Inequality Symbols. is more. than. Writing an Inequality. A number w minus 3.5 is less than or equal to 2. Write this sentence as an inequality.

Transcription

1 . Lesson Key Vocabulary inequality solution of an inequality solution set graph of an inequality An inequality is a mathematical sentence that compares epressions. It contains the symbols <, >,, or. To write an inequality, look for the following phrases to determine where to place the inequality symbol. Inequality Symbols Symbol < > Key Phrases is less than is fewer than is greater than is more than is less than or equal to is at most is no more than is greater than or equal to is at least is no less than EXAMPLE Writing an Inequality A number w minus. is less than or equal to. Write this sentence as an inequality. A number w minus. is less than or equal to. An inequality is w.. w. Eercises 6 9 Write the word sentence as an inequality. 7. A number b is fewer than 0... Twice a number k is at least 0. A solution of an inequality is a value that makes the inequality true. An inequality can have more than one solution. The set of all solutions of an inequality is called the solution set. Value of + Is the inequality true? Reading 6 6 +? yes The symbol / means is not greater than or equal to ? yes 8 8 +? no 98 Chapter Linear Inequalities

2 EXAMPLE Checking Solutions Tell whether is a solution of the inequality. a. + 8 < b.. > + 8 < Write the inequality.. > + 8 <? Substitute for..( ) >? </ 8 > is not less than. 8 is greater than. So, is not a solution of the inequality. So, is a solution of the inequality. Eercises 6 Tell whether 6 is a solution of the inequality.. c + <. m 0.. The graph of an inequality shows all of the solutions of the inequality on a number line. An open circle is used when a number is not a solution. A closed circle is used when a number is a solution. An arrow to the left or right shows that the graph continues in that direction. EXAMPLE Graphing an Inequality Graph y. Use a closed circle because is a solution Test a number to the left of. y = is a solution. Test a number to the right of. y = 0 is not a solution. Shade the number line on the side where you found the solution Eercises 7 0 Graph the inequality on a number line. 6. b > 8 7. g. 8. r < 9. v 0.09 Section. Writing and Graphing Inequalities 99

3 . Eercises. VOCABULARY Would an open circle or a closed circle be used in the graph of the inequality k < 0? Eplain.. DIFFERENT WORDS, SAME QUESTION Which is different? Write both inequalities. w is greater than or equal to 7. w is no less than 7. w is no more than 7. w is at least 7.. REASONING Do 9 and 9 represent the same inequality? Eplain. 9+(-6)= +(-)= +(-9)= 9+(-)= Write an inequality for the graph. Then, in words, describe all the values of that make the inequality true Write the word sentence as an inequality. 6. A number is no less than. 7. A number y added to. is less than. 8. A number b multiplied by is at most. 9. A number k minus 8. is greater than ERROR ANALYSIS Describe and correct the error in writing the word sentence as an inequality. Twice a number c is at least 9. c 9 Tell whether the given value is a solution of the inequality.. s + 6 ; s =. n > ; n =. a..6; a =...q > ; q =.6. h ; h = 6. p < ; p = 6 Graph the inequality on a number line. 7. g 6 8. q >. 9. z < 0. w 89. DRIVING When you are driving with a learner s license, a licensed driver who is years of age or older must be with you. Write an inequality that represents this situation. 00 Chapter Linear Inequalities

4 Tell whether the given value is a solution of the inequality.. p > + p; p =. y y ; y = 8. VIDEO GAME RATINGS Each rating is matched with the inequality that represents the recommended ages of players. Your friend is old enough to play E 0+ games. Is your friend old enough to play T games? Eplain The ESRB rating icons are registered trademarks of the Entertainment Software Association.. SCUBA DIVING Three requirements for a scuba diving training course are shown. a. Write and graph three inequalities that represent the requirements. b. You can swim 0 lengths of a -yard pool. Do you satisfy the swimming requirement of the course? Eplain. 6. LUGGAGE On an airplane, the maimum sum of the length, width, and height of a carry-on bag is inches. Find three different sets of dimensions that are reasonable for a carry-on bag. 7. A number m is less than another number n. The number n is less than or equal to a third number p. a. Write two inequalities representing these relationships. b. Describe the relationship between m and p. c. Can m be equal to p? Eplain. w h Solve the equation. Check your solution. 8. r = p =. 0. n π = 7π. MULTIPLE CHOICE Which linear function relates y to? A y = 0. B y = + C y = 0. D y = 0 y Section. Writing and Graphing Inequalities 0

5 . Lesson Study Tip You can solve inequalities the same way you solve equations. Use inverse operations to get the variable by itself. Addition Property of Inequality Words If you add the same number to each side of an inequality, the inequality remains true. Numbers < Algebra > < 6 > 7 Subtraction Property of Inequality Words If you subtract the same number from each side of an inequality, the inequality remains true. Numbers < Algebra + 7 > < > 7 These properties are also true for and. EXAMPLE Solving an Inequality Using Addition Solve 6 0. Graph the solution. Study Tip Undo the subtraction. To check a solution, you check some numbers that are solutions and some that are not. 6 0 Write the inequality Add 6 to each side. The solution is Check: = is not a solution. Check: = 0 is a solution.. b > 9. m.8. > y 0 Chapter Linear Inequalities

6 EXAMPLE Solving an Inequality Using Subtraction Solve 8 >. +. Graph the solution. 8 >. + Write the inequality. Reading Undo the addition. The inequality 9. > is the same as < Subtract. from each side. 9. > The solution is < 9.. < Eercises 6 7. k +. 6 z + 6. p >. EXAMPLE Real-Life Application On a train, carry-on bags can weigh no more than 0 pounds. Your bag weighs.8 pounds. Write and solve an inequality that represents the amount of weight you can add to your bag. Words Variable Weight of your bag plus amount of weight you can add is no more than Let w be the possible weight you can add. the weight limit. Inequality.8 + w w 0 Write the inequality..8.8 Subtract.8 from each side. w. You can add no more than. pounds to your bag. 7. WHAT IF? Your carry-on bag weighs. pounds. Write and solve an inequality that represents the possible weight you can add to your bag. Section. Solving Inequalities Using Addition or Subtraction 0

7 . Eercises. REASONING Is the inequality r 8 the same as 8 r? Eplain.. WHICH ONE DOESN T BELONG? Which inequality does not belong with the other three? Eplain your reasoning. 7 c + 7 c + c + 7 c 7 9+(-6)= +(-)= +(-9)= 9+(-)= Graph the inequality.. 6. p <. w 6. y 7 7. t 8 > 8. n a + 7 > 0. < v. > d +. g 7. m +.. k h.7 < > s + π 7. u. 8. ERROR ANALYSIS Describe and correct the error in graphing the solution of the inequality PELICAN The maimum volume of a great white pelican s bill is about 700 cubic inches. a. A pelican scoops up 00 cubic inches of water. Write and solve an inequality that represents the additional volume the bill can contain. b. A pelican s stomach can contain about one-third the maimum amount that its bill can contain. Write an inequality that represents the volume of the pelican s stomach. 0 Chapter Linear Inequalities

8 Write and solve an inequality that represents the value of. 0. The perimeter is less. The base is greater. The perimeter is less than 6 feet. than the height. than or equal to feet. 0 in. 0 in. ft 0 m in. in. ft + TIME LEFT: min.. REASONING The solution of w + c 8 is w. What is the value of c?. FENCE The hole for a fence post is feet deep. The top of the fence post needs to be at least feet above the ground. Write and solve an inequality that represents the required length of the fence post. CURRENT SCORE: 00. VIDEO GAME You need at least,000 points to advance to the net level of a video game. a. Write and solve an inequality that represents the number of points you need to advance. b. You find a treasure chest that increases your score by 60%. How does this change the inequality? 6. POWER A circuit overloads at 800 watts of electricity. A microwave that uses 00 watts of electricity is plugged into the circuit. Appliance Watts a. Write and solve an inequality that represents the Clock radio 0 additional number of watts you can plug in without Blender 00 overloading the circuit. Hot plate 00 b. In addition to the microwave, what two appliances in the table can you plug in without overloading the circuit? Toaster The maimum surface area of the solid is π square millimeters. Write and solve an inequality that represents the height of the cylinder. mm h Solve the equation = 9. r = 0. c =. 8 = b. MULTIPLE CHOICE Which fraction is equivalent to.8? A 9 B 9 C D Section. Solving Inequalities Using Addition or Subtraction 0

9 Study Help You can use a four square to organize information about a topic. Each of the four squares can be a category, such as definition, vocabulary, eample, non-eample, words, algebra, table, numbers, visual, graph, or equation. Here is an eample of a four square for an inequality. Definition: A mathematical sentence that uses an inequality symbol (<). Vocabulary: Key phrases that can represent an inequality: i s less than i s fewer than Eample: Words: A number is less than. Symbols: < INEQUALITY (<) Graph: To graph <, draw an open circle at =. Then draw an arrow pointing to the left. 0 Make a four square to help you study these topics.. inequality ( > ). inequality ( ). inequality ( ). solving an inequality using addition. solving an inequality using subtraction After you complete this chapter, make four squares for the following topics. 6. solving an inequality using multiplication 7. solving an inequality using division Sorry, but I have limited space in my four square. I needed pet names with only three letters. 06 Chapter Linear Inequalities

10 .. Quiz Write the word sentence as an inequality.. A number plus is less than.. A number t minus.6 is at most 9. Tell whether the given value is a solution of the inequality.. n < ; n =. y + < ; y = 7 Graph the inequality on a number line.. > 0 6. y 7. w < 6.8 Solve the inequality. 8. < 9. g h 9. s + > 7. v < 0. < p +. WATERCRAFT You must be at least years old to operate a personal watercraft in Florida. Write an inequality that represents this situation.. REASONING The solution of a > is >. What is the value of a? 6. MP PLAYER Your MP player can store up to 8 gigabytes of media. You transfer. gigabytes of media to the MP player. Write and solve an inequality that represents the amount of memory available on the MP player. LIFEGUARDS NEEDED Take Our Training Course NOW!!! Lifeguard Training Requirements Swim at least 00 yards Tread water for at least minutes Swim 0 yards or more underwater without taking a breath 7. LIFEGUARD Three requirements for a lifeguard training course are shown. a. Write and graph three inequalities that represent the requirements. b. You can swim 0 feet. Do you satisfy the swimming requirement of the course? Eplain. Sections.. Quiz 07

11 . Lesson Remember Multiplication and division are inverse operations. Multiplication and Division Properties of Inequality (Case ) Words If you multiply or divide each side of an inequality by the same positive number, the inequality remains true. Numbers 6 < 8 6 > 8 ( 6) < 8 < 6 6 > 8 > Algebra < 9 > < ( 9) > < 8 > These properties are also true for and. EXAMPLE Solving an Inequality Using Multiplication Solve 8 >. Graph the solution. Undo the division. 8 > Write the inequality. 8 8 > 8 ( ) Multiply each side by 8. > 0 The solution is > 0. > Check: = 80 is not a solution. Check: = 0 is a solution. n. a < w 08 Chapter Linear Inequalities

12 EXAMPLE Solving an Inequality Using Division Solve. Graph the solution. Undo the multiplication. 8 The solution is 8. Write the inequality. Divide each side by Check: = 0 is a solution. Check: = 0 is not a solution. Eercises 0 8. b 6. k > >.q Common Error A negative sign in an inequality does not necessarily mean you must reverse the inequality symbol. Only reverse the inequality symbol when you multiply or divide both sides by a negative number. Multiplication and Division Properties of Inequality (Case ) Words If you multiply or divide each side of an inequality by the same negative number, the direction of the inequality symbol must be reversed for the inequality to remain true. Numbers 6 < 8 6 > 8 ( ) ( 6) > ( ) 6 8 < 8 Algebra > 6 < 6 < > > 6 < 0 > 8 < 6 These properties are also true for and. Section. Solving Inequalities Using Multiplication or Division 09

13 EXAMPLE Undo the division. Solving an Inequality Using Multiplication y Solve >. Graph the solution. y > y < Write the inequality. Multiply each side by. Reverse the inequality symbol. y < 6 The solution is y < 6. y < Check: y = 9 is a solution. Check: y = 0 is not a solution. EXAMPLE Solving an Inequality Using Division Solve 7y. Graph the solution. Undo the multiplication. 7y 7y 7 7 y Write the inequality. Divide each side by 7. Reverse the inequality symbol. The solution is y. y Check: y = 0 is not a solution. Check: y = 6 is a solution. Eercises 7 p 7. < z 0. 9m > 6. r. 0.y 0 Chapter Linear Inequalities

14 . Eercises. VOCABULARY Eplain how to solve 6 <.. WRITING Eplain how solving < 8 is different from solving < 8.. OPEN-ENDED Write an inequality that is solved using the Division Property of Inequality where the inequality symbol needs to be reversed. 9+(-6)= +(-)= +(-9)= 9+(-)= Use a table to solve the inequality.. <. 6. > > n > 8. c 9..m <. >. w.6. <.k y > b. 9. ERROR ANALYSIS Describe and correct the error in solving the inequality. Write the word sentence as an inequality. Then solve the inequality. 0. The quotient of a number and is at most.. A number divided by 8 is less than.. Four times a number is at least.. The product of and a number is greater than 0. < > ( ) > 0. CAMERA You earn $9.0 per hour at your summer job. Write and solve an inequality that represents the number of hours you need to work in order to buy a digital camera that costs $7. Section. Solving Inequalities Using Multiplication or Division

15 . COPIES You have $.6 to make copies. Write and solve an inequality that represents the number of copies you can make. 6. SPEED LIMIT The maimum speed limit for a school bus is miles per hour. Write and solve an inequality that represents the number of hours it takes to travel 6 miles in a school bus. 7. n 0 8. w > 0 9. h <. y <. 7d 6 m.. >. k b >.6 6. ERROR ANALYSIS Describe and correct the error in solving the inequality. 7. CRITICAL THINKING Are all numbers greater than zero solutions of > 0? Eplain. m 6 m 6 m 8. TRUCKING By law, the maimum height (including freight) of a vehicle in Florida is. feet. a. Write and solve an inequality that represents the number of crates that can be stacked vertically on the bed of the truck. b. Five crates are stacked vertically on the bed of the truck. Is this legal? Eplain. 8 in.. ft Not drawn to scale Write and solve an inequality that represents the value of. 9. Area 0 cm 0. Area < 0 ft cm 0 ft Chapter Linear Inequalities

16 . TRIP You and three friends are planning a trip. You want to keep the cost below $80 per person. Write and solve an inequality that represents the total cost of the trip.. REASONING Eplain why the direction of the inequality symbol must be reversed when multiplying or dividing by the same negative number.. PROJECT Choose two musical artists to research. a. Use the Internet or a magazine to complete the table. b. Find the average number of copies sold per month for each CD. c. Use the release date to write and solve an inequality that represents the minimum average number of copies sold per month for each CD. d. In how many months do you epect the number of copies of the second top selling CD to surpass the current number of copies of the top selling CD? Artist Name of CD Release Date Current Number of Copies Sold.. Describe all numbers that satisfy both inequalities. Include a graph with your description.. m > and m <. n and n 6. and 7. m > and m < 0 Solve the equation. 8. w + = 9. ( ) = 0. v 6 7 =. m + 00 = 96. MULTIPLE CHOICE Which measure can have more than one value for a given data set? A mean B median C mode D range Section. Solving Inequalities Using Multiplication or Division

17 . Lesson You can solve multi-step inequalities the same way you solve multi-step equations. EXAMPLE Solving Two-Step Inequalities Step : Undo the subtraction. Step : Undo the multiplication. a. Solve. Graph the solution. Write the inequality. + + Add to each side. The solution is. Divide each side by Check: = 0 is not a solution. Check: = is a solution. y b. Solve < 9. Graph the solution. y < 9 Write the inequality. 7 7 Subtract 7 from each side. y 6 < 6 y 6 > 6 Multiply each side by 6. Reverse the inequality symbol. y > The solution is y >. y > Eercises 0. b < c 8. n + > Chapter Linear Inequalities

18 EXAMPLE Standardized Test Practice Which graph represents the solution of 7( + ) 8? A B C D ( + ) 8 Write the inequality. 7 8 Use Distributive Property. + + Add to each side Divide each side by 7. Reverse the inequality symbol. The correct answer is B. Remember EXAMPLE The mean in Eample is equal to the sum of the game scores divided by the number of games. Real-Life Application You need a mean score of at least 90 to advance to the net round of the trivia game. What score do you need on the fifth game to advance? Use the definition of mean to write and solve an inequality. Let be the score on the fifth game. The phrase at least means greater than or equal to Multiply each side by. + 0 Subtract from each side. 98 You need at least 98 points to advance to the net round. Eercises 7. (k ) < 6. (n 0) < 6. 0.(8 + y) 7. WHAT IF? In Eample, you need a mean score of at least 88 to advance to the net round of the trivia game. What score do you need on the fifth game to advance? Section. Solving Multi-Step Inequalities

19 . Eercises. WRITING Compare and contrast solving multi-step inequalities and solving multi-step equations.. OPEN-ENDED Describe how to solve the inequality (a + ) < 9. 9+(-6)= +(-)= +(-9)= 9+(-)=. For what values of k will the. For what values of h will the perimeter of the octagon be surface area of the solid be less than or equal to 6 units? greater than 6 square units? k k k k h m. 7b + 6. v < < w 9..8 < 0..p 0..r ERROR ANALYSIS Describe and correct the error in solving the inequality ( g + ) 8. ( y ) 6. 0 (h ). (u + ) > 6..7 > 0.9(n.7) 7. 0 >.(z.) 8. ATM Write and solve an inequality that represents the number of $0 bills you can withdraw from the account without going below the minimum balance. 6 Chapter Linear Inequalities

20 b b +. > 8.. TYPING One line of tet on a page uses about of an inch. 6 There are -inch margins at the top and bottom of a page. Write and solve an inequality to find the number of lines that can be typed on a page that is inches long.. WOODWORKING A woodworker builds a cabinet in 0 hours. The cabinet is sold at a store for $00. Write and solve an inequality that represents the hourly wage the store can pay the woodworker and still make a profit of at least $00. 7 ft. FIRE TRUCK The height of one story of a building is about 0 feet. The bottom of the ladder on the fire truck must be at least feet away from the building. Write and solve an inequality to find the number of stories the ladder can reach. 8 ft. DRIVE-IN A drive-in movie theater charges $.0 per car. The drive-in has already admitted 00 cars. Write and solve an inequality to find the number of cars the drive-in needs to admit to make at least $00.. For what values of r will the area of the shaded region be greater than or equal to 9(π )? r Find the area of the circle mm in. 66 m 9. MULTIPLE CHOICE What is the volume of the cube? A 8 ft B 6 ft C ft D ft ft Section. Solving Multi-Step Inequalities 7

21 .. Quiz. >. n 6. y 60.. Write the word sentence as an inequality. Then solve the inequality.. The quotient of a number and 6 is more than Five times a number is at most m n 6 8 j 9. > 7 0. > 7 w p. FLOWERS A soccer team needs to raise $00 for new uniforms. The team earns $0.0 for each flower sold. Write and solve an inequality to find the number of flowers it must sell to meet or eceed its fundraising goal.. PARTY You buy lunch for guests at a party. You can spend no more than $00. You will spend $0 on beverages and $0 per guest on sandwiches. Write and solve an inequality to find the number of guests you can invite to the party.. BOOKS You have a gift card worth $0. You want to buy several eral paperback books that cost $6 each. Write and solve an inequality to find the number of books you can buy and still have at least $0 on the gift card. b. GARDEN The area of the triangular garden must be less than square feet. Write and solve an inequality that represents the value of b. 0 ft 8 Chapter Linear Inequalities

22 Chapter Test Write the word sentence as an inequality.. A number j plus 0. is greater than or equal to 0.. A number r multiplied by is less than. 7 Tell whether the given value is a solution of the inequality.. v 7; v = 9. 0 p < 0; p = 0. n 6; n = 7 6. n > b 0 9. y 0. v 7.. (t + ) < 60. VOTING U.S. citizens must be at least 8 years of age on Election Day to vote. Write an inequality that represents this situation.. GARAGE The vertical clearance for a hotel parking garage is 0 feet. Write and solve an inequality that represents the height (in feet) of the vehicle. in.. LUNCH BILL A lunch bill, including ta, is divided equally among you and five friends. Everyone pays less than $8.7. Write and solve an inequality that describes the total amount of the bill. h. TRADING CARDS You have $ to buy trading cards online. Each pack of cards costs $.0. Shipping costs $.9. Write and solve an inequality to find the number of packs of trading cards you can buy. 6. SCIENCE QUIZZES The table shows your scores on four science quizzes. What score do you need on the fifth quiz to have a mean score of at least 80? Test Score (%) ? Chapter Test 9