2.8}. So, 2 is the greatest number of CDs Rita can buy.

Transcription

1 If four people plan to use the canoe with 60 pounds of supplies, write and solve an inequality to find the allowable average weight per person. The canoe holds a maximum of 800 pounds. The weight of 4 people plus 60 pounds is to be in the canoe. Let n weight of each person. Write an inequality. The solution set is {n n Rita is ordering a movie for $11.95 and a few CDs. She has $50 to spend. Shipping and sales tax will be $10. If each CD costs $9.99, write and solve an inequality to find the greatest number of CDs that she can buy. Rita has $50 to spend on an $11.95 movie plus a $10 fee plus an unknown number of $9.99 CDs. Let x represent the maximum number of CDs she can buy. Write an inequality. The solution set is {x x 2.8}. So, 2 is the greatest number of CDs Rita can buy. Solve each inequality. Graph the solution on a number line. 6h The solution set is {h h 7}. r+9 Page 1

2 The solution set is {x x 2.8}. So, 2 is the greatest number of CDs Rita can buy. Solve each inequality. Graph the solution on a number line. 6h The solution set is {h h 7}. r+9 The solution set is {r r 18}. 3x + 7 > 43 The solution set is {x x < 12}. 4m 17 < 6m + 25 Page 2

3 The solution set is {x x < 12}. 4m 17 < 6m + 25 The solution set is {m m > 21}. Define a variable, write an inequality, and solve each problem. Then check your solution. Four times a number minus 6 is greater than eight plus two times the number. Let n = the number. The solution set is {n n > 7}. To check this answer, substitute a number greater than 7 into the original inequality. Let n = 8. Negative three times a number plus 4 is less than five times the number plus eight. Let n = the number. Page 3

4 5-3 Solving Multi-Step Inequalities So, the solution checks. Negative three times a number plus 4 is less than five times the number plus eight. Let n = the number. The solution set is. To check this answer, substitute a number greater than n = 0. Solve each inequality. Graph the solution on a number line. v 2) The solution set is {v v 0}. To check this answer, substitute a number greater than or equal to 0 into the original inequality. Let v = 1. Page 4

5 Solve each inequality. Graph the solution on a number line. v 2) The solution set is {v v 0}. To check this answer, substitute a number greater than or equal to 0 into the original inequality. Let v = 1. 5(g + 4) > 3(g 4) The solution set is {g g < 1} To check this answer, substitute a number less than 1 into the original inequality. Let g = 2. Page 5

6 5-3 Solving Multi-Step Inequalities So, the solution checks. 5(g + 4) > 3(g 4) The solution set is {g g < 1} To check this answer, substitute a number less than 1 into the original inequality. Let g = x 4x) Since the inequality results in a false statement, the solution set is the empty set, Solve each inequality. Graph the solution on a number line. 5b 11 Page 6

7 Since the inequality results in a false statement, the solution set is the empty set, Solve each inequality. Graph the solution on a number line. 5b 11 The solution set is {b b 2}. 21 > a The solution set is {a a < 3}. m+7 The solution set is {m m 40}. 13 > 6 Page 7

8 The solution set is {a a < 3}. m+7 The solution set is {m m 40}. 13 > 6 The solution set is {w w > 56}. a The solution set is {a a 1}. 37 < 7 10w Page 8

9 The solution set is {w w > 56}. a The solution set is {a a 37 < 7 1}. 10w The solution set is {w w < 3}. 8 The solution set is {z z 9}. p + 6 < 12 Page 9

10 The solution set is {z z 9}. p + 6 < 12 The solution set is 3b. b The solution set is {b b 15h + 30 < 10h 1}. 45 The solution set is {h h < 15}. Page 10

11 The solution set is {b b 1}. 15h + 30 < 10h 45 The solution set is {h h < 15}. Define a variable, write an inequality, and solve each problem. Check your solution. Three fourths of a number decreased by nine is at least forty-two. Let n = the number. The solution set is {n n inequality. Let n = }. To check this answer, substitute a number greater than or equal to 68 into the original Two thirds of a number added to six is at least twenty-two. Let n = the number. Page 11

12 5-3 Solving Multi-Step Inequalities So, the solution checks. Two thirds of a number added to six is at least twenty-two. Let n = the number. The solution set is {n n inequality. Let n = }. To check this answer, substitute a number greater than or equal to 24 into the original Seven tenths of a number plus 14 is less than forty-nine. Let n = the number. The solution set is {n n < 50}. To check this answer, substitute a number less than 50 into the original inequality. Let n = 10. Page 12

13 Seven tenths of a number plus 14 is less than forty-nine. Let n = the number. The solution set is {n n < 50}. To check this answer, substitute a number less than 50 into the original inequality. Let n = 10. Eight times a number minus twenty-seven is no more than the negative of that number plus eighteen. Let n = the number. The solution set is {n n inequality. Let n = 1. 5}. To check this answer, substitute a number less than or equal to 5 into the original Ten is no more than 4 times the sum of twice a number and three. Page 13

14 Eight times a number minus twenty-seven is no more than the negative of that number plus eighteen. Let n = the number. The solution set is {n n inequality. Let n = 1. 5}. To check this answer, substitute a number less than or equal to 5 into the original Ten is no more than 4 times the sum of twice a number and three. Let n = the number. The solution set is. To check this answer, substitute a number greater than or equal to n = 1. Page 14

15 Ten is no more than 4 times the sum of twice a number and three. Let n = the number. The solution set is. To check this answer, substitute a number greater than or equal to n = 1. Three times the sum of a number and seven is greater than five times the number less thirteen. Let n = the number. The solution set is {n n < 17}. To check this answer, substitute a number less than 17 into the original inequality. Let n = 1. Page 15

16 Three times the sum of a number and seven is greater than five times the number less thirteen. Let n = the number. The solution set is {n n < 17}. To check this answer, substitute a number less than 17 into the original inequality. Let n = 1. The sum of nine times a number and fifteen is less than or equal to the sum of twenty-four and ten times the number. Let n = the number. The solution set is {n n 9}. To check this answer, substitute a number greater than or equal to 9 into the original inequality. Let n = 1. Solve each inequality. Check your solution. Page 16

17 The sum of nine times a number and fifteen is less than or equal to the sum of twenty-four and ten times the number. Let n = the number. The solution set is {n n 9}. To check this answer, substitute a number greater than or equal to 9 into the original inequality. Let n = 1. Solve each inequality. Check your solution. 3(7n + 3) < 6n The solution set is. To check this answer, substitute a number greater than n = 1. Page 17

18 Solve each inequality. Check your solution. 3(7n + 3) < 6n The solution set is. To check this answer, substitute a number greater than n = 1. a 7) + 9 The solution set is {a a 11}. To check this answer, substitute a number less than or equal to 11 into the original inequality. Let n = 7. Page 18

19 5-3 Solving Multi-Step Inequalities So, the solution checks. a 7) + 9 The solution set is {a a 11}. To check this answer, substitute a number less than or equal to 11 into the original inequality. Let n = 7. 2y + 4 > 2(3 + y) Since the inequality results in a false statement, the solution set is the empty set, 3(2 b) < 10 3(b 6) Since all values of x make the inequality true, all real numbers are the solution. {b b is a real number.} 7+t t + 3) + 2 Page 19

20 5-3 Solving Inequalities Since allmulti-step values of x make the inequality true, all real numbers are the solution. {b b is a real number.} 7+t t + 3) + 2 The solution set is {t t 1}. To check this answer, substitute a number greater than or equal to 1 into the original inequality. Let t = 1. 8a + 2(1 5a The solution set is {a a 9}. To check this answer, substitute a number greater than or equal to 9 into the original inequality. Let a = 1. Define a variable, write an inequality, and solve each problem. Then interpret your solution. A car salesperson is paid a base salary of $35,000 a year plus 8% of sales. What are the sales needed to havemanual an annual income greater than $65,000? esolutions - Powered by Cognero Page 20 Let s = the amount of sales made.

21 Define a variable, write an inequality, and solve each problem. Then interpret your solution. A car salesperson is paid a base salary of $35,000 a year plus 8% of sales. What are the sales needed to have an annual income greater than $65,000? Let s = the amount of sales made. The solution set is {s s > 375,000}. The amount of sales needed to have an annual income greater than $65,000 must be more than $375,000. Keith s dog weighs 90 pounds. A healthy weight for his dog would be less than 75 pounds. If Keith s dog can lose an average of 1.25 pounds per week on a certain diet, after how long will the dog reach a healthy weight? Let w = the number of weeks. The solution set is {w w > 12}. It will take more than 12 weeks for the dog to reach a healthy weight. Solve 6(m 3) > 5(2m + 4). Show each step and justify your work. 6(m 3) > 5(2m + 4) 6m 18 > 10m m 18 6m > 10m m 18 > 4m > 4m > 4m 9.5 > m Original inequality Distributive Property Subtract 6m from each side. Simplify. Subtract 20 from each side. Simplify. Divide each side by 4. Simplify. The solution set is {m m < 9.5}. Solve 8(a a + 2). Show each step and justify your work. 8(a a + 2) Original inequality Page 21

22 9.5 > m Simplify. The solution set is {m m < 9.5}. Solve 8(a a + 2). Show each step and justify your work. 8(a a a 16 8a 16 a + 2) a + 20 a a a + 20 a a Simplify. a The solution set is{a a Original inequality Distributive Property Subtract 8a from each side. Simplify. Subtract 20 from each side. Simplify. Divide each side by 2. 18}. A high school drama club is performing a musical to benefit a local charity. Tickets are $5 each. They also received donations of $565. They want to raise at least $1500. Write an inequality that describes this situation. Then solve the inequality. Graph the solution. a. Let t = the number of tickets that need to be sold. The solution set is {t t 187}. b. Benito has $6 to spend. A sundae costs $3.25 plus $0.65 per topping. Write and solve an inequality to find how many toppings he can order. Let t = the number of toppings Benito can order. Benito can order 4 or fewer toppings. Page 22

23 The solution set is {t t 187}. b. Benito has $6 to spend. A sundae costs $3.25 plus $0.65 per topping. Write and solve an inequality to find how many toppings he can order. Let t = the number of toppings Benito can order. Benito can order 4 or fewer toppings. Write an inequality that represents a camel s body temperature at noon if the camel had no water. If C represents degrees Celsius, then. Write and solve an inequality to find the camel s body temperature at noon in degrees Celsius. t > 104 b. The camel Find all sets of three consecutive positive even integers with a sum no greater than 36. Create sets of 3 positive even integers that start with numbers less than or equal to 10. 2, 4, 6; 4, 6, 8; 6, 8, 10; 8, 10, 12; 10, 12, 14 Find all sets of four consecutive positive odd integers whose sum is less than 42. Page 23

24 5-3 Solving Multi-Step Inequalities Create sets of 3 positive even integers that start with numbers less than or equal to 10. 2, 4, 6; 4, 6, 8; 6, 8, 10; 8, 10, 12; 10, 12, 14 Find all sets of four consecutive positive odd integers whose sum is less than 42. Create sets of 4 positive odd integers that start with numbers less than , 3, 5, 7; 3, 5, 7, 9; 5, 7, 9, 11; 7, 9, 11, 13 Solve each inequality. Check your solution. 2(x x 6) The solution set is {x x inequality. Let x = 9. 8}. To check this answer, substitute a number greater than or equal to 8 into the original 5x + 2 Page 24

25 5-3 Solving Multi-Step Inequalities So, the solution checks. 5x + 2 The solution set is. To check this answer, substitute a number greater than or equal to x = z < 2.5z 4.7 The solution set is {z z < 2}. To check this answer, substitute a number less than 2 into the original inequality. Let z = 10. Page 25

26 5.6z < 2.5z 4.7 The solution set is {z z < 2}. To check this answer, substitute a number less than 2 into the original inequality. Let z = (2m The solution set is {m m 18}. To check this answer, substitute a number greater than or equal to 18 into the original inequality. Let m = 20. 3x + 7 > 4x + 9 Use a graphing calculator to solve the inequality. intersect function from the CALC menu to find the intersection ofpage the 26 Let Y1 = 2x + 7 andbyy2 = 4x esolutions Manual - Powered Cognero

27 3x + 7 > 4x + 9 Use a graphing calculator to solve the inequality. intersect function from the CALC menu to find the intersection of the Let Y1 = 2x + 7 and Y2 = 4x [ 7, 3] scl: 1 by [ 5, 5] scl: 1 At x = 2, Y1 = Y2 or 2x + 7 = 4x + 9. When x < 2, 2x + 7 > 4x + 9. Thus the solution set is {x x < 2}. 13x x + 37 Use a graphing calculator to solve the inequality. intersect function from the CALC menu to find the intersection of Let Y1 = 13x 11and Y2 = 7x [ 5, 15] scl: 1 by [ 2,100] scl: 10 At x = 8, Y1 = Y2 or 13x 2(x 11 = 7x When x x x Thus the solution set is {x x 8}. 3) < 3(2x + 2) Use a graphing calculator to solve the inequality. intersect function from the CALC menu to find the intersection of Let Y1 = 2(x 3) and Y2 = 3(2x [ 5, 5] scl: 1 by [ 17, 3] scl: 2 Page 27

28 [ 5, 15] scl: 1 by [ 2,100] scl: 10 At x = 8, Y1 = Y2 or 13x 11 = 7x When x 2(x x Thus the solution set is {x x x 8}. 3) < 3(2x + 2) Use a graphing calculator to solve the inequality. intersect function from the CALC menu to find the intersection of Let Y1 = 2(x 3) and Y2 = 3(2x [ 5, 5] scl: 1 by [ 17, 3] scl: 2 At x = 3, Y1 = Y2 or 2(x x 3)= 3(2x + 2). When x > 3, 2(x 3)< 3(2x + 2). Thus the solution set is {x x > 3}. 9 < 2x Use a graphing calculator to solve the inequality. Let Y1 = x 9 and Y2 = 2x intersect function from the CALC menu to find the intersection of the [ 9, 1] scl: 1 by [ 18, 2] scl: 2 At x = 6, Y1 = Y2 x 9 = 2x. When x > x 9 < 2x. Thus the solution set is {x x > 6}. Use a graphing calculator to solve the inequality. Let Y1 = Y2 = x intersect function from the CALC menu to find the intersection of the Page 28

29 [ 9, 1] scl: 1 by [ 18, 2] scl: Solving Y1 = Y2 Inequalities At x = 6,Multi-Step x 9 = 2x. When x > x 9 < 2x. Thus the solution set is {x x > 6}. Use a graphing calculator to solve the inequality. Let Y1 = Y2 = x intersect function from the CALC menu to find the intersection of the [ 35, 5] scl: 2 by [ 55, 5] scl: 4 At x =, Y1 = Y2. When x,. Thus the solution set is. (4x x+2 Use a graphing calculator to solve the inequality. Let Y1 = (4x Y2 = x + 2. Use the intersect function from the CALC [ 10, 10] scl: 1 by [ 10, 10] scl: 1 At x = 1.5, Y1 = Y2 (4x x + 2. When x 1.5, (4x x + 2. Thus the solution set is {x x 1.5}. In this problem, you will solve compound inequalities. A number x is greater than 4, and the same number is less than 9. Write two separate inequalities for the statement. Graph the solution set for the first inequality in red. Graph the solution set for the second inequality in blue. Highlight the portion of the graph in which the red and blue overlap. Make a table using ten points from your number line, including points from each section. Use one column for- each inequality and a third column titled Both are True. Complete the table by writing true or false.page 29 esolutions Manual Powered by Cognero Describe the relationship between the colored regions of the graph and the chart. Make a prediction of what the graph of 4 < x < 9 looks like.

30 [ 10, 10] scl: 1 by [ 10, 10] scl: Solving At x = 1.5Multi-Step, Y1 = Y2 Inequalities (4x x + 2. When x 1.5, (4x x + 2. Thus the solution set is {x x 1.5}. In this problem, you will solve compound inequalities. A number x is greater than 4, and the same number is less than 9. Write two separate inequalities for the statement. Graph the solution set for the first inequality in red. Graph the solution set for the second inequality in blue. Highlight the portion of the graph in which the red and blue overlap. Make a table using ten points from your number line, including points from each section. Use one column for each inequality and a third column titled Both are True. Complete the table by writing true or false. Describe the relationship between the colored regions of the graph and the chart. Make a prediction of what the graph of 4 < x < 9 looks like. x > 4; x < 9 b. c. The points that make x > 4 a true statement are in the blue section. The points that make x < 9 a true statement are in the red section. The points that make both statements true are in the highlighted section. The graph would be the highlighted section of the number line. Explain how you could solve 3p 2 without multiplying or dividing each side by a negative number. Add 3p and 2 to each side. The inequality becomes 9 how you know. 3p. Then divide each side by 3 to get 3 p. If ax + b < ax + c is true for all values of x, what will be the solution of ax + b > ax + c? Explain The solution of ax + b > ax + c would be the empty set, ax from each side of ax + b < ax + c leaves b < c. If the original inequality has infinitely many solutions, then b must be a real number that is less than c. Subtracting ax from each side of the second inequality ax + b > ax + c leaves b > c. If b is always less than c, then esolutions Manual - Powered Cognero Page 30 this inequality has nobysolutions. CHALLENGE Solve each inequality for x. Assume that a > 0.

31 Explain how you could solve 3p 2 without multiplying or dividing each side by a negative number. Add 3p and 2 to each side. The inequality becomes 9 how you know. 3p. Then divide each side by 3 to get 3 p. If ax + b < ax + c is true for all values of x, what will be the solution of ax + b > ax + c? Explain The solution of ax + b > ax + c would be the empty set, ax from each side of ax + b < ax + c leaves b < c. If the original inequality has infinitely many solutions, then b must be a real number that is less than c. Subtracting ax from each side of the second inequality ax + b > ax + c leaves b > c. If b is always less than c, then this inequality has no solutions. CHALLENGE Solve each inequality for x. Assume that a > 0. a. b. c. a. b. c. WHICH ONE DOESN Name the inequality that does not belong. Explain. Page 31

32 WHICH ONE DOESN Name the inequality that does not belong. Explain. 4y + 9 > 3 does not belong. It is the only inequality that does not have a solution set of {y y > 3}. Explain when the solution set of an inequality will be the empty set or the set of all real numbers. Show an example of each. Sample answer: The solution set for an inequality that results in a false statement is the empty set, as in 12 < 15. The solution set for an inequality in which any value of x What is the solution set of the inequality 4t + 2 < 8t {t t < 6.5} {t t > 6.5} {t t < 4} {t t > 4} (6t 10)? Page 32

33 numbers. Show an example of each. Sample answer: The Inequalities solution set for an inequality that results in a false statement is the empty set, as in 12 < Solving Multi-Step The solution set for an inequality in which any value of x What is the solution set of the inequality 4t + 2 < 8t {t t < 6.5} {t t > 6.5} {t t < 4} {t t > 4} (6t 10)? The solution set is {t t < 4}, so the correct choice is C. The section of Liberty Ave. between 5th St. and King Ave. is temporarily closed. Traffic is being detoured right on 5th St., left on King Ave. and then back on Liberty Ave. How long is the closed section of Liberty Ave.? 100 ft 120 ft 144 ft 180 ft The closed section of Liberty Ave. is 120 feet, so the correct choice is G. Rhiannon is paid $52 for working 4 hours. At this rate, how many hours will it take her to earn $845? Find the rate. Page 33

34 The closed section of Liberty Ave. is 120 feet, so the correct choice is G. Rhiannon is paid $52 for working 4 hours. At this rate, how many hours will it take her to earn $845? Find the rate. Solve for the number of hours using r = 13. It will take Rhiannon 65 hours of work to earn $845. Classify the triangle. right parallel obtuse equilateral All three sides of the triangle are equal in length, so it is equilateral. The correct choice is D. Solve each inequality. Check your solution. (Lesson 5-2) 5 The solution set is {y y 10}. To check this answer, substitute a number less than or equal to 10 into the original inequality. Let y = 30. Page 34

35 obtuse equilateral All three sides of the triangle are equal in length, so it is equilateral. The correct choice is D. Solve each inequality. Check your solution. (Lesson 5-2) 5 The solution set is {y y 10}. To check this answer, substitute a number less than or equal to 10 into the original inequality. Let y = b > 48 The solution set is{b b > 4}. To check this answer, substitute a number greater than 4 into the original inequality. Let b = 0. t 30 The solution set is {t t 45}. Page 35

36 t 30 The solution set is {t t 45}. To check this answer, substitute a number greater than or equal to 45 into the original inequality. Let t = 48. Solve each inequality. Check your solution, and graph it on a number line. 6 h> 8 The solution set is {h h < 14}. To check this answer, substitute a number less than 14 into the original inequality. Let h = 10. p 9<2 The solution set is {p p < 11}. Page 36

37 p 9<2 The solution set is {p p < 11}. To check this answer, substitute a number less than 11 into the original inequality. Let p = 10. m The solution set is {m m To check this answer, substitute a number greater than or equal to 1 into the original inequality. Let m = 4. Solve each equation by graphing. Verify your answer algebraically. 2x 7 = 4x + 9 Page 37

38 Solve each equation by graphing. Verify your answer algebraically. 2x 7 = 4x x = 7x 11 esolutions - Powered 2(x Manual 3) = 5x + 12 by Cognero Page 38

39 2(x 3) = 5x + 12 THEME PARKS In a recent year, 70.9 million people visited the top 5 theme parks in North America. That represents an increase of about 1.14% in the number of visitors from the prior year. About how many people visited the top 5 theme parks in North America in the prior year? About 70.1 million people visited theme parks in North America in the previous year. If f (x) = 4x f ( 2) 2 3 and g(x) = 2x + 5, find each value. Page 39

40 About 70.1 million people visited theme parks in North America in the previous year. If f (x) = 4x f ( 2) g(2) 2 3 and g(x) = 2x + 5, find each value. 5 f (c + 3) On average, a barber received a tip of $4 for each of 12 haircuts. Write and evaluate an expression to determine the total amount that she earned. The total amount the hair stylist earned is $ Graph each set of numbers on a number line. { 4, 2, 2, 4} Draw a solid dot for each number in the set. { 3, 0, 1, 5} Draw a solid dot for each number in the set. {integers less than 3} Page 40

41 Draw a solid dot for each number in the set. { 3, 0, 1, 5} Draw a solid dot for each number in the set. {integers less than 3} Draw a solid dot for several representative numbers in the set. {integers greater than or equal to 2} Draw a solid dot for several representative numbers in the set. {integers between 3 and 4} Draw a solid dot for each number in the set. {integers less than 1} Draw a solid dot for several representative numbers in the set. Page 41