Solving Linear Inequalities

Transcription

1 Solving Linear Inequalities. Writing and Graphing Inequalities. Solving Inequalities Using Addition or Subtraction. Solving Inequalities Using Multiplication or Division. Solving Multi-Step Inequalities.5 Graphing Linear Inequalities in Two Variables Here is a math quiz, Descartes. Tell me about these smbols... than ou have! eats

2 What You Learned Before Complete the number sentence with <, >, or =. Eample = 0 0, 0. = 0 = 9 0 Because 0 0 is greater than 9 0, is greater than 0.. So, > Eample 6.5 the LARGER number. Use a calculator to estimate Because.5 is less than.5, 6 is less than.5. So, 6 <.5. Complete the number sentence with <, >, or = π 0 Eample Graph. Use a closed circle because is a solution. Shade the number line on the side where ou found the solution Test a number to the left of. = 0 is not a solution. Test a number to the right of. = 6 is a solution. Eample Graph <. Shade the number line on the side where ou found the solution. Use an open circle because is not a solution Test a number to the left of. 0 is a solution. Test a number to the right of. is not a solution. Graph the inequalit < > 0

3 . Writing and Graphing Inequalities real-life statement? How can ou use an inequalit to describe a ACTIVITY: Writing and Graphing Inequalities Work with a partner. Write an inequalit for the statement. Then sketch the graph of all the numbers that make the inequalit true. a. Statement: The temperature t in Minot, North Dakota has never been below 6 F. Inequalit: Graph: b. Statement: The elevation e in Wisconsin is at most 95.5 feet above sea level. Inequalit: Graph: ACTIVITY: Writing and Graphing Inequalities COMMON CORE Writing Inequalities In this lesson, ou will write and graph inequalities. Learning Standards A.CED. A.CED. Work with a partner. Write an inequalit for the graph. Then, in words, describe all the values of that make the inequalit true. a. b. c d. 0 0 Chapter Solving Linear Inequalities

4 Math Practice Construct Arguments How can ou use results from this activit to write a rule? ACTIVITY: Triangle Inequalit Work with a partner. Use 8 to 0 pieces of spaghetti. Break one piece of spaghetti into three parts that can be used to form a triangle. Form a triangle and use a centimeter ruler to measure each side. Round the side lengths to the nearest tenth. in. 5 6 L M S cm Record the side lengths in a table. Repeat the process with two other pieces of spaghetti. Side Lengths That Form a Triangle Small Medium Large S + M Repeat the eperiment b breaking pieces of spaghetti into three pieces that do not form a triangle. Record the lengths in a table. Side Lengths That Do Not Form a Triangle Small Medium Large S + M INDUCTIVE REASONING Write a rule that uses an inequalit to compare the lengths of three sides of a triangle. Use our rule to decide whether the following triangles are possible. Eplain. a. 7 5 b. 0 5 c IN YOUR OWN WORDS How can ou use an inequalit to describe a real-life statement? Give two eamples of real-life statements that can be represented b inequalities. Use what ou learned about writing and graphing inequalities to complete Eercises and 5 on page 08. Section. Writing and Graphing Inequalities 05

5 . Lesson Lesson Tutorials Ke Vocabular inequalit, p. 06 solution of an inequalit, p. 06 solution set, p. 06 graph of an inequalit, p. 07 An inequalit is a mathematical sentence that compares epressions. It contains the smbol <, >,, or. To write an inequalit, look for the following phrases to determine where to place the inequalit smbol. Inequalit Smbols Smbol < > Ke 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 Inequalit A number w minus.5 is less than or equal to. Write this sentence as an inequalit. A number w minus.5 is less than or equal to. w.5 An inequalit is w.5. Eercises 6 9 Write the word sentence as an inequalit.. A number b is fewer than 0... Twice a number k is at least 7 0. A solution of an inequalit is a value that makes the inequalit true. An inequalit can have more than one solution. The set of all solutions of an inequalit is called the solution set. Value of + 5 Is the inequalit true? ? es Reading ? es The smbol means is not greater than or equal to ? no 06 Chapter Solving Linear Inequalities

6 EXAMPLE Checking Solutions Tell whether is a solution of each inequalit. a. + 8 < b..5 > + 8 < Write the inequalit..5 > + 8 <? Substitute for..5( ) >? </ Simplif. 8 > is not less than. 8 is greater than. So, is not a solution of the inequalit. So, is a solution of the inequalit. Eercises 6 Tell whether 6 is a solution of the inequalit.. c + <. 5 m The graph of an inequalit shows all of the solutions of the inequalit 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 Inequalit Graph. Use a closed circle because is a solution Test a number to the left of. = is a solution. Test a number to the right of. = 0 is not a solution. Shade the number line on the side where ou found the solution Eercises 7 0 Graph the inequalit on a number line. 6. b > 8 7. g. 8. r < 9. v 6 Section. Writing and Graphing Inequalities 07

7 . Eercises Help with Homework. VOCABULARY Would an open circle or a closed circle be used in the graph of the inequalit k < 50? 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 inequalit? Eplain. 9+(-6)= +(-)= +(-9)= 9+(-)= Write an inequalit for the graph. Then, in words, describe all the values of that make the inequalit true Write the word sentence as an inequalit. 6. A number is no less than. 7. A number added to 5. is less than. 8. A number b multiplied b 5 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 inequalit. Twice a number c is at least 9. c 9 Tell whether the given value is a solution of the inequalit.. s + 6 ; s =. 5n > ; n =. a.5.6; a =...q > ; q = h ; h = 5 6. p < ; p = 6 Graph the inequalit on a number line. 7. g 6 8. q >.5 9. z < 0. w 6. DRIVING When ou are driving with a learner s license, a licensed driver who is ears of age or older must be with ou. Write an inequalit that represents this situation. 08 Chapter Solving Linear Inequalities

8 Tell whether the given value is a solution of the inequalit.. p > 5 + p; p =. ; = 8. LOGIC Each video game rating is matched with the inequalit that represents the suggested ages of plaers. Your friend is old enough to pla E 0+ games. Is our friend old enough to pla T games? Eplain The ESRB rating icons are registered trademarks of the Entertainment Software Association. 5. 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 5-ard pool. Do ou satisf the swimming requirement of the course? Justif our answer. 6. REPEATED REASONING On an airplane, the maimum sum of the length, width, and height of a carr-on bag is 5 inches. Find three different sets of dimensions that are reasonable for a carr-on bag. Use a diagram to justif our answer. 7. A number m is less than another number n. The number n is less than or equal to a third number p. w h a. Write two inequalities representing these relationships. b. Describe the relationship between m and p. c. Can m be equal to p? Eplain. Solve the equation. Check our solution. (Section.) 8. r = p =.5 0. n π = 7π. MULTIPLE CHOICE Which of the following is the equation of the line in slope-intercept form? (Section.5) A = + B = C = + D = + (0, ) (, ) Section. Writing and Graphing Inequalities 09

9 . Solving Inequalities Using Addition or Subtraction solve an inequalit? How can ou use addition or subtraction to ACTIVITY: Quarterback Passing Efficienc Work with a partner. The National Collegiate Athletic Association (NCAA) uses the following formula to rank the passing efficienc P of quarterbacks. 8.Y + 00C + 0T 00N P = A Y = total length of all completed passes (in Yards) C = Completed passes T = passes resulting in a Touchdown N = intercepted passes A = Attempted passes M = incomplete passes Attempts Completed Intercepted Incomplete Touchdown Not Touchdown Which of the following equations or inequalities are true relationships among the variables? Eplain our reasoning. a. C + N < A b. C + N A c. T < C d. T C e. N < A f. A > T g. A C M h. A = C + N + M COMMON CORE Solving Inequalities In this lesson, ou will write and solve inequalities using addition or subtraction. solve real-life problems. Learning Standards A.CED. A.CED. A.REI. ACTIVITY: Quarterback Passing Efficienc Work with a partner. Which of the following quarterbacks has a passing efficienc rating that satisfies the inequalit P > 00? Show our work. Plaer Attempts Completions Yards Touchdowns Interceptions A B C D Chapter Solving Linear Inequalities

10 ACTIVITY: Finding Solutions of Inequalities Math Practice Find General Methods What method did ou use to choose the values for the formula? Wh? Work with a partner. Use the passing efficienc formula to create a passing record that makes the inequalit true. Then describe the values of P that make the inequalit true. a. P < 0 Attempts Completions Yards Touchdowns Interceptions b. P Attempts Completions Yards Touchdowns Interceptions c. 80 < P 50 Attempts Completions Yards Touchdowns Interceptions d. P Attempts Completions Yards Touchdowns Interceptions e. P 50 > 80 Attempts Completions Yards Touchdowns Interceptions. Write a rule that describes how to solve inequalities like those in Activit. Then use our rule to solve each of the inequalities in Activit. 5. IN YOUR OWN WORDS How can ou use addition or subtraction to solve an inequalit? 6. How is solving the inequalit + < similar to solving the equation + =? How is it different? Use what ou learned about solving inequalities using addition or subtraction to complete Eercises 5 on page. Section. Solving Inequalities Using Addition or Subtraction

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

12 EXAMPLE Solving an Inequalit Using Subtraction Solve 8 >. +. Graph the solution. 8 >. + Write the inequalit. Reading Undo the addition. The inequalit 9. > is the same as < Subtract. from each side. 9. > Simplif. The solution is < 9.. < Eercises 6 7 Solve the inequalit. Graph the solution.. k z + 6. p >. EXAMPLE Real-Life Application On a train, carr-on bags can weigh no more than 50 pounds. Your bag weighs.8 pounds. Write and solve an inequalit that represents the amount of weight ou can add to our bag. Words Variable Weight of our bag plus amount of weight ou can add is no more than Let w be the possible weight ou can add. the weight limit. Inequalit.8 + w w 50 Write the inequalit..8.8 Subtract.8 from each side. w 5. Simplif. You can add no more than 5. pounds to our bag. 7. WHAT IF? Your carr-on bag weighs.5 pounds. Write and solve an inequalit that represents the possible weight ou can add to our bag. Section. Solving Inequalities Using Addition or Subtraction

13 . Eercises Help with Homework. REASONING Is the inequalit r 5 8 the same as 8 r 5? Eplain.. WHICH ONE DOESN T BELONG? Which inequalit does not belong with the other three? Eplain our reasoning. c + 7 c + 7 c + 7 c 7 9+(-6)= +(-)= +(-9)= 9+(-)= Use the formula in Activit to create a passing record that makes the inequalit true.. P 80. P + 0 < P 0 Solve the inequalit. Graph the solution t 8 > 8. n a + 7 > 0. 5 < v. 5 > d + 5. g. m k h.7 < > s + π 7. 5 u.5 8. ERROR ANALYSIS Describe and correct the error in graphing the solution of the inequalit PROBLEM SOLVING 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 inequalit that represents the additional volume the pelican s bill can contain. b. A pelican s stomach can contain about one-third the maimum amount that its bill can contain. Write an inequalit that represents the volume of the pelican s stomach. Chapter Solving Linear Inequalities

14 Write and solve an inequalit 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 5 feet. 0 in. 0 in. ft ft 0 m + in. in. 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 inequalit that represents the required length of the fence post. CURRENT SCORE: VIDEO GAME You need at least,000 points to advance to the net level of a video game. a. Write and solve an inequalit that represents the number of points ou need to advance. b. You find a treasure chest that increases our score b 60%. Eplain how this changes the inequalit. 6. MODELING A circuit overloads at 800 watts of electricit. A microwave that uses 00 watts of electricit is plugged into the circuit. Appliance Watts a. Use a model to write and solve an inequalit that Clock radio 50 represents the additional number of watts ou can Blender 00 plug in without overloading the circuit. Hot plate 00 b. In addition to the microwave, what two appliances in the table can ou plug in without overloading the circuit? Toaster 800 Eplain. h mm 7. The maimum surface area of the solid is 5π square millimeters. Write and solve an inequalit that represents the height of the clinder. Solve the equation. (Section.) 8. 6 = 9. r 5 = 0. c = 5. 8 = b Find the square root. (Skills Review Handbook) Section. Solving Inequalities Using Addition or Subtraction 5

15 . Solving Inequalities Using Multiplication or Division How can ou use multiplication or division to solve an inequalit? ACTIVITY: Using a Table to Solve an Inequalit Work with a partner. Cop and complete the table. Decide which graph represents the solution of the inequalit. Write the solution of the inequalit. a ? b. > 5 0 >? COMMON CORE Solving Inequalities In this lesson, ou will solve inequalities using multiplication or division. solve real-life problems. Learning Standards A.CED. A.CED. A.REI ACTIVITY: Writing a Rule Work with a partner. Use a table to solve each inequalit. a. > b. c. 6 d. 5 < 0 Write a rule that describes how to solve inequalities like those in Activit. Then use our rule to solve each of the four inequalities above. 6 Chapter Solving Linear Inequalities

16 ACTIVITY: Using a Table to Solve an Inequalit Math Practice Look for Patterns How do the patterns help ou complete this activit? Work with a partner. Cop and complete the table. Decide which graph represents the solution of the inequalit. Write the solution of the inequalit. a. 0 5? b. < 5 0 <? ACTIVITY: Writing a Rule Work with a partner. Use a table to solve each inequalit. a. b. < c. > d. 5 5 Write a rule that describes how to solve inequalities like those in Activit. Then use our rule to solve each of the four inequalities above. 5. IN YOUR OWN WORDS How can ou use multiplication or division to solve an inequalit? Use what ou learned about solving inequalities using multiplication or division to complete Eercises 9 on page. Section. Solving Inequalities Using Multiplication or Division 7

17 . Lesson Lesson Tutorials Remember Multiplication and division are inverse operations. Multiplication and Division Properties of Inequalit (Case ) Words If ou multipl or divide each side of an inequalit b the same positive number, the inequalit 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 Inequalit Using Multiplication Solve > 5. Graph the solution. 8 Undo the division. > 5 Write the inequalit > 8 ( 5) Multipl each side b 8. > 0 Simplif. The solution is > 0. > Check: = 80 is not a solution. Check: = 0 is a solution. Solve the inequalit. Graph the solution.. a <. n w 5 8 Chapter Solving Linear Inequalities

18 EXAMPLE Solving an Inequalit Using Division Solve. Graph the solution. Undo the multiplication. 8 The solution is 8. Write the inequalit. Divide each side b. Simplif Check: = 0 is a solution. Check: = 0 is not a solution. Eercises 0 8 Solve the inequalit. Graph the solution.. b 6 5. k > >.5q Common Error A negative sign in an inequalit does not necessaril mean ou must reverse the inequalit smbol. Onl reverse the inequalit smbol when ou multipl or divide both sides b a negative number. Multiplication and Division Properties of Inequalit (Case ) Words If ou multipl or divide each side of an inequalit b the same negative number, the direction of the inequalit smbol must be reversed for the inequalit to remain true. Numbers 6 < 8 6 > 8 Algebra ( ) ( 6) > ( ) 8 6 < 8 > 6 < < 5 > > < 0 5 > 8 < 6 These properties are also true for and. Section. Solving Inequalities Using Multiplication or Division 9

19 EXAMPLE Solving an Inequalit Using Multiplication Solve >. Graph the solution. > Write the inequalit. Undo the division. < < 6 The solution is < 6. < 6 Multipl each side b. Reverse the inequalit smbol. Simplif Check: = 9 is a solution. Check: = 0 is not a solution. EXAMPLE Solving an Inequalit Using Division Solve 7 5. Graph the solution. Undo the multiplication Write the inequalit. Divide each side b 7. Reverse the inequalit smbol. Simplif. The solution is Check: = 0 is not a solution. Check: = 6 is a solution. Eercises 7 5 Solve the inequalit. Graph the solution. 7. p < z 0. 9m > 6 0. r Chapter Solving Linear Inequalities

20 . Eercises Help with Homework. VOCABULARY Eplain how to solve 6 < 5.. WRITING Eplain how solving < 8 is different from solving < 8.. OPEN-ENDED Write an inequalit that is solved using the Division Propert of Inequalit where the inequalit smbol needs to be reversed. 9+(-6)= +(-)= +(-9)= 9+(-)= Use a table to solve the inequalit.. < > > Solve the inequalit. Graph the solution. 0. n > 8. c 9..m <. >. w <.5k > b. 9. ERROR ANALYSIS Describe and correct the error in solving the inequalit. Write the word sentence as an inequalit. Then solve the inequalit. 0. The quotient of a number and is at most.. A number divided b 8 is less than.. Four times a number is at least.. The product of 5 and a number is greater than 0. < 5 > ( 5) > 0. CAMERA You earn $9.50 per hour at our summer job. Write and solve an inequalit that represents the number of hours ou need to work in order to bu a digital camera that costs $7. Section. Solving Inequalities Using Multiplication or Division

21 5. COPIES You have $.65 to make copies. Write and solve an inequalit that represents the number of copies ou can make. 6. SPEED LIMIT The maimum speed limit for a school bus is 55 miles per hour. Write and solve an inequalit that represents the number of hours it takes to travel 65 miles in a school bus. Solve the inequalit. Graph the solution. 7. n w > 0 9. h <. <. 7d 56.. > m 5. k > b.6 6. ERROR ANALYSIS Describe and correct the error in solving the inequalit. 7. CRITICAL THINKING Are all numbers greater than zero solutions of > 0? Eplain. m 6 m 6 m 8. TRUCKING In man states, the maimum height (including freight) of a vehicle is.5 feet. a. Five crates are stacked verticall on the bed of the truck. Is this legal? Eplain. b. Write and solve an inequalit to justif our answer to part (a). 8 in..5 ft Not drawn to scale Write and solve an inequalit that represents the value of. 9. Area 0 cm 0. Area < 0 ft cm 0 ft Chapter Solving Linear Inequalities

22 . TRIP You and three friends are planning a trip. You want to keep the cost below $80 per person. Write and solve an inequalit that represents the total cost of the trip.. PRECISION Eplain wh the direction of the inequalit smbol must be reversed when multipling or dividing b the same negative number.. PROJECT Choose two musical artists to research. a. Use the Internet or a magazine to complete the table. b. Find and compare the average number of copies sold per month for each CD. Which CD do ou consider to be the most successful? Eplain. c. Assume each CD continues to sell at the average rate. Write and solve an inequalit that represents the number of months it will take for the total number of copies sold to eceed twice the current number sold. Artist Name of CD Release Date Current Number of Copies Sold.. Describe all numbers that satisf both inequalities. Include a graph with our description.. m > and m < 5. n and n 6. and 7. m > 5 and m < 0 Solve the equation. (Section.) 8. w + 5 = 9. ( ) = 50. v 6 7 = 5. m + 00 = MULTIPLE CHOICE Which of the following is not a solution of p.9 0.8? (Section.) A p =.5 B p =.7 C p =.75 D p = 5 Section. Solving Inequalities Using Multiplication or Division

23 Stud Help Graphic Organizer You can use a four square to organize information about a topic. Each of the four squares can be a categor, such as definition, vocabular, eample, non-eample, words, algebra, table, numbers, visual, graph, or equation. Here is an eample of a four square for an inequalit. Definition: A mathematical sentence that uses an inequalit smbol (<). Vocabular: Ke phrases that can represent an inequalit: is less than is fewer than Eample: Words: A number is less than. Smbols: < INEQUALITY (<) Graph: To graph <, draw an open circle at =. Then draw an arrow pointing to the left. 0 Make four squares to help ou stud these topics.. inequalit ( ). solving an inequalit using addition. solving an inequalit using subtraction. solving an inequalit using multiplication 5. solving an inequalit using division After ou complete this chapter, make four squares for the following topics. 6. solving a compound inequalit 7. graphing an inequalit in two variables Sorr, but I have limited space in m four square. I needed pet names with onl three letters. Chapter Solving Linear Inequalities

24 .. Quiz Progress Check Write the word sentence as an inequalit. (Section.). 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 inequalit. (Section.). n < ; n =. + < ; = 7 Graph the inequalit on a number line. (Section.) 5. > 0 6. w < 6.8 Solve the inequalit. Graph the solution. (Section. and Section.) 7. < 8. g h 9 0. < p +. n Write the word sentence as an inequalit. Then solve the inequalit. (Section.). The quotient of a number and 6 is more than 9.. Five times a number is at most 0. LIFEGUARDS NEEDED Take Our Training Course NOW!!! Lifeguard Training Requirements Swim at least 00 ards Tread water for at least 5 minutes Swim 0 ards or more underwater without taking a breath 5. LIFEGUARD Three requirements for a lifeguard training course are shown. (Section.) a. Write and graph three inequalities that represent the requirements. b. You can swim 50 feet. Do ou satisf the swimming requirement of the course? Eplain. 6. REASONING The solution of a > is >. What is the value of a? (Section.) b 7. GARDEN The area of the triangular garden must be less than 5 square feet. Write and solve an inequalit that represents the value of b. (Section.) 0 ft Sections.. Quiz 5

25 . Solving Multi-Step Inequalities How can ou use an inequalit to describe the area and perimeter of a composite figure? ACTIVITY: Areas and Perimeters of Composite Figures Work with a partner. a. For what values of will the area of the blue region be greater than square units? b. For what values of will the sum of the inner and outer perimeters of the blue region be greater than 0 units? 5 c. For what values of will the area of the trapezoid be less than or equal to 0 square units? d. For what values of will the perimeter of the trapezoid be less than or equal to 6 units? COMMON CORE Solving Inequalities In this lesson, ou will write and solve multi-step inequalities. solve real-life problems. Learning Standards A.CED. A.CED. A.REI. e. For what values of w will the area of the red region be greater than or equal to 6 square units? f. For what values of w will the sum of the inner and outer perimeters of the red region be greater than 7 units? 0 6 g. For what values of will the area of the ellow region be less than π square units? h. For what values of will the sum of the inner and outer perimeters of the ellow region be less than π + 0 units? w 8 6 Chapter Solving Linear Inequalities

26 ACTIVITY: Volume and Surface Area of a Composite Solid Work with a partner. Math Practice a. For what values of will the volume of the solid be greater than or equal to cubic units? Use Operations Which operations will ou use to find the volume and surface area of the composite solid? b. For what values of will the surface area of the solid be greater than 7 square units? ACTIVITY: Planning a Budget Work with a partner. You are building a patio. You want to cover the patio with Spanish tile that costs $5 per square foot. Your budget for the tile is $700. How wide can ou make the patio without going over our budget? Tiles are needed under the plants. No tile is needed under the hot tub. 6 ft 6 ft ft. IN YOUR OWN WORDS How can ou use an inequalit to describe the area and perimeter of a composite figure? Give an eample. Include a diagram with our eample. Use what ou learned about solving multi-step inequalities to complete Eercises and on page 0. Section. Solving Multi-Step Inequalities 7

27 . Lesson Lesson Tutorials You can use the properties of inequalit to solve multi-step inequalities the same wa ou use the properties of equalit to solve multi-step equations. EXAMPLE Undo the addition. Undo the division. Solving a Multi-Step Inequalit Solve + 7 < 9. Graph the solution < 9 Write the inequalit Subtract 7 from each side. 6 < 6 6 > 6 Simplif. Multipl each side b 6. Reverse the inequalit smbol. > Simplif. The solution is >. > Eercises 5 0 Solve the inequalit. Graph the solution.. b < c 8. n + > When solving an inequalit, if ou obtain an inequalit that is true, such as 5 < 0, then the solution is the set of all real numbers. If ou obtain an inequalit that is false, such as, then the inequalit has no solutions. EXAMPLE Solving an Inequalit with No Solution Solve 8 > ( + ). 8 > ( + ) Write the inequalit. 8 > 8 + Distributive Propert 8 8 Subtract 8 from each side. > Simplif. The inequalit > is false. So, there are no solutions. 8 Chapter Solving Linear Inequalities

28 EXAMPLE Solving an Inequalit with Infinitel Man Solutions Which graph represents the solution of (5 ) 7 + 0? A 0 B 0 C 0 D 0 Stud Tip The graph of the set of all real numbers is the entire number line. (5 ) Write the inequalit Distributive Propert 0 0 Subtract 0 from each side. 7 Simplif. The inequalit 7 is true. So, the solution is the set of all real numbers. The correct answer is B. Game : Game : Game : Game : Remember EXAMPLE The mean in Eample is equal to the sum of the game scores divided b 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 ou need on the fifth game to advance? Use the definition of mean to write and solve an inequalit. Let be the score on the fifth game Simplif Multipl each side b Simplif. 5 5 Subtract 5 from each side. 98 The meaning of the phrase at least is greater than or equal to. Simplif. You need at least 98 points to advance to the net round. Eercises 0 Solve the inequalit, if possible.. (k 5) < k (n ) > n WHAT IF? In Eample, ou need a mean score of at least 88 to advance to the net round of the trivia game. What score do ou need on the fifth game to advance? Section. Solving Multi-Step Inequalities 9

29 . Eercises Help with Homework. WRITING Compare and contrast solving multi-step inequalities and solving multi-step equations.. WRITING How do ou know when an inequalit has no solutions? How do ou know when the solution of an inequalit is the set of all real numbers? 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 Solve the inequalit. Graph the solution. 5 h 5. 7b + 6. v < 8 7. m < w 5. ERROR ANALYSIS Describe and correct the error in solving the inequalit. Solve the inequalit, if possible < 0.5.p 0..r ( g + ) 8. ( ) 9. 0 (h ) 5. (u + ) > > 0.9(n.7) 7. 0 >.5(z.) 8. 5(w + ) 5w (6 ) < c 5 > (c + ). ATM Write and solve an inequalit that represents the number of $0 bills ou can withdraw from the account without going below the minimum balance. 0 Chapter Solving Linear Inequalities

30 Solve the inequalit. Graph the solution 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 inequalit to find the number of lines that can be tped on a page that is inches long. 5. WOODWORKING A woodworker builds a cabinet in 0 hours. The cabinet is sold at a store for $500. Write and solve an inequalit that represents the hourl wage the store can pa the woodworker and still make a profit of at least $00. 7 ft 6. FIRE TRUCK The height of one stor of a building is about 0 feet. The bottom of the ladder on the fire truck must be at least feet awa from the building. Write and solve an inequalit to find the number of stories the ladder can reach. 8 ft 7. REASONING A drive-in movie theater charges $.50 per car. a. The drive-in has alread admitted 00 cars. Write and solve an inequalit to find how man more cars the drive-in needs to admit to earn at least $500. b. The theater increases the price b $ per car. How does this affect the total number of cars needed to earn $500? Eplain. 8. For what values of r will the area of the shaded region be greater than or equal to 9(π )? r Graph the linear equation. (Section.) 9. = 0. =. = 5. = +. MULTIPLE CHOICE Which of the following is shown in the graph? (Section.) A + = B = 5 C + = D = Section. Solving Multi-Step Inequalities

31 Etension. Solving Compound Inequalities Lesson Tutorials Ke Vocabular compound inequalit, p. absolute value inequalit, p. A compound inequalit is an inequalit formed b joining two inequalities with the word and or the word or. Solutions of a compound inequalit with and consist of numbers that are solutions of both inequalities. Solutions of a compound inequalit with or consist of numbers that are solutions of at least one of the inequalities < > 0 and < 5 or > < Stud Tip EXAMPLE A compound inequalit with and can be written as a single inequalit. For eample, ou can write > 8 and as 8 <. Writing and Graphing Compound Inequalities Write each word sentence as an inequalit. Graph the inequalit. a. A number is greater than 8 and less than or equal to. A number is greater than 8 and less than or equal to. > 8 and b. A number is at most 0 or at least 7. A number is at most 0 or at least 7. 0 or In Eercises, write the word sentence as an inequalit. Graph the inequalit.. A number k is more than and less than 9.. A number n is greater than or equal to 6 and no more than.. A number w is fewer than 0 or no less than 6.. A number z is less than or equal to 5 or more than. 5. Write an inequalit to describe the graph The world s longest human life span is ears. Write and graph a compound inequalit that describes the ages of all humans. Chapter Solving Linear Inequalities

32 You can solve compound inequalities b solving two inequalities separatel. When a compound inequalit with and is written as a single inequalit, ou can solve the inequalit b performing the same operation on each epression. Stud Tip EXAMPLE You can also solve the inequalit in Eample b solving the inequalities < + and + 9 separatel. Solving a Compound Inequalit with And Solve < + 9. Graph the solution. < + 9 Write the inequalit. Subtract from each epression. < 8 Simplif. > 8 > Simplif. The solution is <. Divide each epression b. Reverse the inequalit smbols EXAMPLE COMMON CORE Solving Inequalities In this etension, ou will write, solve, and graph compound inequalities. write, solve, and graph absolute value inequalities. Appling Standards A.CED. A.CED. A.REI. Solving a Compound Inequalit with Or Solve 5 < 8 or > 5. Graph the solution. 5 < 8 or > 5 Write the inequalit Addition Propert of Inequalit < or > 6 Simplif. < or > 6 < or > Simplif. The solution is < or >. Division Propert of Inequalit Solve the inequalit. Graph the solution. 7. < 5 < < > or 5. 7 < 5 or Etension. Solving Compound Inequalities

33 Stud Tip When an absolute value epression is on the left side of an inequalit, use an and statement for < and, and an or statement for > and. An absolute value inequalit is an inequalit that contains an absolute value epression. For eample, < and > are absolute value inequalities. The distance between and 0 is less than. < The distance between and 0 is greater than. > 0 0 The graph of < is > and <. The graph of > is < or >. You can solve these tpes of inequalities b solving a compound inequalit. Solving Absolute Value Inequalities To solve a + b < c for c > 0, solve the compound inequalit a + b > c and a + b < c. To solve a + b > c for c > 0, solve the compound inequalit a + b < c or a + b > c. In the inequalities above, ou can replace < with and > with. EXAMPLE Solving Absolute Value Inequalities a. Solve + 7. Graph the solution. Use + 7 to write a compound inequalit. Then solve. + 7 and + 7 Write compound inequalit Subtract 7 from each side. 9 and 5 Simplif. The solution is 9 and b. Solve 8 < 0. The absolute value of an epression must be greater than or equal to 0. The epression 8 cannot be less than 0. So, the inequalit has no solution. Chapter Solving Linear Inequalities

34 EXAMPLE 5 Solving an Absolute Value Inequalit Solve 5 + > > 9 Write the inequalit. 5 > 7 Isolate the absolute value epression. Use 5 > 7 to write a compond inequalit. Then solve. 5 < 7 or 5 > 7 Write compound inequalit Add 5 to each side. < or > Simplif. < or > < or > 6 Simplif. Divide each side b. EXAMPLE 6 Real-Life Application In a poll, 7% of voters sa the plan to reelect the maor. The poll has a margin of error of ± percentage points. Write and solve an absolute value inequalit to find the least and greatest percents of voters who plan to reelect the maor. Words Actual percent of voters minus percent of voters in poll is less than or equal to the margin of error. Variable Let represent the actual percent of voters who plan on reelecting the maor. Inequalit 7 7 and 7 Write compound inequalit Add 7 to each side. 5 and 9 Simplif. The least percent of voters who plan to reelect the maor is 5%. The greatest percent of voters who plan to reelect the maor is 9%. Solve the inequalit. Graph the solution, if possible < < > 7 8. NUMBER SENSE What is the solution of 6? Eplain. 9. MODELING In Eample 6, % of the voters sa the plan to reelect the maor. The poll has a margin of error of ± percentage points. Use a model to write and solve an absolute value inequalit to find the least and greatest percents of voters who plan to reelect the maor. Etension. Solving Compound Inequalities 5

35 .5 Graphing Linear Inequalities in Two Variables How can ou use a coordinate plane to solve problems involving linear inequalities? ACTIVITY: Graphing Inequalities Work with a partner. a. Graph = + in a coordinate plane. b. Choose three points that lie above the graph of = +. Substitute the values of and of each point in the inequalit > +. If the substitutions result in true statements, plot the points on the graph. c. Choose three points that lie below the graph of = +. Substitute the values of and of each point in the inequalit > +. If the substitutions result in true statements, plot the points on the graph. O d. To graph > +, would ou choose points above or below = +? e. Choose a point that lies on the graph of = +. Substitute the values of and in the inequalit > +. What do ou notice? Do ou think the graph of > + includes the points that lie on the graph of = +? Eplain our reasoning. f. Eplain how ou could change the inequalit so that it includes the points that lie on the graph of = +. ACTIVITY: Writing and Graphing Inequalities COMMON CORE Graphing Inequalities In this lesson, ou will graph linear inequalities in two variables. Learning Standard A.REI. O Work with a partner. The graph of a linear inequalit in two variables shows all the solutions of the inequalit in a coordinate plane. An ordered pair (, ) is a solution of an inequalit if the inequalit is true when the values of and are substituted in the inequalit. a. Write an equation for the graph of the dashed blue line. b. The solutions of an inequalit are represented b the shaded region. In words, describe the solutions of the inequalit. c. Write an inequalit for the graph. Which inequalit smbol did ou use? Eplain our reasoning. 6 Chapter Solving Linear Inequalities

36 Math Practice Recognize Usefulness of Tools When do ou think it would be useful to use a graphing calculator? EXAMPLE: Using a Graphing Calculator Use a graphing calculator to graph. a. Enter the equation = into our calculator. b. The inequalit contains the smbol. So, the region to be shaded is above the graph of =. Adjust our graphing calculator so that the region above the graph will be shaded. For some calculators, this icon represents the region above the graph. c. Graph on our calculator Some graphing calculators alwas use a solid line when graphing inequalities. In this case, ou will have to decide whether the line should be dashed or solid. 0. Use a graphing calculator to graph each inequalit in a standard viewing window. a. > + 5 b. + c. 5. IN YOUR OWN WORDS How can ou use a coordinate plane to solve problems involving linear inequalities? Give an eample of a real-life problem that can be represented b an inequalit in two variables. Use what ou learned about writing and graphing inequalities to complete Eercises 8 0 on page. Section.5 Graphing Linear Inequalities in Two Variables 7

37 .5 Lesson Lesson Tutorials Ke Vocabular linear inequalit in two variables, p. 8 solution of a linear inequalit, p. 8 graph of a linear inequalit, p. 8 half-planes, p. 8 A linear inequalit in two variables and can be written as a + b < c a + b c a + b > c a + b c where a, b, and c are real numbers. A solution of a linear inequalit in two variables is an ordered pair (, ) that makes the inequalit true. EXAMPLE Checking Solutions of a Linear Inequalit Tell whether the ordered pair is a solution of the inequalit. a. + < ; (, 9) + < Write the inequalit. ( ) + 9 <? Substitute for and 9 for. 7 < Simplif. 7 is not less than. So, (, 9) is not a solution of the inequalit. b. 8; (, ) 8 Write the inequalit. ( )? 8 Substitute for and for. 8 8 Simplif. 8 is equal to 8. So, (, ) is a solution of the inequalit. Eercises 8 Tell whether the ordered pair is a solution of the inequalit.. + > 0; (, ). 5; (0, 0). 5 ; (, ). < 5; (5, 7) Reading A dashed boundar line means that points on the line are not solutions. A solid boundar line means that points on the line are solutions. The graph of a linear inequalit in two variables shows all of the solutions of the inequalit in a coordinate plane. All solutions of < lie on one side of the boundar line. The boundar line divides the coordinate plane into two half-planes. The shaded halfplane is the graph of <. 8 Chapter Solving Linear Inequalities

38 Graphing a Linear Inequalit in Two Variables Step Step Step Graph the boundar line for the inequalit. Use a dashed line for < or >. Use a solid line for or. Test a point that is not on the boundar line to determine if it is a solution of the inequalit. If the test point is a solution, shade the half-plane that contains the point. If the test point is not a solution, shade the half-plane that does not contain the point. It is convenient to use the origin as a test point because it is easil substituted. However, ou must choose a different test point if the origin is on the boundar line. EXAMPLE Graphing Linear Inequalities in One Variable a. Graph in a coordinate plane. Step : Graph =. Use a solid line because the inequalit smbol is. Step : Test (0, 0). Write the inequalit. 0 Substitute. Step : Because (0, 0) is a solution, shade the half-plane that contains (0, 0). (0, 0) b. Graph > in a coordinate plane. Step : Graph =. Use a dashed line because the inequalit smbol is >. Step : Test (0, 0). > Write the inequalit. 0 > Substitute. Step : Because (0, 0) is not a solution, shade the half-plane that does not contain (0, 0). (0, 0) Eercises Graph the inequalit in a coordinate plane. 5. > > Section.5 Graphing Linear Inequalities in Two Variables 9

39 EXAMPLE Graphing Linear Inequalities in Two Variables Check 5 Graph + > in a coordinate plane. Step : Graph + =, or = +. Use a dashed line because the inequalit smbol is >. Step : Test (0, 0). (0, 0) + > (0) + (0) >? Write the inequalit. Substitute. 0 > Simplif. Step : Because (0, 0) is not a solution, shade the half-plane that does not contain (0, 0). EXAMPLE Real-Life Application You can spend at most $0 on grapes and apples for a fruit salad. Grapes cost $.50 per pound and apples cost $ per pound. Write and graph an inequalit for the amounts of grapes and apples ou can bu. Identif and interpret two solutions of the inequalit. Words Cost per pound of grapes times Pounds of grapes plus Cost per pound of apples times Pounds of apples is at most Amount ou can spend Variables Let be pounds of grapes and be pounds of apples. Inequalit (, 6) (, 5) Step : Graph.5 + = 0, or = Use a solid line because the inequalit smbol is. Step : Test (0, 0) Write the inequalit..5(0) + 0? 0 Substitute. 0 0 Simplif. Step : Because (0, 0) is a solution, shade the half-plane that contains (0, 0). Two possible solutions are (, 6) and (, 5). So, ou can bu pound of grapes and 6 pounds of apples, or pounds of grapes and 5 pounds of apples. Eercises 5 0 Graph the inequalit in a coordinate plane < Chapter Solving Linear Inequalities

40 .5 Eercises Help with Homework. VOCABULARY How can ou tell whether an ordered pair is a solution of an inequalit?. OPEN-ENDED Write an eample of an inequalit in two variables.. WRITING Compare the graph of a linear inequalit in two variables with the graph of a linear equation in two variables.. REASONING Wh do ou onl need to test one point when graphing a linear inequalit? Match the inequalit with its graph. 5. > 6. > 7. < A. B. C. O O O 9+(-6)= +(-)= +(-9)= 9+(-)= In words, describe the solutions of the inequalit. 8. > < Tell whether the ordered pair is a solution of the inequalit.. + < 7; (6, ). 0; (, 5). + ; (, ). + > 6; (0, 0) ; (, 5) ; (, ) 7. 6 > ; ( 8, ) 8. 8 < 5; ( 6, ) Tell whether the ordered pair is a solution of the inequalit whose graph is shown. 9. (0, ) 0. (0, 0). (, ). (, ). (, ). (, ) 5. FABRIC You can spend at most $60 on lace. Cotton lace is $ per ard and linen lace is $ per ard. Write an inequalit for the amounts of lace ou can bu. Can ou bu ards of cotton lace and 5 ards of linen lace? Eplain. Section.5 Graphing Linear Inequalities in Two Variables

41 In Eercises 6 8, use the inequalit + <. 6. Write the equation of the boundar line in slope-intercept form. 7. Tell whether ou would use a solid line or a dashed line to graph the boundar line. Then graph the boundar line. 8. Test the point (0, 0) in the inequalit. Is the test point a solution? If so, shade the half-plane that contains the point. If not, shade the half-plane that does not contain the point. Match the inequalit with its graph < 6. 6 A. B. C. O O O Graph the inequalit in a coordinate plane.. < 5.. < > < 0. + > ERROR ANALYSIS Describe and correct the error in graphing the inequalit.. < +. O O. CRITICAL THINKING When graphing a linear inequalit in two variables, wh must ou choose a test point that is not on the boundar line?. MODELING In order for the drama club to cover the epenses of producing a pla, at least $500 worth of tickets must be sold. a. Use a model to write an inequalit that represents this situation. b. Graph the inequalit. c. Eight adults and 0 students attend the pla. Does the drama club cover its epenses? Eplain. Chapter Solving Linear Inequalities

42 Tell whether the ordered pair is a solution of the inequalit. 5. < + ; (6, ) ; (0.5,.5) ; (0,.) 8. > 5; ( ), Write an inequalit that represents the graph. 9. O 50. O 5. O 5. REASONING How man solutions does the inequalit + 5 have? 5. PROBLEM SOLVING After buing our admission ticket, ou have $9 to spend at the movies. Arcade games cost $0.75 per game and drinks cost $.5. a. Write and graph an inequalit that represents the numbers of arcade games ou can pla and drinks ou can bu. b. Identif and interpret two solutions of the inequalit. 5. Large boes weigh 75 pounds and small boes weigh 0 pounds. a. Write and graph an inequalit that represents the numbers of large and small boes a 00-pound deliver person can take on the elevator. b. Identif and interpret two solutions of the inequalit that are on the boundar line. c. Eplain wh the solutions in part (b) might not be practical in real life. Weight Limit 000 lb Multipl. (Skills Review Handbook) ( ) ( ) ( ) MULTIPLE CHOICE Which graph represents the solution of 5( 9) 5? (Section.) A B C D Section.5 Graphing Linear Inequalities in Two Variables

43 ..5 Quiz Progress Check Solve the inequalit. Graph the solution. (Section.). m + 7. n 6 8. j 5 > 7. 5 > w 7 Write the word sentence as an inequalit. Graph the inequalit. (Section.) 5. A number h is greater than and less than A number q is less than or equal to or at least. Solve the inequalit. Graph the solution, if possible. (Section.) 7. 7 > + 5 > 8. z + 0 or z b 0. r + 7 < 9 Graph the inequalit in a coordinate plane. (Section.5). 8. < 6. + > PARTY You bu lunch for guests at a part. You can spend no more than $00. You will spend $0 on beverages and $0 per guest on sandwiches. Write and solve an inequalit to find the number of guests ou can invite to the part. (Section.) 6. BOOKS You have a gift card worth $50. You want to bu several paperback books that cost $6 each. Write and solve an inequalit to find the number of books ou can bu and still have at least $0 on the gift card. (Section.) 7. SUPPLIES You have $6 to spend on pens and notebooks. Pens cost $0.75 each and notebooks cost $.50 each. Write and graph an inequalit that represents the numbers of pens and notebooks ou can bu. Identif and interpret a solution of the inequalit. (Section.5) Chapter Solving Linear Inequalities

44 Chapter Review Review Ke Vocabular inequalit, p. 06 solution of an inequalit, p. 06 solution set, p. 06 graph of an inequalit, p. 07 compound inequalit, p. absolute value inequalit, p. linear inequalit in two variables, p. 8 Review Eamples and Eercises Vocabular Help solution of a linear inequalit, p. 8 graph of a linear inequalit, p. 8 half-planes, p. 8. Writing and Graphing Inequalities (pp. 0 09) a. Four plus a number w is at least. Write this sentence as an inequalit. Four plus a number w is at least. + w An inequalit is + w. b. Graph m >. Use an open circle because is not a solution Test a number to the left of. m is not a solution. Test a number to the right of. m 5 is a solution. Shade the number line on the side where ou found the solution Write the word sentence as an inequalit.. A number v is less than.. A number minus is no more than. Tell whether the given value is a solution of the inequalit.. 0 q < ; q = 6. m ; m = Graph the inequalit on a number line. 5. p <. 6. n > 0 Chapter Review 5

45 . Solving Inequalities Using Addition or Subtraction (pp. 0 5) Solve < n. Graph the solution. Undo the subtraction. < n Write the inequalit. + + Add to each side. < n Simplif. The solution is n >. n Check: n is not a solution. Check: n is a solution. Solve the inequalit. Graph the solution. 7. b + < Solving Inequalities Using Multiplication or Division (pp. 6 ) Solve 8a 8. Graph the solution. Undo the multiplication. 8a 8 8a a 6 Write the inequalit. Divide each side b 8. Reverse the inequalit smbol. Simplif. The solution is a 6. a Check: a 0 is a solution. Check: a 8 is not a solution. Solve the inequalit. Graph the solution. 0.. z <. q 8 6 Chapter Solving Linear Inequalities