Inequalities and Relationships

Transcription

1 Inequalities and Relationships MODULE 13? ESSENTIAL QUESTION How can you use inequalities and relationships to solve real-world problems? LESSON 13.1 Writing Inequalities LESSON 13.2 Addition and Subtraction Inequalities 6.9.A, 6.9.B, 6.10.B 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 6.9.B, 6.10.A, 6.10.B Real-World Video 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. 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

2 Are YOU Ready? Complete these exercises to review skills you will need for this chapter. Understand Integers Personal Math Trainer Online Assessment and Intervention EXAMPLE A water well was drilled 735 feet into the ground Decide whether the integer is positive or negative: into the ground negative Write the integer. Write an integer to represent each situation. 1. a loss of $75 2. a football player s gain of 9 yards Integer Operations 3. spending $1,200 on a flat screen TV 4. a climb of 2,400 feet EXAMPLE 3 8 = (-5) = 6 7 (-4) = = -8 The product or quotient of two integers is positive if the signs of the integers are the same. The product or quotient of two integers is negative if the signs of the integers are different. Find the product or quotient (-5) (10) 9. 3 (-7) (-2) Solve Multiplication Equations Solve. EXAMPLE 3_ 4 h = 15 4_ 3 _ 3 4 h = 15 4_ 3 h = h = p = Write the equation. h is multiplied by 3. Multiply both sides by the 4 reciprocal, 4, to isolate the variable. 3 Simplify. 3_ 5 n = _ 7 k = e = Unit 4

3 Reading Start-Up Visualize Vocabulary Use the words to complete the graphic. >, < 3x - 5 Understand Vocabulary Match the term on the left to the correct expression on the right. 1. solution of an inequality Evaluating Expressions 4x 4 = 12; x = A. A value or values that make the inequality true. 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 coefficient (coeficiente) constant (constante) solution of an inequality (solución de una desigualdad) variable (variable) 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. 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

4 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. 6.9.B Represent solutions for onevariable, one-step equations and inequalities on number lines. Key Vocabulary equation (ecuación) A mathematical sentence that shows that two expressions are equivalent. 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. What It Means to You You will learn to graph the solution of an inequality on a number line. UNPACKING EXAMPLE 6.9.B The temperature in a walk-in freezer must stay under 5 C. Write and graph an inequality to represent this situation. Write the inequality. Let t represent the temperature in the freezer. The temperature must be less than 5 C. t < 5 Graph the inequality A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. Visit to see all the unpacked. 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 3 x > 9 Use algebra tiles to model 3x > 9 and solve the inequality. x > 3 Donny spent more than $3 on each binder. > Image Credits: Image Source/Corbis 348 Unit 4

5 ? 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? EXPLORE ACTIVITY 6.9.A 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. B C D E 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. How do the temperatures in B compare to -2? How can you see this relationship on the number line? How many other numbers have the same relationship to -2 as the temperatures in B? Give some examples. Suppose you could graph all of the possible answers to D on a number line. What would the graph look like? F Let x represent all the possible answers to D. Complete this inequality: x -2 Lesson

6 Math On the Spot Graphing the Solutions of an Inequality 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 B Graph the solutions of each inequality. Check the solutions. A y -3 Math Talk Mathematical Processes Is -4 1_ 4 a solution of y -3? Is -5.6? STEP 1 STEP 2 STEP 3 Draw a solid circle at -3 to show that -3 is a solution. Shade the number line to the left of -3 to show that numbers less than -3 are solutions Check your solution. Use a solid circle for an inequality that uses or. Choose a number that is on the shaded section of the number line, such as -4. Substitute -4 for y is less than -3, so -4 is a solution. B 1 < m STEP 1 STEP 2 Draw an empty circle at 1 to show that 1 is not a solution. Shade the number line to the right of 1 to show that numbers greater than 1 are solutions. Use an open circle for an inequality that uses > or <. STEP Check your answer. Substitute 2 for m. 1 < 2 1 is less than 2, so 2 is a solution. Reflect 1. How is x < 5 different from x 5? 350 Unit 4

7 YOUR TURN 2. Graph the solution of the inequality t 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. EXAMPLE A, 6.10.B Personal Math Trainer Online Assessment and Intervention Math On the Spot 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. The sum of y and 2 is greater than 5. y 2 > 5 Animated Math STEP 2 Graph the solution. For y 2 to have a value greater than 5, y must be a number greater than 3. Use an open circle at 3 and shade to the right of B STEP 3 Check your solution by substituting a number greater than 3, such as 4, into the original inequality. 4 2 > 5 6 > 5 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 STEP 2 Write the inequality. Let t represent the temperature in the lab. t < 5 Graph the inequality. Substitute 4 for y. 6 is greater than 5, so 4 is a solution. The temperature must be less than 5 C Lesson

8 YOUR TURN Personal Math Trainer Online Assessment and Intervention 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. Write and graph an inequality to represent each situation. 4. The highest temperature in February was 6 F Each package must weigh more than 2 ounces Guided Practice 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) Graph -3 > z. Check the graph using substitution. (Example 1) 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) ? 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) 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? Unit 4

9 Name Class Date 13.1 Independent Practice 6.9.A, 6.9.B, 6.10.B Personal Math Trainer Online Assessment and Intervention 6. Which of the following numbers are solutions to x 0? -5, 0.03, -1, 0, 1.5, -6, 1_ 2 Graph each inequality. 7. t < h 9. x n > _ 2 >x Write an inequality that matches the number line model A child must be at least 48 inches tall to ride a roller coaster. a. Write and graph an inequality to represent this situation b. Can a child who is 46 inches tall ride the roller coaster? Explain. Lesson

10 Write and graph an inequality to represent each situation. 17. The stock is worth at least $ The temperature is less than 3.5 F The goal of the fundraiser is to make more than $ FOCUS ON HIGHER ORDER THINKING Work Area 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 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? Unit 4

11 ? LESSON 13.2 ESSENTIAL QUESTION Addition and Subtraction Inequalities How can you solve an inequality involving addition or subtraction? 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. EXPLORE ACTIVITY 6.10.A 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. Temperature at dawn Increase in 8 temperature Image Credits: Janusz Wrobel/ Alamy B C 8 The model shows 5 x 8. How many tiles must you remove from each side to isolate x on one side of the inequality? Circle these tiles x 8 What values of x make this inequality true? Graph the solution of the inequality on the number line. x 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

12 Using Properties of Inequalities Addition and Subtraction Properties of Inequality Math On the Spot Addition Property of Inequality You can add the same number to both sides of an inequality and the inequality will remain true. Subtraction Property of Inequality You can subtract the same number from both sides of an inequality and the inequality will remain true. EXAMPLE B, 6.10.B Solve each inequality. Graph and check the solution. A x 5 < -12 STEP 1 STEP 2 Solve the inequality. x 5 < -12 Use the Subtraction Property of Inequality. - 5 x < Subtract 5 from both sides. Graph the solution. Math Talk Mathematical Processes What would it tell you if the inequality is false when you check the solution? B STEP 3 8 y Check the solution. Substitute a solution from the shaded part of your number line into the original inequality <? -12 Substitute -18 for x into x 5 < < -12 The inequality is true. STEP 1 STEP 2 STEP 3 Solve the inequality. 8 y - 3 _ 3 _ 3 11 y Graph the solution. Use the Addition Property of Inequality. Add 3 to both sides. You can rewrite 11 y as y Check the solution. Substitute a solution from the shaded part of your number line into the original inequality. 8? 12-3 Substitute 12 for y in 8 y The inequality is true. 356 Unit 4

13 YOUR TURN Solve each inequality. Graph and check the solution. 2. y > 12 x Personal Math Trainer Online Assessment and Intervention Interpreting Inequalities as Comparisons You can write a real-world problem for a given inequality. Examine each number and mathematical operation in the inequality. EXAMPLE 2 Write a real-world problem for the inequality 60 w 5. Then solve the inequality. 6.9.C Math On the Spot STEP 1 Examine each part of the inequality. w is the unknown quantity. 5 is added to w. 60 is greater than or equal to a number added to 5. Image Credits: Lew Robertson/Corbis STEP 2 STEP 3 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? Solve the inequality. 60 w w 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

14 YOUR TURN Personal Math Trainer Online Assessment and Intervention 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. 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 > z t 5 > y - 4 < 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)? ESSENTIAL QUESTION CHECK-IN 7. Explain how to solve 7 x 12. Tell what property of inequality you would use. 358 Unit 4

15 Name Class Date 13.2 Independent Practice 6.9.B, 6.9.C, 6.10.A, 6.10.B Personal Math Trainer Online Assessment and Intervention Solve each inequality. Graph and check the solution. 8. x - 35 > y y z Write an inequality to solve each problem. 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 1_ whole pizzas left over. How many pizzas 2 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 $ How much more does Tom need to spent to get his free meal? Lesson

16 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 $ 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 18. Critique Reasoning Kim solved y and got y 2. What might Kim have done wrong? 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, Unit 4

17 LESSON 13.3 Multiplication and Division Inequalities with Positive Numbers 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 multiplication or division with positive numbers? EXPLORE ACTIVITY 6.10.A 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 > 9 > 9 B The model shows the inequality from A. There are x-tiles, so draw circles to separate the tiles into equal groups. > C Reflect How many units are in each group? What values make the inequality you wrote in A true? Graph the solution of the inequality Analyze Relationships Is 3.25 a solution of the inequality you wrote in A? If so, does that solution make sense for the situation? Represent Real-World Problems Rewrite the situation in A to represent the inequality 3x < 9. Lesson

18 Math On the Spot Solving Inequalities Involving Multiplication and Division You can use properties of inequality to solve inequalities involving multiplication and division with positive integers. 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 B, 6.10.B Solve each inequality. Graph and check the solution. A 12x < 24 STEP 1 Solve the inequality. Math Talk Mathematical Processes Are all negative numbers solutions to 12x < 24? Explain. STEP 2 12x 12 < x < 2 Graph the solution Divide both sides by Use an open circle to show that 2 is not a solution. STEP 3 Check the solution by substituting a solution from the shaded part of the graph into the original inequality.? 12(0) < 24 Substitute 0 for x in the original inequality. B 0 < 24 y_ 3 5 STEP 1 Solve the inequality. STEP 2 STEP 3 3 ( y _ 3 ) 3(5) y 15 Graph the solution The inequality is true. Multiply both sides by 3. Use a closed circle to show that 15 is a solution Check the solution by substituting a solution from the shaded part of the graph into the original inequality. 18? 3 5 Substitute 18 for x in the original inequality. 6 5 The inequality is true. 362 Unit 4

19 YOUR TURN Solve each inequality. Graph and check the solution. 3. 5x z_ 4 < Personal Math Trainer Online Assessment and Intervention Solving Real-World Problems You can use multiplication and division inequalities to model and solve real-world problems. EXAMPLE 2 Problem Solving 6.10.A Math On the Spot 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. Justify Solve and Evaluate 4x 22 4x x 5.5 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 Let x represent a side length. Divide both sides by 4. The side lengths must be greater than or equal to 5.5 in. Substitute 6 for x The statement is true. Cy s flag could have a side length of 6 inches. Lesson

20 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. 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) 2. 8y < r_ 3 11? 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

21 Name Class Date 13.3 Independent Practice 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. Personal Math Trainer 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. 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? 8. In a litter of 7 kittens, each kitten weighs more than 3.5 ounces. Find the possible total weight of the litter. 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. 9. To cover his rectangular backyard, Will needs at least square feet of sod. The length of Will s yard is 15.5 feet. What are the possible widths of Will s yard? Solve each inequality. Graph and check the solution x t_ 2 > 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. p t > y x 9.5 < Lesson

22 The sign shows some prices at a produce stand. 19. Tom has $10. What is the greatest amount of spinach he can buy? Produce Price per Pound Onions $1.25 Yellow Squash $0.99 Spinach $3.00 Potatoes $ 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 23. Critique Reasoning A student solves r_ 5 2_ 2 and gets 5 r. What is the 25 correct solution? What mistake might the student have made? 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 1_ inch. The volume of the prism is at most 2 65 cubic inches. Find all possible heights of the prism. Show your work. 366 Unit 4

23 LESSON 13.4 Multiplication and Division Inequalities with Rational Numbers Expressions, equations, and relationships 6.9.B Represent solutions for one-step inequalities on number lines. Also 6.10.A, 6.10.B? ESSENTIAL QUESTION 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. Inequality Multiply each side by: 3 < > > New inequality New inequality is true or false? B Inequality Divide each side by: 4 < > 5-5 New inequality New inequality is true or false? 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

24 Math On the Spot Multiplication and Division Properties of Inequality Recall that you can multiply or divide both sides of an inequality by the same positive number, and the statement will still be true. 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 B, 6.10.B Solve each inequality. Graph and check the solution. My Notes A -4x > 52 STEP 1 Solve the inequality. STEP 2-4x > 52-4x -4 < 52-4 x < -13 Graph the solution. Divide both sides by -4. Reverse the inequality symbol STEP 3 Check your answer using substitution.? -4(-15) > 52 Substitute -15 for x in -4x > > 52 The statement is true. B - y _ 3 < -5 STEP 1 STEP 2 Solve the inequality. - y _ 3 < -5-3 ( - y _ 3 ) > -3(-5) y > 15 Graph the solution. Multiply both sides by -3. Reverse the inequality symbol STEP 3 Check your answer using substitution. - 18? 3 < -5-6 < -5 Substitute 18 for y in - y 3 < -5. The inequality is true. 368 Unit 4

25 YOUR TURN Solve each inequality. Graph and check the solution y < t Personal Math Trainer Online Assessment and Intervention 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. EXAMPLE 2 Problem Solving 6.10.A Math On the Spot 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 Rewrite the question as a statement. Find the number of seconds that the submersible decends below sea level. Image Credits: Jeffrey L. Rotman/Peter Arnold Inc/Getty Images 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 Solve and Evaluate -5t < -40-5t -5 > t > 8 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.? -5(9) < -40 Substitute 9 for t in the inequality -5t < < -40 Rate of change Time < Maximum elevation Divide both sides by -5. Reverse the inequality symbol. The statement is true. Lesson

26 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? Guided Practice Solve each inequality. Graph and check the solution. (Explore Activity and Example 1) 1. -7z t 4 > x < t 10 > 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) 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.? 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

27 Name Class Date 13.4 Independent Practice 6.9.B, 6.10.A, 6.10.B Personal Math Trainer Online Assessment and Intervention Solve each inequality. Graph and check your solution q x < 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? y < -6r 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? > 2x x 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? b. Suppose Parav flips the counter and gets five 6s and twelve -6s when he plays the game. Does he win? Explain 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? Lesson

28 Solve each inequality x x x < < -x x < x Use the order of operations to simplify the left side of the inequality below. What values of x make the inequality a true statement? - 1_ 2 ( 3 2 7)x > 32 FOCUS ON HIGHER ORDER THINKING Work Area 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. 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

29 MODULE QUIZ Ready 13.1 Writing Inequalities Write an inequality to represent each situation, then graph the solutions. Personal Math Trainer Online Assessment and Intervention 1. There are fewer than 8 gallons of gas in the tank There are at least 3 pieces of gum left in the pack The valley was at least 4 feet below sea level Addition and Subtraction Inequalities Solve each inequality. Graph the solution. 4. c - 28 > v Today s high temperature of 80 F is at least 16 warmer than yesterday s high temperature. What was yesterday s high temperature? 13.3, 13.4 Multiplication and Division Inequalities Solve each inequality. Graph the solution. 7. 7f a 2 < g k -3 < Module

30 MODULE 13 MIXED REVIEW Texas Test Prep Personal Math Trainer Online Assessment and Intervention 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 Which number line represents the inequality r > 6? A B C D For which inequality below is z = 3 a solution? A z 5 9 B z 5 > 9 C z 5 8 D z 5 < 8 4. What is the solution to the inequality 6x < 18? A x > 3 B x < 3 C x 3 D x The number line below represents the solution to which inequality? A m 4 > 2.2 C m 3 > 2.5 B 2m < 17.6 D 5m > Which number line shows the solution to w - 2 8? A B C D Gridded Response 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? Unit 4