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1 UNIT 5 INEQUALITIES CCM6+/7+ Name: Math Teacher: Topic(s) Page(s) Unit 5 Vocabulary 2 Writing and Graphing Inequalities 3 8 Solving One-Step Inequalities 9 15 Solving Multi-Step Inequalities Inequalities with Variables on Both Sides Notes Inequality Word Problems Unit 5 Study Guide Projected Test Date:

2 Common Core Math 6+/7+ Unit 5 Vocabulary Definitions of Critical Vocabulary and Underlying Concepts rational numbers A number expressible in the form a/b or -a/b for some fraction a/b. The rational numbers include the integers. integers A number expressible in the form a or -a for some whole number a. constant A number that does not change. expression A mathematical phrase that contains operations, numbers, and/or variables. additive inverses Two numbers whose sum is 0 are additive inverses of one another. additive identity property of zero The property that states the sum of zero and any number is that number. addition property of opposites The property that states that the sum of a number and its opposite equals zero. addition property of equality The property that states that if you add the same number to both sides of an equation, the new equation will have the same solution. subtraction property of equality The property that states that if you subtract the same number from both sides of an equation, the new equation will have the same solution. multiplication property of The property that states that if you multiply both sides of an equation by equality the same number, the new equation will have the same solution. division property of equality The property that states that if you divide both sides of an equation by the same nonzero number, the new equation will have the same solution. Inequality A mathematical sentence that shows the relationship between quantities that are not equivalent. algebraic inequality An inequality that contains at least one variable. solution set The set of values that make a statement true. 2 P a g e

3 Writing and Graphing Inequalities Key Vocabulary: Inequality: Algebraic Inequality: Solution Set: Part I: Inequality Key Words and Phrases Symbol Phrases Symbol Phrases < > Using the key words and phrases, write an inequality. 1. There are fewer than 25 students (s) in the class. 2. Above 18 people (p) in the room had brown hair. 3. There are at least 15 dogs (d) in the park. 4. No more than 50 people (p) were on the bus. 5. There are at most 31 days (d) in a month. 6. There is more than \$282 (d) in the bank. 3 P a g e

4 Part II: Graphing Inequalities: An Open Circle: on the number line means the inequality sign is or. A Closed Circle: on the number line means the inequality sign is or. Graph the following inequalities. x < -2 g > 7-3 y m 0 Looking at the graphs, write an inequality. Inequality: Inequality: Inequality: Inequality: Using the inequalities you wrote in Part I, graph each inequality. 1. There are fewer than 25 students (s) in the class. 2. Above 18 people (p) in the room had brown hair. 3. There are at least 15 dogs (d) in the park. 4. No more than 50 people (p) were on the bus. 5. There are at most 31 days (d) in a month. 6. There is more than \$282 (d) in the bank. 4 P a g e

5 INEQUALITIES Understanding and Graphing Inequalities A simple equation has just one answer. For the equation x = 3, the only number that makes the equation true is the number 3. The equation x = 3 can be shown on the number line by using a dot. An inequality is a mathematical sentence that shows the relationship between quantities that are not equal. An inequality may contain a variable, as in the inequality x > 3. Values of the variable that make the inequality true are solutions of the inequality. x x > 3 Solution? 0 0 > 3? No; 0 is not greater than 3, so 0 is not a solution. 3 3 > 3? No; 3 is not greater than 3, so 3 is not a solution. 4 4 > 3? Yes; 4 is greater than 3, so 4 is a solution > 3? Yes; 12 is greater than 3, so 12 is a solution This table shows that an inequality may have more than one solution. You can use a number line to show all of the solutions. There are an infinite number of points that can be drawn on the number line that are greater than 3. Numbers such as 4, 5, 6, and 7 are all greater than three as are decimals like 4.78, 5.2, 8.4, and It would take an eternity to draw each point individually, so the accepted method of showing all the answers is starting from the number 3 and shading over all the numbers greater than 3. The graph above has an open circle on the 3 and is shaded on all the numbers to the right of the 3. The numbers that have been shaded are solutions to x > 3. The number line appears to end at 8, so the arrow to the right can be shaded to show that the solutions continue indefinitely for numbers larger than 8. 5 P a g e

6 Symbols and Their Meanings < means less than and is represented with an open circle on the number line. > means greater than and is represented with an open circle on the number line. means less than or equal to and is represented with a closed circle on the number line. means greater than or equal to and is represented with a closed circle on the number line. **** An open circle on the critical point shows that the critical point is not a solution. **** **** A closed circle on the critical point shows that the critical point is a solution. **** The process of drawing the graph of an inequality can be broken down into three steps. The three steps for the inequality b 5 are shown below. 6 P a g e

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9 Solving One-Step Inequalities Notes Review: Solving One-Step EQUATIONS Ex. 1) c (-3) = 4 Ex. 2) n + 14 = -5 Ex. 3) -8m = -32 Ex. 4) r 3 = 7 Use addition or subtraction to solve each inequality. Graph the solutions. Ex 1. n ( 8) 19 Ex 2. t ( 6) 44 Ex y 14 Ex k 9 P a g e

10 Use multiplication or division to solve each inequality. Graph the solutions. KEY RULE: Ex 1. 8x 112 Ex m Ex 3. 9t 36 Ex 4. m 12 4 Ex 5. r 13 Ex 6. t Identify the accurate simplified inequality. -45 n 12 A. n -33 B. n -33 C. n -57 D. n -57-3n > 18 A. n > 6 B. n < 6 C. n > -6 D. n < -6 Match the inequality to the graph: 1. x (-3) A. B x -3 C. D. 3. x x P a g e

11 Skills Check: Solving One-Step Inequalities Solve each inequality, show the steps of your work k < x y < k (-15) Skills Check: Writing and Graphing Inequalities Write an inequality for each situation described. 1. A city had more than 2 feet of snow in one snowfall. (Let f = feet of snow) 2. There are less than 35 students in a class. (Let x = students) 3. A student s grade on a test is at least 85. (Let g = grade) 4. The number of minutes for a cell phone plan is at most 500. (Let m = minutes) Identify the graph that matches the simplified inequality. 5. 3a k 22 A. A. B. B. C. C. D. D. 11 P a g e

12 Solving and Graphing Inequalities The rules of algebra apply the same way to inequalities as they to do equations. Basically, the inequality 3x 2 < 10 can be solved in the same way as the equation 3x 2 = 10. Example: Simplify the inequality 8x The graph of the example shows that any number that is three or less is a solution to the original inequality. Any number greater than three does not work. To check your answer, try a few points and see if they work. The points x = 2 and x = -1 are solutions to the inequality as expected from the graph. Likewise, doing the math for the point x = 6 demonstrates that it is not a solution to the inequality. 12 P a g e

13 Solving Inequality Rules 13 P a g e

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16 Solve each two-step inequality. Notes: Solving Multi-Step Inequalities 1.) -9x 8 > 28 2.) 12 3x < 9 3.) r ) -4 4(-x + 2) 5.) w c ) Identify the accurate simplified inequality n n 7 > -19 A. n -17 B. n -17 C. n 17 D. n 17 A. n > 3 B. n < 3 C. n > -3 D. n < -3 Solve each inequality. Match it to the appropriate graph in the final column. 16 P a g e

17 r 3 > x 23 A. B < x n - 3 > 9 C. D. Notes: Solving Two-Step Inequalities including Combining Like Terms and Distributive Property Solve each inequality. 1.) -9x 8 > 28 2.) 3(4 x) < 9 3.) 3 + r ) -4 4(-2x + 3) 5.) w ) -4c c Identify the accurate simplified inequality. 17 P a g e

18 n n 7 > -19 A. n -17 B. n -17 C. n 17 D. n 17 A. n > 3 B. n < 3 C. n > -3 D. n < -3 Solve each inequality. Match it to the appropriate graph in the final column r 3 > x 23 A. B < x n n > 9 C. D. 18 P a g e

19 Skills Check: Multi-Step Inequalities Write an inequality for each situation described. 1. A city had at least 10 inches of snow in one snowfall. (Let i = inches of snow) 2. There are at most 240 students in the cafeteria. (Let x = students) 3. A student s grade on a test is greater than an 85. (Let g = grade) 4. The number of minutes for taking a test is fewer than 60. (Let m = minutes) Identify the graph that matches each simplified inequality. 5. Letter Answer: 6. Letter Answer: m n n > Letter Answer: 40 < 5(-7 + 3x) 8. Letter Answer: m -1 > 4-3 A. B. C. D. Identify the simplified inequality. 9. Letter Answer: -45-2n n A. n -29 B. n -29 C. n 29 D. n P a g e

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21 Extension: Inequalities with Variables on Both Sides Notes Solve and graph the inequalities that contain variables on both sides. Example 1: Example 2: 15b + 6 > 9b 36-12k + 28 < 18k + 88 Example 3: c 6 3c Example 4: 2 (n 5) 30-8n 21 P a g e

22 Notes: One and Two-Step Inequality Word Problems with Integers 22 P a g e

23 Define a variable (V), write an inequality (I), and solve (S) & graph the inequality that represents the solution set. 1. A store makes a profit of \$25 on each watch it sells. What solution set will represent how many watches the store must sell to make a profit of at least \$375? 2. The sum of three integers is at most -57. If one integer is 33 and another is -23, what is the solution set for the third integer? 3. The difference of four times a number and -8 is greater than What is the solution set for the number? 4. 3 times the difference of a number and 8 is no more than -93. What is the solution set for the number? 5. Working at Burger in a Box, you are paid \$25 each week plus \$6 per hour. This week, you want your pay to be at least \$133. What solution set represents the amount of hours that you must work to earn enough? 6. The 12 members of the Filmmaking Club need to raise at least \$1400 to make a short film. They already have raised \$650. What solution set represents the amount that each member must raise to reach their team goal? 23 P a g e

24 Practice Inequalities Word Problems Define a variable (V), write an inequality (I), solve (S) the inequality, and graph the solution set. 1. The perimeter of the Jacob s square backyard is no more than 72 meters. What is the solution set for the length that his backyard can be? 2. To get an A, you need more than 200 points on a two-part test. You score 109 points on the first part. What solution set represents how many more points you need? 3. Marcus wants to buy 4 baseballs. He has \$68. What solution set represents what each baseball can cost? 4. 3 times a number, plus 5, is at most -28. What solution set represents what the number can be? times a number, minus 6, is at least 22. What solution set represents what the number can be? 6. You buy want to buy several candy bars for \$2 each and one newspaper for \$3. What solution set would represent the number of candy bars can you buy if you only have \$19? 7. A telephone company charges a \$27 monthly service, plus \$2 for each long-distance call that you make. If you budget \$65 for your telephone bill each month, what solution set represents how many long distance phone calls you can make during the month without going over your budget? 24 P a g e

25 Extension: Inequality Word Problems with Variables on Both Sides Define a variable (V), write an inequality (I), and solve (S) & graph the inequality to represent the solution set. 14 times a number, plus 5 is less than 11 times the same number, minus 19. What is the solution set for the number? V: 5 times a number is greater than 8 times the same number plus 120. What is the solution set for the number? V: S: S: Create your own!!! V: S: 25 P a g e

26 1. Admission to the state fair costs \$7. In addition, each ride costs \$2. If Ahmed wants to spend no more than \$63 at the fair, what is the solution set for the number of rides he can take? 1. Admission to the state fair costs \$7. In addition, each ride costs \$2. If Ahmed wants to spend no more than \$63 at the fair, what is the solution set for the number of rides he can take? 2. Gabrielle went to the movie theatre with her friends. She had \$25.00 to spend. The movie ticket cost \$6. What solution set represents how much money she had to spend on snacks? 2. Gabrielle went to the movie theatre with her friends. She had \$25.00 to spend. The movie ticket cost \$6. What solution set represents how much money she had to spend on snacks? 3. Pet Mart purchases dog food from PureFresh for \$15 per bag. They sell the bags to their customers for \$22 per bag. If the store wants to make a profit of no less than \$4,410 on this dog food, what solution set represents how many bags of dog food that it needs to sell? 4. Negative eight plus four times a number is greater than negative 64. What is the solution set for this number? 5. Sondra has \$207 to spend on new school clothes. First, she purchased a pair of shoes for \$45. The store was charging \$18 for all shirts, pants, and other items of clothing. What solution represents the number of items of clothing that she can purchase? less than the quotient of a number and -6 is greater than or equal to 17. What is the solution set for this number? 7. *Challenge* Negative four times the sum of a number and negative 12 is at least 212. What is the solution set for this number? 3. Pet Mart purchases dog food from PureFresh for \$15 per bag. They sell the bags to their customers for \$22 per bag. If the store wants to make a profit of no less than \$4,410 on this dog food, what solution set represents how many bags of dog food that it needs to sell? 4. Negative eight plus four times a number is greater than negative 64. What is the solution set for this number? 5. Sondra has \$207 to spend on new school clothes. First, she purchased a pair of shoes for \$45. The store was charging \$18 for all shirts, pants, and other items of clothing. What solution represents the number of items of clothing that she can purchase? less than the quotient of a number and -6 is greater than or equal to 17. What is the solution set for this number? 7. *Challenge* Negative four times the sum of a number and negative 12 is at least 212. What is the solution set for this number? 8. *Challenge* Two times the difference of a number and four is no more than What is the solution set for this number? 8. *Challenge* Two times the difference of a number and four is no more than What is the solution set for this number? 26 P a g e

27 Pass the Card: Inequalities Define a variable (V), write an inequality (I), and solve (S) & graph the inequality to represent the solution set P a g e

28 Study Guide: Inequalities Write an inequality for each situation described below. 1. Today s attendance (a) will be at least 250 people. 2. Tomorrow s attendance (a) will be less than 200 people. 3. Last weekend, there were more than 75 birds (b) in the sanctuary. 4. Next weekend, there will be at most 90 birds (b) in the sanctuary. 5. Each prize (p) is worth over \$ You can walk there in 20 minutes (m) or less. Solve and graph each of the following inequalities. 7. k (-178) > -231 k 8. < g + (-49) x 8 < m g + (-12) (7g + 10) a 19 > (k 5) P a g e

29 Define a variable (V), write an inequality (I), and solve (S) & graph the inequality to represent the solution set. 16. The school record for the most points scored in a football season is 85. Lawrence has 44 points so far this season. What is the solution set for how many more points he needs to break the record? 17. A ride at an amusement park requires a height of at least 48 inches. Your little brother is 37 inches tall. What is the solution set for how many more inches must he grow in order to go on the ride? 18. Each month, Marnie pays a service fee of \$12 plus \$0.06 per kilowatt-hour for electricity. This month, she has budgeted \$84 for her electricity. What is the solution set for the number of kilowatt-hours Marnie can use and stay within budget? 19. Your mom gave you \$25 to go to the movies. You spend \$8 on your ticket and \$5 on a small bag of popcorn. You want to spend the rest of your money on boxes of candy to share with your friends. If each box of candy costs \$3, what solution set represents the number of boxes of candy you can buy? 20. Five times the difference of a number and 8 is at most 105. What solution set represents what the number can be? 29 P a g e

30 **You need to be careful to when you or by a negative number. 21) A number is no less than ) Graph the solution to -8 < x. 23) What inequality is graphed? 24) 3x ) 7(x 8) 3x + 4 > 10x 40 Short Answer 26) When should you put an open circle on a graph? 27) When should you put a solid/closed circle on a graph? 28) Make up and solve a problem where you have to reverse the inequality to solve the problem. Graph your solution. 9) Two times the difference of a number and four is no more than What is the solution set for this number? I: 30 P a g e

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