8-6 Solving Rational Equations and Inequalities. Solve each equation. Check your solution. ANSWER: 11 ANSWER: 9 ANSWER: 7 ANSWER: 3 ANSWER: 8
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1 Solve each equation. Check your solution esolutions Manual - Powered by Cognero Page 1
2 CCSS STRUCTURE Sara has 10 pounds of dried fruit selling for $6.25 per pound. She wants to know how many pounds of mixed nuts selling for $4.50 per pound she needs to make a trail mix selling for $5 per pound. a. Let m = the number of pounds of mixed nuts. Complete the following table. b. Write a rational equation using the last column of the table. c. Solve the equation to determine how many pounds of mixed nuts are needed. a. b m = m c. 25 esolutions Manual - Powered by Cognero Page 2
3 10. DISTANCE Alicia s average speed riding her bike is 11.5 miles per hour. She takes a round trip of 40 miles. It takes her 1 hour and 20 minutes with the wind and 2 hours and 30 minutes against the wind. a. Write an expression for Alicia s time with the wind. b. Write an expression for Alicia s time against the wind. c. How long does it take to complete the trip? d. Write and solve the rational equation to determine the speed of the wind. a. b. c. 3 h and 50 min d. esolutions Manual - Powered by Cognero Page 3
4 11. WORK Kendal and Chandi wax cars. Kendal can wax a particular car in 60 minutes and Chandi can wax the same car in 80 minutes. They plan on waxing the same car together and want to know how long it will take. a. How much will Kendal complete in 1 minute? b. How much will Kendal complete in x minutes? c. How much will Chandi complete in 1 minute? d. How much will Chandi complete in x minutes? e. Write a rational equation representing Kendal and Chandi working together on the car. f. Solve the equation to determine how long it will take them to finish the car. a. b. c. d. e. f. about 34.3 min Solve each inequality. Check your solutions < x < 1.15 esolutions Manual - Powered by Cognero Page 4
5 Solve each equation. Check your solutions esolutions Manual - Powered by Cognero Page 5
6 CHEMISTRY How many milliliters of a 20% acid solution must be added to 40 milliliters of a 75% acid solution to create a 30% acid solution? 180 ml 23. GROCERIES Ellen bought 3 pounds of bananas for $0.90 per pound. How many pounds of apples costing $1.25 per pound must she purchase so that the total cost for fruit is $1 per pound? 1.2 lb esolutions Manual - Powered by Cognero Page 6
7 24. BUILDING Bryan s volunteer group can build a garage in 12 hours. Sequoia s group can build it in 16 hours. How long would it take them if they worked together? about 6.86 hours Solve each inequality. Check your solutions. 25. x < 0 or x > < a < x < 2, or 2 < x < < x < x < 5 or 4 < x < esolutions Manual - Powered by Cognero Page 7
8 30. 2 < x, 2 < x < 1, x < AIR TRAVEL It takes a plane 20 hours to fly to its destination against the wind. The return trip takes 16 hours. If the plane s average speed in still air is 500 miles per hour, what is the average speed of the wind during the flight? mph 32. FINANCIAL LITERACY Judie wants to invest $10,000 in two different accounts. The risky account earns 9% interest, while the other account earns 5% interest. She wants to earn $750 interest for the year. Of tables, graphs, or equations, choose the best representation needed and determine how much should be invested in each account. $6250 at 9% and $3750 at 5% esolutions Manual - Powered by Cognero Page 8
9 33. MULTIPLE REPRESENTATIONS Consider a. ALGEBRAIC Solve the equation for x. Were any values of x extraneous? b. GRAPHICAL Graph and on the same graph for 0 < x < 5. c. ANALYTICAL For what value(s) of x do they intersect? Do they intersect where x is extraneous for the original equation? d. VERBAL Use this knowledge to describe how you can use a graph to determine whether an apparent solution of a rational equation is extraneous. a. 1; yes; 3 b. c. 1; no d. Graph both sides of the equation. Where the graphs intersect, there is a solution. If they do not, then the possible solution is extraneous. Solve each equation. Check your solutions esolutions Manual - Powered by Cognero Page 9
10 OPEN ENDED Give an example of a rational equation that can be solved by multiplying each side of the equation by 4(x + 3)(x 4). Sample answer: 37. CHALLENGE Solve all real numbers except 5, 5, CCSS TOOLS While using the table feature on the graphing calculator to explore the values 2 and 3 say ERROR. Explain its meaning. Sample answer: The values 2 and 3 are undefined values. On the graph of f (x) there would be vertical asymptotes at these values. 39. WRITING IN MATH Why should you check solutions of rational equations and inequalities? Sample answer: Multiplying each side of a rational equation or inequality by the LCD can result in extraneous solutions. Therefore, you should check all solutions to make sure that they satisfy the original equation or inequality. esolutions Manual - Powered by Cognero Page 10
11 40. Nine pounds of mixed nuts containing 55% peanuts were mixed with 6 pounds of another kind of mixed nuts that contain 40% peanuts. What percent of the new mixture is peanuts? A 58% B 51% C 49% D 47% C 41. Working alone, Dato can dig a 10-foot by 10-foot hole in five hours. Pedro can dig the same hole in six hours. How long would it take them if they worked together? F 1.5 hours G 2.34 hours H 2.52 hours J 2.73 hours J 42. An aircraft carrier made a trip to Guam and back. The trip there took three hours and the trip back took four hours. It averaged 6 kilometers per hour on the return trip. Find the average speed of the trip to Guam. A 6 km/h B 8 km/h C 10 km/h D 12 km/h B esolutions Manual - Powered by Cognero Page 11
12 43. SHORT RESPONSE If a line is perpendicular to a segment CD at point F and CF = FD, how many points on line are the same distance from point C as from point D? all of the points Determine whether each relation shows direct or inverse variation, or neither. 44. inverse 45. direct 46. neither esolutions Manual - Powered by Cognero Page 12
13 Graph each function esolutions Manual - Powered by Cognero Page 13
14 WEATHER The atmospheric pressure P, in bars, of a given height on Earth is given by using the formula In the formula, a is the surface pressure on Earth, which is approximately 1 bar, k is the altitude for which you want to find the pressure in kilometers, and H is always 7 kilometers. a. Find the pressure for 2, 4, and 7 kilometers. b. What do you notice about the pressure as altitude increases? a bars; 0.56 bars; 0.37 bars b. The pressure decreases as the altitude increases. 51. COMPUTERS Since computers have been invented, computational speed has multiplied by a factor of 4 about every three years. a. If a typical computer operates with a computational speed s today, write an expression for the speed at which you can expect an equivalent computer to operate after x three-year periods. b. Suppose your computer operates with a processor speed of 2.8 gigahertz and you want a computer that can operate at 5.6 gigahertz. If a computer with that speed is currently unavailable for home use, how long can you expect to wait until you can buy such a computer? a. b. 0.5 three-yr periods or 1.5 yr esolutions Manual - Powered by Cognero Page 14
15 Determine whether the following are possible lengths of the sides of a right triangle , 12, 13 yes , 80, 100 yes 54. 7, 24, 25 yes esolutions Manual - Powered by Cognero Page 15
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