. 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 Wor 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? 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
ACTIVITY: Volume and Surface Area of a Composite Solid Wor 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 Wor with a partner. You are building a patio. You want to cover the patio with Spanish tile that costs $ per square foot. Your budget for the tile is $700. How wide can ou mae 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
. 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. 6 + 7 < 9 Write the inequalit. 6 7 7 Subtract 7 from each side. 6 < 6 6 > 6 Multipl each side b 6. Reverse the inequalit smbol. > The solution is >. > 8 6 0 8 6 0 Eercises 0 Solve the inequalit. Graph the solution.. b < 7. 8 + 9c 8. n + > When solving an inequalit, if ou obtain an inequalit that is true, such as < 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. > The inequalit > is false. So, there are no solutions. 8 Chapter Solving Linear Inequalities
EXAMPLE Solving an Inequalit with Infinitel Man Solutions Which graph represents the solution of ( ) 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. ( ) 7 + 0 Write the inequalit. 0 7 + 0 Distributive Propert 0 0 Subtract 0 from each side. 7 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. 9 + 9 + 77 + 89 + 90 + 90 + 90 Multipl each side b. + 0 Subtract from each side. 98 The meaning of the phrase at least is greater than or equal to. You need at least 98 points to advance to the net round. Eercises 0 Solve the inequalit, if possible.. ( ) < +. (n ) > n +. 6. 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
. Eercises Help with Homewor. WRITING Compare and contrast solving multi-step inequalities and solving multi-step equations.. WRITING How do ou now when an inequalit has no solutions? How do ou now when the solution of an inequalit is the set of all real numbers? 9+(-6)= +(-)= +(-9)= 9+(-)=. For what values of 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? Solve the inequalit. Graph the solution. h. 7b + 6. v < 8 7. m 6 8. < w. ERROR ANALYSIS Describe and correct the error in solving the inequalit. Solve the inequalit, if possible. 9..8 < 0..p 0..r + 9.6.8 + 6 + 6 6. 6( g + ) 8. ( ) 9. 0 (h ). (u + ) > 6..7 > 0.9(n.7) 7. 0 >.(z.) 8. (w + ) w + 0 9. (6 ) < 7. 0. c > (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
Solve the inequalit. Graph the solution.. + 7 + 0. 7b 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.. WOODWORKING A woodworer builds a cabinet in 0 hours. The cabinet is sold at a store for $00. Write and solve an inequalit that represents the hourl wage the store can pa the woodworer and still mae 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 truc 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 $.0 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 $00. b. The theater increases the price b $ per car. How does this affect the total number of cars needed to earn $00? 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. =. =. = +. MULTIPLE CHOICE Which of the following is shown in the graph? (Section.) A + = B = C + = D = Section. Solving Multi-Step Inequalities