Chapter 2: Solving Linear Inequalities Chapter 2: Solving Linear Inequalities Assignment Sheet Date Topic 2.1.1-­‐ Comparing and Ordering Numbers 2.1.2-­‐ Writing and Graphing Inequalities 2.2-­‐ Solving Inequalities using Addition and Subtraction 2.3-­‐ Solving Inequalities using Multiplication and Division Review Assignment pg. 51 # 7-­‐13 Completed pg. 58 # 2, 5-­‐25 (odd), 27, 29-­‐43 (odd) pg. 65 # 1-­‐31 (odd) pg. 71 # 1-­‐29 (odd) pg. 80 # 1-­‐10, 14-­‐
16(a) Quiz 2.1-­‐2.3 NO HOMEWORK 2.4.1-­‐ Solving Multi-­‐Step pg. 77 # 3-­‐16 Inequalities 2.4.2-­‐ Solving Multi-­‐Step pg. 77 # 17-­‐28, 31, 33 Inequalities 2.5.1-­‐ Solving Compound pg. 85 # 1-­‐11 Inequalities 2.5.2-­‐ Solving Compound pg. 85 # 13-­‐19 (odd), Inequalities 25-­‐31 (odd) 2.6-­‐ Solving Absolute pg. 91 # 3-­‐27 (odd), Value Inequalities 31 Review pg. 94 # 1-­‐30 Review Cumulative Review: choose 3 problems from each bold-­‐
headed section EXAM NO HOMEWORK 2.1.1-­‐ Comparing and Ordering Numbers Complete the statement with ˂ , ˃ , or = 1) 6 ___ 5 2) -­‐2 ____ 3 4) -­‐8 ___ -­‐5 5) −5 ____ 5 7) √14 ___3.75 8) −√15___ -­‐4 3) -­‐4 ____ -­‐7 6) -­‐7 ____ −6 9) 2 ! ___ 2. 3 !
Order the values from least to greatest. 10) 3, − −2 , −2 , 0 , −1 !
11) 𝜋, 5.16, 5 ! , 25, 5.25 12) −1, 0.11, 0, −11, 1.1, 1 4
3
13) 5−3 , 7 , 5, 8 , 0.003 Homework: pg. 51 # 7-­‐13 2.1.2-­‐ Writing and Graphing Inequalities Inequality-­‐ Inequality Symbols-­‐ Solution of an inequality-­‐ Graph of an inequality-­‐ Complete the following chart. Words Algebra Graph x is less than 2 x is greater than 2 x is less than or equal to 2 x is greater than or equal to 2 Examples: Write the sentence as an inequality. a. A number w minus 3.5 is less than or equal to -­‐2. b. Three is less than a number n plus 5. c. Zero is greater than or equal to twice a number x plus 1. d. The temperature t in Sweden is at least -­‐10 degrees C. e. The elevation e of Alabama is at most 2407 feet. f. An upcoming marathon’s qualifying time for males is 3 hours. Will a runner with a time of 3 hours and 9 minutes qualify for this marathon? You try: 1) You must be at least 16 to get your driver's permit. 2) You can earn at most 100% on your report card. 3) 12 is greater than or equal to five times a number n. 4) Seven is less than or equal to the difference of a number q and 6. Tell whether the value is a solution of the inequality. a. x + 8 ˂ -­‐3; x = -­‐4 b. -­‐4.5x ˃ -­‐21; x = -­‐4 You try: 1) c + 4 ˂ -­‐1; c = -­‐6 2) 10 ≤ 3 – m; m = -­‐6 3) 4x – 25 ˃ -­‐2; x = -­‐6 Graph the inequality. a. y ≤ -­‐3 b. 2 ˂ x c. x ˃ 0 You try: 1) r ˂ ½ 2) x ≥ 3 3) 1.4 ≥ g Modeling with Mathematics a. The graphs show the height restrictions h (in inches) for two rides at an amusement park. Write an inequality that represents the height restrictions for each ride. b. On a fishing trip, you catch two fish. The weight of the first fish is 1.2lb. The second fish weighs at least 0.5lb more than the first fish. Write the inequality that represents the possible weights of the second fish. c. There are 430 people in a wave pool. The maximum capacity is 600. Write an inequality that represents how many more people can enter the pool. d. The winning swim team earned 245 points. The other teams earned at least 72 points less. Write an inequality that represents the points that the other teams earned. Homework: pg. 58 # 2, 5-­‐25 (odd), 27, 29-­‐43 (odd) 2.2-­‐ Solving Inequalities Using Addition and Subtraction Addition Property of Inequality: ________________________ the same number on __________________________________________________ of an inequality. Subtraction Property of Inequality: ______________________________ the same number on _________________________________________________ of an inequality. ***Using these properties creates ________________________________ _________________________________. Examples: a. 𝑥 − 6 ≥ −10 b. 𝑦 + 8 ≤ 5 c. −8 < 1.4 + 𝑚 d. 𝑠 − −1 ≥ 2 e. 15 − 7𝑝 + 8𝑝 > 15 − 2 You try: 1) 𝑘 + 5 ≤ −3 !
!
2) ! ≤ 𝑧 + ! 3) −3 > −3 + ℎ 4) 12 ≤ 4𝑐 − 3𝑐 + 10 5) 6 − 9 + 𝑢 < −2 Modeling with Mathematics 6) A circuit overloads at 1800 watts of electricity. You plug a microwave oven that uses 1100 watts of electricity into a circuit. Write and solve an inequality that represents how many watts you can add to the circuit without overloading the circuit. Which of the following appliances can you plug in to the circuit (in addition to the microwave)? Clock radio 50 watts Blender 300 watts Toaster 800 watts 7) A circuit overloads at 1800 watts of electricity. You plug a microwave oven that uses 1000 watts of electricity into a circuit. Write and solve an inequality that represents how many watts you can add to the circuit without overloading the circuit. Can you plug in a toaster oven that requires 800 watts (in addition to the microwave)? 8) You have $15 to spend on groceries. You have 12.25 worth of groceries already in your cart. Write an inequality that represents how much more money m you can spend on groceries. 9) You and your friend are planning to walk across an old bridge. The bridge can hold at most 1000 pounds. The total weight of people currently on the bridge is 675 pounds. You weigh 156 pounds. Write and solve an inequality that represents how much your friend can weight within the limits of the bridge. If your friend weighs 182 pounds, can you and your friend both walk on the bridge? Homework: pg. 65 # 1-­‐31 (odd) 2.3-­‐ Solving Inequalities Using Multiplication and Division **Prior knowledge: Compare the following two inequalities: 𝑥 < 5 and 5 > 𝑥 What numbers are solutions of the above inequalities? Multiplication and Division Property of Inequality (c ˃ 0): _________________________ or ___________________________ by the same positive number produces an equivalent inequality. Examples: a. 4𝑏 ≥ 36 b. −24 ≥ 3𝑥 !
c. ! > −5 !
d. −6.4 ≥ ! 𝑤 Multiplication and Division Property of Inequality (c ˂ 0): When _________________________ or ___________________________ by the same negative number, the direction of the inequality symbol must be reversed to produce an equivalent inequality. **Common error**: a negative sign in an inequality does not necessarily mean you reverse the inequality symbol! Only reverse the inequality symbol when you multiply or divide each side by a negative number. Examples: a. −7𝑦 ≤ 35 b. −15 ≤ −3𝑐 !
c. !! ≤ −5 !
d. 2 < !! !
e. 1 ≥ − !" 𝑧 You try: Solve the inequality. Graph the solution. 1) −18 > 1.5𝑞 !!
2) −22 ≥ ! ℎ !
3) ! ≥ −1 !
4) !! < 7 5) −9𝑚 ≥ 63 Modeling with Mathematics 6) You earn $9.50 per hour at your summer job. Write and solve an inequality that represents the number of hours you need to work to buy a digital camera that costs $247. 7) The maximum speed limit for a school bus is 55 miles per hour. Write and solve an inequality that represents the number of hours it takes to travel 165 miles in a school bus. 8) You earn $8.50 at your after school job. Write and solve an inequality that represents the number of hours you need to work to earn $187 to buy a tablet computer. 9) You run at a speed of 6.3 miles per hour. Your friend says if you continue to run at this speed, you will not be able to complete a marathon in less than 4 hours. If the marathon is 26.2 miles, is your friend correct? Use formula 𝑑 ≤ 𝑟𝑡. Homework: pg. 71 # 1-­‐29 (odd) 2.4.1-­‐ Solving Multi-­‐Step Inequalities How can you solve a multi-­‐step inequality? Step 1: __________________________________________ each side of the inequality, if necessary. Step 2: Then use _______________________ ____________________________ to isolate the variable. **Be sure to reverse the ______________________________ if multiplying or dividing by a negative number. Examples: Solve each inequality. Graph each solution. a. Solve 2𝑣 − 4 ≥ 8 Graph your solution. !
b. Solve !! + 7 < 9 Graph your solution. c. Solve 5 4 − 𝑦 < 25 Graph your solution. Solving an Inequality with Variables on Both Sides d. Solve 6𝑥 − 5 < 2𝑥 + 11. Graph your solution. e. Solve −5𝑛 − 4 > 7𝑛 + 20. Graph your solution. You try: 1) 3𝑥 − 2 < 10 2) 8 − 9𝑥 ≥ −28 3) 5𝑥 − 12 ≤ 3𝑥 − 4 4) 5 − 2𝑛 > 8 − 4𝑛 Inequalities with Special Solutions Case 1: Solve 8𝑏 − 3 > 4(2𝑏 + 3). Graph the solution. When solving an inequality, if you obtain an equivalent inequality that is false, the inequality has no solution. Case 2: Solve 2 5𝑤 − 1 ≤ 7 + 10𝑤 When solving an inequality, if you obtain an equivalent inequality that is true, the solutions of the inequality are all real numbers. If you obtain an equivalent inequality that is false, the inequality has no solution. You try: Solve the inequality. Graph the solution. 1) 3 2𝑎 − 1 ≥ 10𝑎 − 11 2) 2 𝑘 − 5 < 2𝑘 + 5 3) −4 3𝑛 − 1 > −12𝑛 + 5.2 4) 4𝑘 − 6 < 3𝑘 + 𝑘 − 1 Homework: pg. 77 # 3-­‐16 2.4.2-­‐ Solving Real-­‐Life Problems 1) You need a score of at least 90 points to advance to the next round of the touch screen trivia game. What scores in the fifth game will allow you to advance? 2) The area of the rectangle shown is at most 140 square centimeters. Write and solve an inequality to find the possible values of x. 3) You must maintain a minimum balance of $50 in your checking account. You currently have a balance of $280. Write and solve an inequality that represents how many $20 bills you can withdraw without going below the minimum balance. 4) The graph shows your budget and the total cost x gallons of gasoline and a car wash. You want to determine the possible amounts of gasoline you can buy within your budget. a. What is your budget? b. How much does a gallon of gasoline cost? c. How much does a car wash cost? d. Write an inequality that represents the possible amounts of gasoline you can buy. e. What is the maximum number of gallons you can buy within your budget? Challenge: Suppose your bank charges an ATM fee of 2.50 which is charged each time you withdraw $20. Write and solve an inequality that represents how many $20 bills you can withdraw without going below the minimum balance in this situation. Homework: pg. 77 # 17-­‐28, 31, 33 2.5.1-­‐ Solving Compound Inequalities Compound Inequality The graph of a compound inequality with “and” is the intersection of the graphs of the inequalities. The graph shows numbers that are solutions of both inequalities. Example: 2 ≤ 𝑥 < 5 2 ≤ 𝑥 𝑎𝑛𝑑 𝑥 < 5 What numbers are possible solutions of this compound inequality? The graph of a compound inequality with “or” is the union of the graphs of the inequalities. The graph shows numbers that are solutions of either inequality. Example: 𝑦 ≤ −2 𝑜𝑟 𝑦 > 1 Examples: Write the inequality that is represented by the graph. Writing and Graphing Compound Inequalities *Recall the following chart from Lesson 2.1 Examples: Write each sentence as a compound inequality. Graph each inequality. e. You are purchasing a new refrigerator. To fit the space, the width of the refrigerator cannot be more than 42 inches. To meet your storage requirements, the width of the refrigerator must be at least 36 inches. f. The zoo waives its entry fee for children under age 3 or for senior citizens ages 65 and older. g. A number y is no less than -­‐2.4 and fewer than 4.2 i. A number y is at most 0 or at least 2. You try: Write the sentence as a compound inequality. Graph the inequality. 1) In Florida, the minimum speed limit on a four lane highway is 40 miles per hour. The maximum speed on a four lane highway is 65 miles per hour. 2) The life zones on Mount Rainier can be approximately classified by elevation. The low elevation forest ranges from above 1700 feet to 2500 feet. 3) A number x is fewer than -­‐6 or no less than -­‐3. 4) A number x is greater than -­‐8 and less than or equal to 4 !
5) A number c is more than -­‐4 or at most −6 ! Homework: pg. 85 # 1-­‐11 2.5.2-­‐ Solving Compound Inequalities Solve the following compound inequalities. Graph your solutions. Example: −4 < 𝑥 − 2 < 3 Example: −3 < −2𝑥 + 1 ≤ 9 Example: 3𝑦 − 5 < −8 𝑜𝑟 2𝑦 − 1 > 5 Example: 2x + 6 < −2 or 4x − 5 > 3 You try: Solve the inequality. Graph the solution. 6) 5 ≤ 𝑚 + 4 < 10 7) −3 < 2𝑘 − 5 < 7 8) 4𝑐 + 3 ≤ −5 𝑜𝑟 𝑐 − 8 > −1 9) 2𝑝 + 1 < −7 𝑜𝑟 3 − 2𝑝 ≤ −1 Homework: pg. 85 # 13-­‐19 (odd), 25-­‐31 (odd) 2.6-­‐ Solving Absolute Value Inequalities Absolute Value Inequality-­‐ is an inequality that contains an absolute value expression Examples: Solving Absolute Value Inequalities Step 1: Isolate the absolute variable expression on one side of the inequality, if necessary. Step 2: Rewrite the absolute variable inequality as an equation and solve. Step 3: Graph you solutions and test a point between them to find which way to shade. a. Solve 𝑥 < 2 This means the distance between x and 0 is less than 2. b. Solve 𝑥 + 7 ≤ 2 c. Solve 2𝑤 − 1 ≤ 11 d. Solve 𝑥 > 2 This means the distance between x and 0 is greater than 2. e. Solve 𝑥 − 1 ≥ 5 f. Solve 4 2𝑥 − 5 + 1 > 21 You try: 1) 𝑥 < 3.5 2) 𝑦 − 3 < 4 3) 3 2𝑥 + 5 + 10 ≤ 37 !
4) ! > 2 Absolute Value Inequalities with Special Solutions Case 1: Solve 𝑘 − 3 < −1 Case 2: Solve 𝑓 + 12 > −4 Case 3 6 𝑘 + 3 + 14 > 14 You try: !
5) ! − 7 < −2 6) 𝑛 + 2 − 3 > −6 Solving Real Life Problems 1) At a certain company, the average starting salary s for a new worker is $25,000. The actual salary has an absolute deviation of at most $1800. Write and solve an inequality to find the range of the starting salaries. 2) You are buying a new computer. The table shows the prices of computers in a store advertisement. You are willing to pay the mean price with an absolute deviation of at most $100. How many of the computer prices meet your condition? Homework: pg. 91 # 3-­‐27 (odd), 31