by the same positive number and the inequality will remain true. EXAMPLE 1 6.9.B, 6.10.B Solve each inequality. Graph and check the solution. A 12x < 24 Solve the inequality. 12x __ ___ < 24 12 12 Math Talk Mathematical Processes Are all negative numbers solutions to 12x < 24? Explain. Divide both sides by 12. x<2 STEP 2 Graph the solution. -5 -4 - 3 - 2 - 1 STEP 3 Use an open circle to show that 2 is not a solution. 0 1 2 3 4 5 Check the solution by substituting a solution from the shaded part of the graph into the original inequality. ? Substitute 0 for x in the original inequality. 12(0) < 24 0 < 24 The inequality is true. y B _3 ≥ 5 STEP 1 Solve the inequality. y 3 ( _3 ) ≥ 3(5) Multiply both sides by 3. y ≥ 15 STEP 2 Use a closed circle to show that 15 is a solution. Graph the solution. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 STEP 3 Check the solution by substituting a solution from the shaded part of the graph into the original inequality. 18 ? __ ≥5 Substitute 18 for x in the original inequality. 3 6≥5 362 Unit 4 The inequality is true. © Houghton Mifflin Harcourt Publishing Company STEP 1 YOUR TURN Solve each inequality. Graph and check the solution. Personal Math Trainer 3. 5x ≥ 100 15 16 17 18 19 20 21 22 23 24 25 Online Assessment and Intervention my.hrw.com 4. _4z < 11 40 41 42 43 44 45 46 47 48 49 50 Solving Real-World Problems You can use multiplication and division inequalities to model and solve real-world problems. EXAMPL 2 EXAMPLE Problem Solving Math On the Spot 6.10.A my.hrw.com Cy is making a square flag. He wants the perimeter to be at least 22 inches. Write and solve an inequality to find the possible side lengths. Analyze Information Find the possible lengths of 1 side of a square that has a perimeter of at least 22 inches. Formulate a Plan Write and solve a multiplication inequality. Use the fact that the perimeter of a square is 4 times its side length. © Houghton Mifflin Harcourt Publishing Company Justify and Evaluate Solve 4x ≥ 22 Let x represent a side length. 4x __ __ ≥ 22 4 4 Divide both sides by 4. x ≥ 5.5 The side lengths must be greater than or equal to 5.5 in. Cy’s flag should have a side length of 5.5 inches or more. Justify and Evaluate Check the solution by substituting a value in the solution set in the original inequality. Try x = 6. ? 4(6) ≥ 22 24 ≥ 22 Substitute 6 for x. The statement is true. Cy’s flag could have a side length of 6 inches. Lesson 13.3 363 Reflect 5. Represent Real-World Problems Write and solve a real-world problem for the inequality 4x ≤ 60. YOUR TURN Personal Math Trainer Online Assessment and Intervention 6. A paperweight must weigh less than 4 ounces. Brittany wants to make 6 paperweights using sand. Write and solve an inequality to find the possible weight of the sand she needs. my.hrw.com Guided Practice 1. Write the inequality shown on the model. Circle groups of tiles to show the solution. Then write the solution. (Explore Activity) Inequality: + + < + + + + + + + + Solution: Solve each inequality. Graph and check the solution. (Example 1) 35 36 37 38 39 40 41 42 43 44 45 3. _3r ≥ 11 30 31 32 33 34 35 36 37 38 39 40 4. Karen divided her books and put them on 6 shelves. There were at least 14 books on each shelf. How many books did she have? Write and solve an inequality to represent this situation. (Example 2) ? ? ESSENTIAL QUESTION CHECK-IN 5. Explain how to solve and check the solution to 5x < 40 using properties of inequalities. 364 Unit 4 © Houghton Mifflin Harcourt Publishing Company 2. 8y < 320 Name Class Date 13.3 Independent Practice Personal Math Trainer 6.9.B, 6.9.C, 6.10.A, 6.10.B Write and solve an inequality for each problem. 6. Geometry The perimeter of a regular hexagon is at most 42 inches. Find the possible side lengths of the hexagon. 7. Tamar needs to make at least $84 at work on Tuesday to afford dinner and a movie on Wednesday night. She makes $14 an hour at her job. How many hours does she need to work on Tuesday? © Houghton Mifflin Harcourt Publishing Company 8. In a litter of 7 kittens, each kitten weighs more than 3.5 ounces. Find the possible total weight of the litter. 9. To cover his rectangular backyard, Will needs at least 170.5 square feet of sod. The length of Will’s yard is 15.5 feet. What are the possible widths of Will’s yard? my.hrw.com Online Assessment and Intervention 12. Steve pays less than $32 per day to rent his apartment. August has 31 days. What are the possible amounts Steve could pay for rent in August? 13. If you were to graph the solution for exercise 12, would all points on the graph make sense for the situation? Explain. 14. Multistep Lina bought 4 smoothies at a health food store. The bill was less than $16. a. Write and solve an inequality to represent the cost of each smoothie. b. What values make sense for this situation? Explain. c. Graph the values that make sense for this situation on the number line. Solve each inequality. Graph and check the solution. 0 1 2 3 4 5 Solve each inequality. 10. 10x ≤ 60 0 1 2 3 4 5 6 7 8 9 10 p 15. __ ≤ 30 13 16. 2t > 324 11. _2t > 0 -5 -4 - 3 - 2 - 1 -1 0 1 2 3 4 5 17. 12y ≥ 1 x 18. ___ < 11 9.5 Lesson 13.3 365 The sign shows some prices at a produce stand. 19. Tom has $10. What is the greatest amount of spinach he can buy? Price per Pound Produce $1.25 Onions $0.99 Yellow Squash $3.00 Spinach $0.50 Potatoes 20. Gary has enough money to buy at most 5.5 pounds of potatoes. How much money does Gary have? 21. Florence wants to spend no more than $3 on onions. Will she be able to buy 2.5 pounds of onions? Explain. 22. The produce buyer for a local restaurant wants to buy more than 30 lb of onions. The produce buyer at a local hotel buys exactly 12 pounds of spinach. Who spends more at the produce stand? Explain. FOCUS ON HIGHER ORDER THINKING Work Area 24. Represent Real-World Problems Write and solve a word problem that can be represented with 240 ≤ 2x. 25. Persevere in Problem Solving A rectangular prism has a length of 13 inches and a width of _12 inch. The volume of the prism is at most 65 cubic inches. Find all possible heights of the prism. Show your work. 366 Unit 4 © Houghton Mifflin Harcourt Publishing Company 2 23. Critique Reasoning A student solves _5r ≤ _25 and gets r ≤ __ . What is the 25 correct solution? What mistake might the student have made? LESSON 13.4 ? Multiplication and Division Inequalities with Rational Numbers ESSENTIAL QUESTION Expressions, equations, and relationships—6.9.B Represent solutions for one-step inequalities on number lines. Also 6.10.A, 6.10.B How do you solve inequalities that involve multiplication and division of integers? EXPLORE ACTIVITY 6.10.A Investigating Inequality Symbols You have seen that multiplying or dividing both sides of an inequality by the same positive number results in an equivalent inequality. How does multiplying or dividing both sides by the same negative number affect an inequality? A Complete the tables. © Houghton Mifflin Harcourt Publishing Company Inequality Multiply each side by: 3<4 2 2 ≥ -3 3 5>2 -1 -8 > -10 -8 Inequality Divide each side by: 4<8 4 12 ≥ -15 New inequality New inequality is true or false? New inequality New inequality is true or false? 3 -16 ≤ 12 -4 15 > 5 -5 B What do you notice when you multiply or divide both sides of an inequality by the same negative number? C How could you make each of the multiplication and division inequalities that were not true into true statements? Lesson 13.4 367 Multiplication and Division Properties of Inequality Math On the Spot Recall that you can multiply or divide both sides of an inequality by the same positive number, and the statement will still be true. my.hrw.com Multiplication and Division Properties of Inequality • If you multiply or divide both sides of an inequality by the same negative number, you must reverse the inequality symbol for the statement to still be true. EXAMPLE 1 6.9.B, 6.10.B Solve each inequality. Graph and check the solution. My Notes A -4x > 52 STEP 1 Solve the inequality. -4x > 52 Divide both sides by -4. Reverse the inequality symbol. -4x 52 ____ < ___ -4 -4 x < -13 STEP 2 Graph the solution. STEP 3 Check your answer using substitution. ? Substitute -15 for x in -4x > 52. -4(-15) > 52 -8 The statement is true. y B - _3 < -5 STEP 1 Solve the inequality. y - _3 < -5 y -3(- _3 ) > -3(-5) Multiply both sides by -3. Reverse the inequality symbol. y > 15 STEP 2 Graph the solution. 10 11 12 13 14 15 16 17 18 19 20 STEP 3 Check your answer using substitution. ? 18 < - ___ -5 3 -6 < -5 368 Unit 4 y Substitute 18 for y in - __ < -5. 3 The inequality is true. © Houghton Mifflin Harcourt Publishing Company 60 > 52 -15 -14 -13 -12 -11 -10 -9 YOUR TURN Solve each inequality. Graph and check the solution. 1. -10y < 60 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 2. 7 ≥ - __t 6 0 1 -47 -46 -45 -44 -43 -42 -41 -40 Personal Math Trainer Online Assessment and Intervention my.hrw.com Solving a Real-World Problem Although elevations below sea level are represented by negative numbers, we often use absolute value to describe these elevations. For example, -50 feet relative to sea level might be described as 50 feet below sea level. EXAMPL 2 EXAMPLE Problem Solving Math On the Spot my.hrw.com 6.10.A A marine submersible descends more than 40 feet below sea level. As it descends from sea level, the change in elevation is -5 feet per second. For how many seconds does it descend? Analyze Information © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jeffrey L. Rotman/Peter Arnold Inc/Getty Images Rewrite the question as a statement. • Find the number of seconds that the submersible decends below sea level. List the important information: • The final elevation is greater than 40 feet below sea level or < -40 feet. • The rate of descent is -5 feet per second. Formulate a Plan Write and solve an inequality. Use this fact: Rate of change in elevation × Time in seconds = Total change in elevation Justify and Evaluate Solve -5t < -40 -5t > ____ -40 ____ -5 -5 t> 8 Rate of change × Time < Maximum elevation Divide both sides by -5. Reverse the inequality symbol. The submersible descends for more than 8 seconds. Justify and Evaluate Check your answer by substituting a value greater than 8 seconds in the original inequality. ? Substitute 9 for t in the inequality -5t < -40. -5(9) < -40 -45 < -40 The statement is true. Lesson 13.4 369 YOUR TURN Personal Math Trainer Online Assessment and Intervention 3. Every month, $35 is withdrawn from Tom’s savings account to pay for his gym membership. He has enough savings to withdraw no more than $315. For how many months can Tony pay for his gym membership? my.hrw.com Guided Practice Solve each inequality. Graph and check the solution. (Explore Activity and Example 1) 1. -7z ≥ 21 2. -__t > 5 4 -10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 -50 -40 -30 -20 -10 3. 11x < -66 t >5 4. -___ 10 0 10 20 30 -8 -7 - 6 - 5 - 4 - 3 - 2 - 1 40 0 50 0 1 2 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 5. For a scientific experiment, a physicist must make sure that the temperature of a metal does not get colder than -80 °C. The metal begins the experiment at 0 °C and is cooled at a steady rate of -4 °C per hour. How long can the experiment run? (Example 2) b. Solve the inequality in part a. How long will it take the physicist to change the temperature of the metal? c. The physicist has to repeat the experiment if the metal gets cooler than -80 °C. How many hours would the physicist have to cool the metal for this to happen? ? ? ESSENTIAL QUESTION CHECK-IN 6. Suppose you are solving an inequality. Under what circumstances do you reverse the inequality symbol? 370 Unit 4 © Houghton Mifflin Harcourt Publishing Company a. Let t represent time in hours. Write an inequality. Use the fact that the rate of change in temperature times the number of seconds equals the final temperature. Name Class Date 13.4 Independent Practice 6.9.B, 6.10.A, 6.10.B Personal Math Trainer my.hrw.com Solve each inequality. Graph and check your solution. q 7. - __ ≥ -1 7 0 1 2 3 4 5 6 7 8 9 10 Online Assessment and Intervention 14. A veterinarian tells Max that his cat should lose no more than 30 ounces. The veterinarian suggests that the cat should lose 7 ounces or less per week. What is the shortest time in weeks and days it would take Max’s cat to lose the 30 ounces? 8. -12x < 60 -10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 y 9. 0.5 ≤ __ 8 0 1 2 3 4 5 6 7 8 9 10 10. 36 < -6r -10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 15. The elevation of an underwater cave is -120 feet relative to sea level. A submarine descends to the cave. The submarine’s rate of change in elevation is no greater than -12 feet per second. How long will it take to reach the cave? 11. -12 > 2x -8 -7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 x ≤ -0.5 12. - __ 6 © Houghton Mifflin Harcourt Publishing Company -5 -4 - 3 - 2 - 1 0 1 2 3 4 5 13. Multistep Parav is playing a game in which he flips a counter that can land on either a -6 or a 6. He adds the point values of all the flips to find his total score. To win, he needs to get a score less than -48. a. Assuming Parav only gets -6s when he flips the counter, how many times does he have to flip the counter? 16. The temperature of a freezer is never greater than -2 °C. Yesterday the temperature was -10 °C, but it increased at a steady rate of 1.5 °C per hour. How long in hours and minutes did the temperature increase inside the freezer? 17. Explain the Error A student's solution to the inequality -6x > 42 was x > -7. What error did the student make in the solution? What is the correct answer? b. Suppose Parav flips the counter and gets five 6s and twelve -6s when he plays the game. Does he win? Explain. Lesson 13.4 371 Solve each inequality. 18. 18 ≤ -2x x < - __ 1 20. - __ 8 2 1 22. 4x < __ 5 x ≤ -23 19. - __ 7 21. 0.4 < -x x ≤ -30 23. - ___ 0.8 24. Use the order of operations to simplify the left side of the inequality below. What values of x make the inequality a true statement? - _12 (32 + 7)x > 32 FOCUS ON HIGHER ORDER THINKING Work Area 26. Communicate Mathematical Thinking Van thinks that the answer to -3x < 12 is x < -4. How would you convince him that his answer is incorrect? 372 Unit 4 © Houghton Mifflin Harcourt Publishing Company 25. Counterexamples John says that if one side of an inequality is 0, you don’t have to reverse the inequality symbol when you multiply or divide both sides by a negative number. Find an inequality that you can use to disprove John’s statement. Explain your thinking. MODULE QUIZ Ready Personal Math Trainer 13.1 Writing Inequalities Online Assessment and Intervention my.hrw.com Write an inequality to represent each situation, then graph the solutions. 1. There are fewer than 8 gallons of gas in the tank. 0 1 2 3 4 5 6 7 8 9 10 2. There are at least 3 pieces of gum left in the pack. 0 1 2 3 4 5 6 7 8 9 10 3. The valley was at least 4 feet below sea level. - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 13.2 Addition and Subtraction Inequalities Solve each inequality. Graph the solution. 4. c - 28 > -32 -10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 5. 0 v + 17 ≤ 20 0 1 2 3 4 5 6 7 8 9 10 6. Today’s high temperature of 80 °F is at least 16 ° warmer than yesterday’s high temperature. What was yesterday’s high temperature? © Houghton Mifflin Harcourt Publishing Company 13.3, 13.4 Multiplication and Division Inequalities Solve each inequality. Graph the solution. 8. __a2 < 4 7. 7f ≤ 35 0 1 2 3 4 5 6 7 8 9 10 k <3 10. ___ -3 9. -25g ≥ 150 -10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 0 -10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 Module 13 373 Personal Math Trainer MODULE 13 MIXED REVIEW Texas Test Prep Selected Response 1. Em saves at least 20% of what she earns each week. If she earns $140 each week for 4 weeks, which inequality describes the total amount she saves? A t > 112 B t ≥ 112 C t < 28 D t ≤ 28 2. Which number line represents the inequality r > 6? A 0 1 2 3 4 5 6 7 8 9 10 my.hrw.com 5. The number line below represents the solution to which inequality? 0 1 2 3 4 5 6 7 8 9 10 m A __ > 2.2 4 C B 2m < 17.6 D 5m > 40 m __ > 2.5 3 6. Which number line shows the solution to w - 2 ≤ 8? A 0 1 2 3 4 5 6 7 8 9 10 B 0 1 2 3 4 5 6 7 8 9 10 C 0 1 2 3 4 5 6 7 8 9 10 B 0 1 2 3 4 5 6 7 8 9 10 Online Assessment and Intervention D 0 1 2 3 4 5 6 7 8 9 10 C 0 1 2 3 4 5 6 7 8 9 10 Gridded Response 0 1 2 3 4 5 6 7 8 9 10 3. For which inequality below is z = 3 a solution? A z+5≥9 7. Hank needs to save at least $150 to ride the bus to his grandparent’s home. If he saves $12 a week, what is the least number of weeks he needs to save? B z+5>9 z+5≤8 0 0 0 0 0 0 D z+5<8 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 C 4. What is the solution to the inequality −6x < −18? A x>3 B x<3 C x≥3 D x≤3 374 . Unit 4 © Houghton Mifflin Harcourt Publishing Company D