Unit 2. Linear Inequalities

Transcription

1 Algebra I Revised 9/16 Unit 2 Linear Inequalities Name:

2 CONTENTS 5.1 Solving Inequalities by Addition and Subtraction 5.2 Solving Inequalities by Multiplication and Division 5.3 Solving Multi-Step Inequalities Assessment 5.6 Graphing Inequalities in Two Variables 6.6 Systems of Inequalities Unit Evaluation 2

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4 Symbols you need to know: < > Solving inequalities is similar to solving equations except that you get more than one answer or solution. Oftentimes we graph the solutions on a number line. Guided Practice: Solve and graph each. Test values in your solution set to check your answer. 1. x x + 5 > x + 2 < x Set-Builder Notation: 4

5 You should also be familiar with set-builder notation as a way to write a solution set. Guided Practice: Solve the inequality. Circle the correct answer. x + 23 < 14 A. {x x < -9} B. {x x > -9} C. {x x < 37} D. {x x > 39} Guided Practice: Twice a number is more than the sum of that number and 9. Write and solve an inequality to model this problem. 5

6 Real-World Example: Guided Practice: A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Define a variable, write an inequality to represent this problem and then solve the inequality. 5-1 Practice ~ Solving Inequalities by Addition and Subtraction Match each inequality with its corresponding graph x 15 a. 2. 4x + 3 < 5x b. 3. 8x > 7x 4 c x 9 d. 6

7 Solve each inequality. Check your solution, and then graph it on a number line. 5. r ( 5) > x + 8 4x 7. z + 3 > 8. c Define a variable, write an inequality, and solve each problem. Check your solution. 9. The sum of a number and 17 is no less than Twice a number minus 4 is less than three times the number. 11. Twelve is at most a number decreased by Eight plus four times a number is greater than five times the number. 7

8 13. ATMOSPHERIC SCIENCE The troposphere extends from the Earth s surface to a height of 6-12 miles, depending on the location and the season. If a plane is flying at an altitude of 5.8 miles, and the troposphere is 8.6 miles deep in that area, how much higher can the plane go without leaving the troposphere? 14. EARTH SCIENCE Mature soil is composed of three layers, the uppermost being topsoil. Jamal is planting a bush that needs a hole 18 centimeters deep for the roots. The instructions suggest an additional 8 centimeters depth for a cushion. If Jamal wants to add even more cushion, and the topsoil in his yard is 30 centimeters deep, how much more cushion can he add and still remain in the topsoil layer? 8

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10 **REMEMBER** When Multiplying and Dividing both sides by a Negative Number switch the sign of the inequality! If both sides of an inequality are multiplied or divided by a negative number, reverse the inequality sign. Guided Practice: Check your solutions to see that this will indeed yield the correct solutions. 1. Solve and graph. -3x + 1 < Solve and graph. x Solve and graph. x

11 5-2 Practice ~ Solving Inequalities by Multiplication and Division Match each inequality with its corresponding statement. 1. 4n 5 a. Negative four times a number is less than five. 2. n > 5 b. Four fifths of a number is no more than five. 3. 4n 5 c. Four times a number is fewer than five. 4. n 5 d. Negative four times a number is no less than five. 5. 4n < 5 e. Four times a number is at most five. 6. 4n < 5 f. Four fifths of a number is more than five. Solve each inequality. Check your solution. 7. < > 13p 11. > < m k b c > < y < 3 11

12 Define a variable, write an inequality, and solve each problem. Check your solution. 19. Negative three times a number is at least Two thirds of a number is no more than Negative three fifths of a number is less than 6. Take each everyday phrase and write it as an inequality using the variable of your choice. 22. Every day I drink at least 4 glasses of Vitamin Water. 23. The temperature during Saturday and Sunday was between 35 degrees and 58 degrees. 24. The amount that I am allowed to spend at Great Adventure on food is at most $ A growing teen is supposed to consume no more than 1800 calories per day. 26. A river is rising at a rate of 3 inches per hour. If the river rises more than 2 feet, it will exceed flood stage. How long can the river rise at this rate without exceeding flood stage? 12

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14 Guided Practice: Write an inequality to represent the problem; solve and interpret your answer. 1. A car salesperson is paid a base salary of $35,000 a year plus 8% of sales. What are the sales (S) needed to have an annual income greater than $65,000? 14

15 2. Keith s dog weighs 90 lbs. A healthy weight for this particular breed is no more than 75 lbs. If Keith s dog can lose about 1.25 lbs per week on a certain diet, then how many weeks will it take for the dog to be 75 lbs or less? 3. 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 $ Lisa has $6 to spend at the ice cream store. A sundae costs $3.25 plus $0.65 per topping. How many toppings can Lisa order for the sundae if she cannot spend more than $6? Write an inequality, solve, and graph. 5. Seven tenths of a number N plus 14 is less than forty-nine. 6. Ten is no more than 4 times the sum of twice a number and three. 7. Eight times a number minus twenty-seven is no more than the negative of that number plus eighteen. 8. Three times the sum of a number and seven is greater than five times the number less thirteen. 15

16 5-3 Practice ~ Solving Multi-Step Inequalities Solve and graph each inequality. Check your solution u 6 6u > a (z + 1) + 11 < 2(z + 13) 8. 3r + 2(4r + 2) 2(6r + 1) 9. 5n 3(n 6) < 3n 6n x + 3 3x x - 3 x 13. 4k - k (5 + 3x) < -4x (5 7x) + 7 < 7+ 4x x < 7( x + 6)+ 5x 16

17 Tell whether each inequality is true or false for the given value x 3 < 7; x = (x 6) > 18; x = 12 Solve and graph each inequality on a number line x 3 > 5x 3x + 3 < > (x 3) > 25 < > Define a variable, write an inequality, and solve each problem. Check your solution. 21. A number is less than one fourth the sum of three times the number and four. 22.Two times the sum of a number and four is no more than three times the sum of the number and seven decreased by four. 23. The area of a triangular garden can be no more than 120 square feet. The base of the triangle is 16 feet. What is the height of the triangle? 17

18 24. Nabuko practices the violin at least 12 hours per week. She practices for three fourths of an hour each session. If Nabuko has already practiced 3 hours in one week, how many sessions remain to meet or exceed her weekly practice goal? 25. You have been saving some of your lunch money on a daily basis. You currently have $45 and are saving 55 cents per day. In how many days will you have more than 125 dollars saved? 26. Your parents lent you $450 and you are paying them back at the rate of $10 per week. In how many weeks will you have less than $20 to pay back? 27. The temperature at night has been 32 degrees. Now that it is almost Spring, the temperature has been 2 degrees warmer than the previous night. In how many nights will the temperature be warmer than 50 degrees? 28. Pet Supplies makes a profit of $5.50 per bag on its line of natural dog food. If the store wants to make a profit of no less than $5225 on natural dog food, how many bags of dog food does it need to sell? 18

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20 Graphing Linear Inequalities on the Coordinate Plane Step 1: Graph the boundary. Use a solid line when the inequality contains or. Use a dashed line when the inequality contains < or >. Step 2: Use a test point to determine which half-plane should be shaded. Step 3: Shade the half-plane that contains the solution

21 Guided Practice: Graph each inequality y > x x y < x + 5y 10 21

22 Real Life Application: 22

23 Guided Practice: 1. A t-shirt company promises to give the freshmen class 10 additional (free) shirts if the freshmen class spends at least $1000 on class t-shirts. Each short-sleeved shirt costs $15 and each long-sleeved shirt costs $20. How many shirts of each type need to be purchased? There are many combinations. Let x = the number of short-sleeved shirts and let y = the number of long-sleeved shirts. Write a two-variable inequality to represent this situation and solve it for y. Graph it by shading all the possibilities. Determine which ordered pairs are part of the solution set for each inequality. 1. 3x + y 6, {(4, 3), ( 2, 4), ( 5, 3), (3, 3)} 2. y x + 3, {(6, 3), ( 3, 2), (3, 2), (4, 3)} 3. 3x 2y < 5, {(4, 4), (3, 5), (5, 2), ( 3, 4)} 23

24 Match each inequality to the graph of its solution. 4. y 2x < 2 a. b. 5. y 3x 6. 2y x 4 7. x + y > 1 c. d. Graph each inequality. 8. y < 1 9. y x y > 3x 11. y 2x y + x > y x 1 24

25 14. 2y x < x 2y y > 2x x > x > 2 x + Use a graph to solve each inequality. 20. y 2x > y 6x y x > 3 25

26 22. MOVING A moving van has an interior height of 7 feet (84 inches). You have boxes in 12 inch and 15 inch heights, and want to stack them as high as possible to fit. Write an inequality that represents this situation. 23. BUDGETING Satchi found a used bookstore that sells pre-owned DVDs and CDs. DVDs cost $9 each, and CDs cost $7 each. Satchi can spend no more than $35. a. Write an inequality that represents this situation. b. Does Satchi have enough money to buy 2 DVDs and 3 CDs? 24. You have been saving some of your lunch money on a daily basis. You currently have $45 and are saving 55 cents per day. In how many days will you have more than 125 dollars saved? 25. Your parents lent you $450 and you are paying them back at the rate of $10 per week. In how many weeks will you have less than $20 to pay back? 26. The temperature at night has been 32 degrees. Now that it is almost Spring, the temperature has been 2 degrees warmer than the previous night. In how many nights will the temperature be warmer than 50 degrees? 26

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28 Guided Practice: Solve each system of inequalities by graphing. 1. y > -2x x + y > 2 y x + 3-4x + 2y < 8 3. y 3 4. x < 6 x + y 1 y > x

29 5. y 3x 5 3x y > -4 Real life Application: Guided Practice: 1. Monica is running for student council. The election rules say that for the election to be valid, at least 80% of the 900 students must vote. Monica knows that needs more than 330 votes to win. a. Define the variables. let x = # of votes required by election rules let y = # of votes Monica needs to win b. Write a system of inequalities to represent this situation. c. Graph the system. d. Find a solution from your graph above and interpret its meaning. 29

30 2. The Theater Club is selling shirts. They have only enough supplies to print 120 shirts. They will sell sweatshirts for $22 and T-shirts for $15, with a goal of at least $2000 in sales. a. Define variables. let x = the number of sweatshirts let y = the number of t-shirts b. Write a system of inequalities. c. Graph the system. d. Name one possible solution. e. Is (45, 30) a solution. Explain. 30

31 6-6 Practice ~ Systems of Inequalities Solve each system of inequalities by graphing. 1. y > x 2 2. y x x + y 1 y x y > 2x + 3 x + 2y > y < 2x 1 5. y > x x y 2 y > 2 x 2x + y 2 x 2y 2 31

32 7. FITNESS Diego started an exercise program in which each week he works out at the gym between 4.5 and 6 hours and walks between 9 and 12 miles. a. Make a graph to show the number of hours Diego works out at the gym and the number of miles he walks per week. b. List three possible combinations of working out and walking that meet Diego s goals. 8. SOUVENIRS Emily wants to buy turquoise stones on her trip to New Mexico to give to at least 4 of her friends. The gift shop sells stones for either $4 or $6 per stone. Emily has no more than $30 to spend. a. Make a graph showing the numbers of each price of stone Emily can purchase. b. List three possible solutions. 32

33 Unit 1 Evaluation Review 1. Solve y 7 5. Then graph your solution on a number line. Solve each inequality k m < < t + 8 3t 3 4. > 8. 3( w 6) < 2(2w + 8) k x < 2.1(3 + x) 33

34 10. Graph y > 3x. 11. Use a graph to solve 2y 4x < Abe has $4500. He wants to buy a boat within $1300 of this amount. Define a variable, write an open sentence, and find the range of boat prices. 13. What inequality has the solution set shown in the graph? 14. Matthew is shopping for shoes and socks. He has $75.00 to spend. The shoes he likes cost $28.00, and the socks cost $4.00. Write an inequality for this situation. Can Matthew buy 2 pairs of shoes and 5 pairs of socks? For Questions 15 and 16, solve the system of inequalities by graphing. 15. y < x x y > 1 y x 1 x y 1