# Define the word inequality

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1 Warm Up: Define the word inequality Agenda: Objective- Students can solve linear inequalities in one variable, including equations with coefficients represented by letters. Define Inequalities One & Two Step Inequalities Reminders: Solving Inequalities 1

2 Students can define inequality. Inequality: is like an equation that uses symbols for "less than" (<), "less than or equal to" ( ), "greater than" (>), and "greater than or equal to" ( ) where an equation uses a symbol for "is equal to" (=). Students can explain inequalities. Explain each situation shown on the balance scales using inequalities vocabulary. 2

3 Students can identify symbolic representation of inequalities. Inequality Symbols Less Than < Greater Than > Less Than or Equal Too < Greater Than or Equal Too > Students can identify symbolic representation of inequalities. Inequality Symbols Less Than < Greater Than > Less Than or Equal Too < Greater Than or Equal Too >

4 Students can identify symbolic representation of inequalities. Set Notation: 3 < x < 10 Number line: 3 10 Interval Notation: (3, 10) Students can identify symbolic representation of inequalities. For each of the following in set notation, a. Create a Number Line b. Write in each of the following in Interval Notation 1. x > 2 2. x < x x < 9 4

5 Students can solve one-step linear inequalities with one variable. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. a) x 4 > 9 b) x + 6 < _ 4 Students can solve one-step linear inequalities with one variable. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. x c) x < 4 d) 9x > 36 3 _ 5

6 Students can solve two-step linear inequalities with one variable. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. a) 5x + 3 < _ 4 b) 10x 3 > 7 Students can solve two-step linear inequalities with one variable. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. c) 5x + 3 > 4 d) x < 1 x 5 2 6

7 Students can solve linear inequalities with a negative coefficient. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. _ a) 6x < 12 b) x < 8 When you mulply or divide by a negave number, you MUST switch the inequality sign. Students can solve linear inequalities with a negative coefficient. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. c) 2x + 6 > 8 d) 17 3x > 2 _ When you mulply or divide by a negave number, you MUST switch the inequality sign. 7

8 YOUR TURN!! Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. x a) 4x + 7 > 5 b) 5x 3 > 0 c) x + 9 > 6 x d) x 2 < 5 e) 4x + 1 < 13 f) 4x > 5 x g) 3x < 12 h) 6x + 1 < 0 i) 5x 2 < 17 When you mulply or divide by a negave number, you MUST switch the inequality sign. Warm Up What was done to both sides of the first inequality to give the second in each of the following below? 1) x 7 < 2 to give x < 9 2) 3x > _ 15 to give x > _ 5 3) x + 10 > 0 to give x > 10 4) 6x < 12 to give x > 2 5) x _ < 8 to give x > _ 8 6) x > 1 to give x < 4 4 8

9 Agenda: Do Now Homework Questions Objective Solving inequalities with the Distributive Property Reminders: Algebra Common Core Students can use the distributive property when solving an inequality. Students can explain each step in solving an inequality as following the equality of numbers asserted in the previous step, starting from the assumption that the original inequality has a solution. Students can write solutions to inequalities in interval notation. 9

10 RECALL: ONE STEP TWO STEP x + 4 > 3 5x 13 _ < 2 Students can solve inequalities with the Distributive Property. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. a) 5(x - 2) < 30 b) 3(6 + x) > -36 c) 2(x + 4) < 9-10

11 Students can solve inequalities with the Distributive Property. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. d) 2(4x - 9) - 6x > 0 e) -(x + 7) > 15 f) 5(x - 2) - 4(x - 1) < 0 - Students can solve inequalities with the Distributive Property. YOUR TURN: Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Write a list of these values. Write it in interval notation. a)3(5x + 4) < 42 b) 5x - 2(2x + 1) > 8 c) 4(r + 5 ) -> 8 d) -6(a+2) +7a < 12 e) -(2a - 6) + 3(a + 2) < 4-11

12 Warm Up Amanda has \$40 to spend on flowers. She wants to buy a pair of red roses for \$18 and the rest on lily flowers. Each lily costs \$11. How many lily flowers can she purchase? 11x + 18 _ < 40 Agenda: Do Now Homework questions Objective Solving Inequalities with Variables on Both Sides Reminders: 12

13 Algebra Common Core Students can explain each step in solving an inequality as following the equality of numbers asserted in the previous step, starting from the assumption that the original inequality has a solution. Students can write solutions to inequalities in interval notation. RECALL: ONE STEP TWO STEP DISTRIBUTIVE x > 8 2x + 12 < 4 2x 2(x 1) < 5 _ 4 13

14 Students can solve inequalities with variables on both sides. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a # line representing these values. Write it in interval notation. a) -2m - 27 _ > -5m - 6 b) 9j + 3 < 8j + 4 c) 5p + 10 < _ p Students can solve inequalities with variables on both sides. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a # line representing these values. Write it in interval notation. d) 3m - 2 < _ 5m + 10 e) 5a - 4 < 6a + 4 f) 2 x + 5 _ > 1 x

15 Students can explain each step when solving an inequality. On each line provided, identify the property being used. -6y + 20 < 3y y < 3y y - 3y -9y < y > 3 Subtraction Property of Equality Subtraction Property of Equality Division Property of Equality Students can solve inequalities with variables on both sides. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Write a list of these values. Write it in interval notation. a) 6-6g _ > 8-5g b) 3t + 4 > 4t + 7 c) 6d - 16 _ < -32-2d d) 5w + 8 > 3w + 14 e) 5k + 7 > 8k - 8 f) 2-6 > h 2 h 15

16 Students can explain each step when solving an inequality. On each line provided, identify the property being used. 7u + 8 < u u < 6 + 9u - 9u - 9u -2u < u > -3 Subtraction Property of Equality Subtraction Property of Equality Division Property of Equality Warm Up What was done to both sides of the both sides of the first inequality to give the second of the following? a) x 7 < 2 to give x < 9 b) x < 9 to give x < 18 2 c) 5x 2 < 17 to give 5x < 15 d) 2(x 6) > 3 x to give 2x 12 > 3 x 16

17 Agenda: Do Now Homework Questions Objective Solving Multi Step Inequalities Reminders: Algebra Common Core Students can explain each step in solving an inequality as following the equality of numbers asserted in the previous step, starting from the assumption that the original inequality has a solution. Students can write solutions to inequalities in interval notation. 17

18 RECALL: ONE STEP TWO STEP 7x < _ 28 x _ < 0 DISTRIBUTIVE VARIABLES ON BOTH SIDES 2(x 5) > 30 2x + 15 > 3 + 4x Students can solve inequalities with Multi Steps. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. a) 3(x + 3) < 2(x + 6) b) x + 5 > 3x - 2(x - 1) 18

19 Students can solve inequalities with Multi Steps. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. c) 4a - 3-2(4 - a) < _ 7a - 10 d) 1 2 (4x + 8) < _ 6x + 44 Students can solve inequalities with Multi Steps. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. e) (x - 4)(x + 1) > x(x - 4) 19

20 YOUR TURN Students can solve inequalities with Multi Steps. Solve the following inequalities for x. Explain which values of x make this algebraic sentence true. Draw a number line representing these values. Write it in interval notation. _ a) 2(5p + 10) < 2( p) b) 5 + 7r > -(3-2r) - 3r _ c) 9z - (6 - z) > 12z -10 d) (x + 2)(x - 3) < x(x - 2) e) 1 (3x - 27) > 2(x - 5) 3 Review for Quiz Warm Up 20

21 Warm Up What happens to the direction of an inequality if... a) the same number is subtracted from both sides? b) both sides are multiplied by the same positive number? c) both sides are divided by the same negative number? Agenda: Do Now Objectives Translating an English sentence in an symbolic inequality Reminders: 21

22 Algebra Common Core Obectives: Students can solve word problems that lead to inequalities. Students can explain each step in solving an inequality as following the equality of numbers asserted in the previous step, starting from the assumption that the original inequality has a solution. Students can write solutions to inequalities in interval notation. Students can solve word problems that lead to inequalities. Word Problems Solved by Inequalities Frequently, word problems involve translating an English sentence into an inequality. For example, Tony can spend at most \$25 on a football. This means that he must spend \$25 or less. Therefore, at most is translated as _ <. Janet needs at least a 90 average for all her subjects to quality for the honor roll. This means that she must have an average of 90 or greater. Therefore, at least is translated as _ >. 22

23 Students can solve word problems that lead to inequalities. Example One Jamal is paid \$150 a week plus a commission of \$40 on each stereo he sells. How many must he sell to make at least \$350 a week? STRATEGY Let x = the number he can sell. Think: salary plus commission is at least 350 salary plus number sold times 40 is at least 350 salary plus x 40 is at least x > 350 Solve the translated inequality... Students can solve word problems that lead to inequalities. Example Two Mona rents a care \$ 175 a week, plus \$0.15 a mile. How far can she travel if she wants to spend at most \$250? STRATEGY Let x = the number of miles she can travel. fixed cost + cost per mile number of miles is at most

24 Students can solve word problems that lead to inequalities. Your Turn An amusement park charges \$6 for admission and \$0.70 for each ride. If you go to the park with \$20, what is the maximum number of rides you can go on? 2. You want to start a business making custom t shirts. It will cost you \$48 for supplies. It costs you \$4.50 to make each t shirt which you will sell for \$8.50. Write an inequality showing how many you must sell to earn a profit of at least \$600. Solve. Explain. 3. You and you friends have a total of \$12.50 to spend on pizza. A large pizza with cheese costs \$10, plus \$0.50 for each additional topping. What's the most toppings you can afford? 4. Brian is paid \$150 a week, plus a commission of \$15 on each camera he sells. How many must he sell to make at least \$600 a week? Warm Up Translate the following algebraic inequalities into sentences. a) 10 times a number is at least 60 b) 39 is less than the difference of 21 and a number. 24

25 Agenda: Do Now Objectives Master of Translating: Translating an English sentences and phrases into algebraic expression, equations, and inequality Reminders: Students can solve word problems that lead to inequalities. Students can explain each step in solving an inequality as following the equality of numbers asserted in the previous step, starting from the assumption that the original inequality has a solution. Students can write solutions to inequalities in interval notation. Students can complete Master of Translating. 25

26 Warm Up Graph the function y = 2x + 3 How many regions does the line separate the coordinate plane into? Agenda: Do Now Objectives Graphing Linear Inequalities Reminders: 26

27 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Graphing the iequality y < 2x + 3 Graphing the inequality y > 2x + 3 Graphing the inequality y < _ 2x + 3 _ Graphing the inequality y > 2x

28 SUMMARY Inequality Symbol Line Shade > Dotted Up < Dotted Down Solid Up Solid Down Graph the inequality: y x 5 Slope? y intercept? Line? Shade? SOLUTIONS: 28

29 Graph the inequality: y < 2x + 1 Slope? y intercept? Line? Shade? SOLUTIONS: Graph the inequality: y < 3x Slope? y intercept? Line? Shade? SOLUTIONS: 29

30 Graph the inequality: y 2x +2 Slope? y intercept? Line? Shade? SOLUTIONS: YOUR TURN 30

31 Warm Up Solve the following inequality for y: 2x 3y < 9 Could the following inequalities be graphed on a coordinate plane? x 6 y > 4 Agenda: Do Now Objectives Graphing Linear Inequalities Practice: Finish Worksheet from yesterday. Reminders: 31

32 Agenda: Do Now Homework Questions Writing linear equations Reminders: 32

33 33

34 Warm Up for THE FOLLOWING linear inequalities, IDENTIFY WHETHER YOU'D HAVE A SOLID OR DOTTED LINE AND IF YOU'D SHADE UP OR DOWN. 1. Y -2X Y < X Y 2 4. Y < 3 5. Y -X - 3 Solving Systems of Linear Inequalities Agenda: Do Now Objectives Graphing Systems of Linear Inequalities Reminders: 34

35 Student's can graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. 35

36 36

37 Warm Up The following linear inequalities can't be graphed in their current state. Make them so you can graph them. 1) 4x - 3y 2) 2x + y 2 Nov 25 10:43 AM Student's can graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Nov 25 10:43 AM 37

38 Nov 25 10:43 AM Nov 25 10:43 AM 38

39 Nov 25 10:43 AM Review for Quiz Warm Up 39

40 Agenda: Objective: > Students can determine where a function is increasing and decreasing. > Students can determine the range given a domain of a piecewise function. Reminders: Nov 21 9:43 AM Increasing/ Decreasing Functions A function is increasing when the slope is positive. A function is decreasing when the slope is negative. A function is constant when the slope is 0. Nov 21 9:43 AM 40

41 Increasing/ Decreasing Functions We will use INTERVAL NOTATION or SET NOTATION when we write the intervals of increase and decrease. Intervals of increase or decreae are always CLOSED!!! Where is f(x) increasing? Where is f(x) decreasing? Where is f(x) constant? Nov 21 9:43 AM Increasing/ Decreasing Functions Where is f(x) increasing? Where is f(x) decreasing? Where is f(x) constant? f(x) Nov 21 9:43 AM 41

42 Increasing/ Decreasing Functions Where is f(x) increasing? Where is f(x) decreasing? Where is f(x) constant? f(x) Nov 21 9:43 AM Increasing/ Decreasing Functions Let's take a look at a piecewise function. Remember, a piecewise function is simply a function made up of 2 or more "pieces." These pieces can be linear, quadratic, exponential, etc. You Try: 1. Evaluate h(3). 2. Evaluate h(0) 3. Evaluate h(1). Nov 21 9:43 AM 42

43 Increasing/ Decreasing Functions When you are asked to evaluate a piecewise function at a specific x value, you must use the closed value for the y value. Example: Evaluate: f(x) a. f( 4) b. f(2) c. f( 2) Nov 21 9:43 AM Increasing/ Decreasing Functions Example: Evaluate: a. y(0) b. y(6) c. y(2) Nov 21 9:43 AM 43

44 Increasing/ Decreasing Functions Exit Ticket: Determine where f(x) is increasing and decreasing. Evaluate f(4). Nov 21 9:43 AM Warm UP Given f(x) = 3x 1 and g(x) = 2 + x 2, evaluate the following: 1. f(2) 2. g(2) 3. g( 1) 4. f( 5) 5. g( 3) Nov 21 2:33 PM 44

45 Agenda: Objective: > Students can determine where a function is increasing and decreasing. > Students can determine the range given a domain of a piecewise function. Reminders: Nov 21 2:33 PM Increasing/ Decreasing Functions Class Work: In pairs or independently, please complete the following problems. Please hand this in before class ends. If you finish, please let me know and I have Quick Checks for you to work on for extra credit. Nov 21 9:43 AM 45

46 Increasing/ Decreasing Functions 1. Determine the intervals of increase and decrease for each function given. 2. Evaluate: a. f(0) b. f(1) Pay attention to the scale on the graph! Nov 21 9:43 AM Increasing/ Decreasing Functions 3. Determine the intervals of increase and decrease for each function given. 4. Evaluate: a. f( 1) b. f(1) Pay attention to the scale on the graph! Nov 21 9:43 AM 46

47 Increasing/ Decreasing Functions 5. Determine the intervals of increase and decrease for each function given. 6. Evaluate: a. f(0) b. f(1) Pay attention to the scale on the graph! Nov 21 9:43 AM Increasing/ Decreasing Functions BONUS (+5 on your next quiz!) *Choose an appropriate scale! Nov 21 9:43 AM 47

48 Do NOW: Review Agenda: Reminders: 48

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