1 Topic 15: Solving linear inequalities 65 SOLVING LINEAR INEQUALITIES Lesson 15.1 Inequalities on the number line 15.1 OPENER Consider the inequality x > List five numbers that make the inequality true. 2. Plot your five numbers on the number line. 3. Does an x value of 7.25 make the inequality true? 4. Does an x value of make the inequality true? 5. Write a sentence that describes all of the numbers that make this inequality true CORE ACTIVITY 1. Graph the solutions for the following mathematical statements: a. x = b. x < - 4 c. x > 0 d. x > - 7 e. x < f. x 5
2 66 Unit 5 Linear equations and inequalities 2. Consider this compound inequality: x < 1 or x > 4 a. Complete the table by determining whether the values for x make the statement true. Value for x Makes the statement true Yes/No π 7 12 b. List five additional numbers that make this compound inequality true. 3. a. Construct a graph on the number line to represent the inequality x < 1. b. Construct a graph on the number line to represent the inequality x > 4. c. Construct a graph on the number line to show ALL the numbers that make the compound inequality x < 1 or x > 4 true. Use the graphs you sketched in 3a and 3b to help you. 4. On a number line, show all of the numbers, x, such that x > - 2 or x < 7.
3 Topic 15: Solving linear inequalities Consider the following compound inequality: x > 2 and x < 7 a. Complete the table by determining whether the values for x make the statement true. Value for x Makes the statement true Yes/No π 7 12 b. List five additional numbers that make this statement true. 6. a. Construct a graph on the number line to represent the inequality x > 2. b. Construct a graph on the number line to represent the inequality x < 7. c. Construct a graph on the number line to show ALL the numbers that make the statement in question 5 true. To help you construct the final graph, consider your work from questions 5a and 5b. 7. On a number line, show all of the numbers, x, such that x < - 3 and x 2.
4 68 Unit 5 Linear equations and inequalities 8. Construct a graph on each number line to show the solution set for each compound inequality. Then, describe the solution set for each compound inequality using one of these four descriptions: (1) some but not all real numbers, (2) an empty set, (3) all real numbers, or (4) exactly one number. Graph a. x < 4 and x > 6 Solution set b. x < 6 or x > 4 c. x < 6 and x > 4 d. x < 4 or x > 6 e. x 5 or x 5 f. x 5 and x CONSOLIDATION ACTIVITY 1. For each card numbered 1 through 10, work with your partner to find a match using the lettered cards. Then, write the letter of the matching card in the table. Inequality card number Number line card letter 2. Write an equation or inequality that would produce the same number line graph as x 4 and x 4 3. Write an equation or inequality that would produce the same number line graph as x < 5 or x Write an equation or inequality that would produce the same number line graph as x < 4 or x > 4.
5 Topic 15: Solving linear inequalities 69 HOMEWORK 15.1 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Match the inequality to its representation on a number line. a. x < i. b. x ii. c. x > iii. d. x = iv. e. x < v. f. x > vi. 2. Write an inequality statement whose solution is an empty set. 3. Write an inequality statement whose solution is some but not all real numbers. Graph the solution on a number line. 4. Write an inequality statement whose solution is all real numbers. Graph the solution on a number line. 5. Write an inequality statement whose solution is exactly one number. Graph the solution on a number line. 6. Write an inequality statement to represent the graph shown. 7. Write an inequality statement to represent the graph shown.
6 70 Unit 5 Linear equations and inequalities STAYING SHARP 15.1 Practicing algebra skills & concepts 1. Select the appropriate symbol (<, >, or =) to describe the relationship between each pair of numbers. a b c List the lines according to the value of their slopes. List them in order from the smallest slope value to the largest slope value. d Which pair of x- and y- coordinates appears in the tables for both functions? Answer: 4. What are the coordinates of the point of intersection of the two lines in the graph? Preparing for upcoming lessons Answer: y = 2x 7 y = 3x Answer: Focus skill: Slope and geometric connections 5. What is the slope of the line? A slope triangle is drawn for you. Answer: 6. Write an equation for the line in Question 5. Answer with supporting work:
7 Topic 15: Solving linear inequalities 71 Lesson 15.2 Introduction to solving linear inequalities 15.2 OPENER A car rental company charges $29.95 plus 16 cents per mile for each mile driven. 1. Write a function rule to describe the relationship between the cost of the rental, r, and the number of miles you drove, m. 2. What input value would result in an output value of $75? 15.2 CORE ACTIVITY 1. Write an inequality to represent the situation below. A car rental company charges $29.95 plus 16 cents per mile for each mile driven. Your boss is very careful with the company's money. She wants you to plan your business trips so you will not spend more than $75 for car rental fees. 2. Solve the inequality you wrote in question 1 to determine how many miles you could drive and spend $75 or less for the trip. Your teacher will assign you and your partner a particular method for solving the inequality. Create a poster presenting your solution. 3. List the four methods you will be using in this topic to solve linear inequalities. 4. Complete a journal entry. Vocabulary term My understanding of what the term means An example that shows the meaning of the term Linear inequality
8 72 Unit 5 Linear equations and inequalities 15.2 CONSOLIDATION ACTIVITY Your boss is still trying to save money on car rentals. You compare Omega Car Rental, which charges $72.00 per day with unlimited free mileage, and Optimal Car Rental, which charges $22.90 plus 30 cents per mile. 1. Some days you drive many miles, and some days you drive fewer miles. Determine the range of daily mileage for which each company is the less expensive choice. 3. Write a report to your boss that includes your recommendations about which car- rental company to use depending on the number of miles you plan to drive. Explain how you arrived at your recommendation. Use computations, graphs, or tables to provide support for your conclusions.
9 Topic 15: Solving linear inequalities 73 HOMEWORK 15.2 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Solve and graph the solutions to the inequalities below using any of the methods from class: a table, a graph, undoing, or algebraic operations. (You may need to use a separate sheet of notebook paper or graph paper.) a. x + 3 > 2 b. 15x 45 c. 3 < 4x - 5 d. 2x e. 5 > x 1 f. 1.5x < The students on the dance committee at Jefferson High School are planning a dance. They hope to make a profit of at least $150 to donate to a local animal shelter. The dance committee has decided to sell tickets for $5. They also know that the cost for the DJ will be $200. a. The inequality 150 5x 200 can be used to determine if the dance committee met its goal. Explain how each part of this inequality fits the problem situation. b. Solve the inequality using a table, a graph, undoing, or algebraic operations. (You may need to use a separate sheet of graph paper.) c. d. In a complete sentence, explain what your solution to part b means in the context of the problem situation.
10 74 Unit 5 Linear equations and inequalities STAYING SHARP 15.2 Practicing algebra skills & concepts 1. Select the appropriate symbol (<, >, or =) to describe the relationship between each pair of numbers. a b c d On the coordinate plane, sketch a line that has an x- intercept of ( 2,0) and a y- intercept of (0,6). Then, calculate the slope of the line. e Slope of line: Preparing for upcoming lessons Complete these tables to answer questions 3 and 4. Table A Table B y = x + 4 y = 0.5x a. As x increases by 1, by how much does y change in Table A? b. As x increases by 1, by how much does y change in Table B? 4. Will there be a common (x,y) pair in the two tables if the tables are continued? Explain. Focus skill: Slope and geometric connections Use the two slope triangles to answer Questions 5 and a. Calculate the slope of the line using the smaller slope triangle. b. Calculate the slope of the line using the larger slope triangle. 6. Explain why the two slopes you calculated are equal.
11 Topic 15: Solving linear inequalities 75 Lesson 15.3 Solving inequalities using graphs and tables 15.3 OPENER Use your graphing calculator to solve the equation!3x!1 = 1 x + 6 using a table and graph. Sketch both the graph display 2 and the table display from your calculator screen CORE ACTIVITY Use the graph you created in the Opener to answer these questions: 1. For which values of x is the inequality!3x!1 < 1 2 x + 6 true? 2. For which values of x is the inequality!3x!1 > 1 2 x + 6 true? 3. How did solving the equation in the Opener help you answer questions 1 and 2? 4. Complete a journal entry explaining how to use each of the given methods to solve an inequality in one variable. Each explanation should include an example. Using table technology to solve inequalities Using graphing technology to solve inequalities 15.3 CONSOLIDATION ACTIVITY
12 76 Unit 5 Linear equations and inequalities 1. Solve the following inequalities using the table and graphing features on your graphing calculator. Sketch the calculator graph you used to solve the inequality. Also, copy the calculator table that you used. State the solution to the inequality. Inequality Graph Table Solution a. 5 2x 9 b. 3x 6 > 8 c. 2x + 1 x 2
13 Topic 15: Solving linear inequalities 77 Inequality Graph Table Solution d. 5x + 6 < 8x In question 1, you used the graphing capability of the graphing calculator and the table feature of the graphing calculator to solve several inequalities. What are the advantages and disadvantages of each method? 3. Did one method seem easier for a particular inequality than the other method? Explain.
14 78 Unit 5 Linear equations and inequalities HOMEWORK 15.3 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Use tables and a graph to find the x- values that make the inequality 2x + 3 > x 1 true. a. Tables: x y = 2x + 3 x y = x Solution: b. Graph: Solution: c. Which method, using a table or a graph, did you find more helpful in solving this inequality? Explain. 2. Nikki and Amber were solving the inequality x + 5 < - 3x 1. They created the tables shown. Nikki said the solution was x - 1. Amber said the solution was x < - 2. Is either of them correct? Explain. x y = x x y = - 3x
15 Topic 15: Solving linear inequalities 79 STAYING SHARP 15.3 Practicing algebra skills & concepts 1. Consider this inequality: 2x > 5x a. List three values you can substitute for x to make the inequality statement true. b. List three values you can substitute for x to make the inequality statement false. 2. List the lines according to the value of their slopes. List them in order from the smallest slope value to the largest slope value. Consider the function rules y = 3x + 1 and y = 6x Graph both functions on the coordinate plane. Label each line with its algebraic rule. Answer: Preparing for upcoming lessons 4. Substitute the x and y values of the intersection point into both function rules to verify that the coordinates make both rules true. Focus skill: Slope and geometric connections A lattice point has x- and y- coordinates that are both integers. 5. Mark two lattice points that are on the line. 6. Draw a slope triangle between the points you marked, and use the slope triangle to calculate the slope of the line. Slope of line:
16 80 Unit 5 Linear equations and inequalities
17 Topic 15: Solving linear inequalities 81 Lesson 15.4 Solving with algebraic operations 15.4 OPENER 1. Complete each inequality statement with the correct symbol, either < or >. 2 < < < < < < < < < What do you notice about the relationship between the answers you get when you add the same number to 2 and to 8? 3. What do you notice about the relationship between the answers you get when you subtract the same number from 2 and from 8? 15.4 CORE ACTIVITY 1. Complete each inequality statement with the correct symbol, either < or > Study the pattern in question 1. When do you need to reverse the inequality sign? 3. Solve the equation 2x + 5 = 11 using algebraic operations. 4. Using the balance scale as a model and the fact that 2x + 5 weighs less than 11, explain why each of the following statements is true. a. 2x will weigh less than 6. b. x will weigh less than 3.
18 82 Unit 5 Linear equations and inequalities 5. Solve 8 2x 14 by first subtracting 8 from both sides. Then complete solving the inequality. 6. This time, solve the inequality 8 2x 14 by first adding 2x to both sides. 7. What do you notice about the processes you used to answer questions 5 and 6? 8. Use inverse operations and algebraic properties to solve the following inequalities. Use substitution to check your solutions. a. 8a 5 1 b. 6b 1 > 3b + 8 c. 0.25(16 12c) 31 d. 1 e e e. 3 < 3(- 5d + 16) f. 5f + 4(f 1) 2 + 5(2 + f) 2
19 Topic 15: Solving linear inequalities CONSOLIDATION ACTIVITY 1. Jacob solved four different inequalities, but he is not sure whether he solved them correctly. In fact, sometimes he makes more than one mistake when solving! Jacob s work is shown below. Decide whether Jacob solved each inequality correctly. If not, find the mistake(s) and explain what Jacob did wrong. Then, solve the inequality correctly. Jacob s solution: a. 3x + 6 > 8 3x ( 6) > 8 + ( 6) 3x < 2 3x 3 < 2 3 x < 2/3 Did Jacob solve correctly? If not, describe his mistake(s): Show the correct solution (if needed): b. 2x + 1 x 2 2x x + 1 x x 2 1x x x -3 c. 4(x + 2) > 5x + 1 4x + 2 > 5x + 1 4x > 5x 1 x > -1! d. 5x + 5 > 10x x > 10x x + 5 > 2x + 30 x ( 5) > 2x ( 5) x > 2x + 25 x x < 2x x < x < x < x 2. On your whiteboard, create your own inequality. Exchange your whiteboard with your partner and solve the problem your partner created. When both you and your partner are finished, discuss your solutions.
20 84 Unit 5 Linear equations and inequalities HOMEWORK 15.4 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Solve each of the following inequalities. Use substitution to check your solutions. a. 4 2x 5 x + 1 b. 4 3 x + 7 > 3 c. 2x 3(x + 3) < 14 d. 9x 24x + 45 > 0 e. 3.1x x f. 3x + 2(4x + 2) 2(6x + 1) 2. Suppose your friend is having trouble solving the inequality 2(3x 8) < 5(4 x). Show each step you would take to solve the inequality, and write an explanation of why you are taking that step. 3. Jessica was solving the inequality 4x She used inverse operations. To check her work, she chose a number greater than 1 and substituted it into the original inequality to see if it made a true inequality. The number she chose for x was 2. When she substituted it into the inequality, however, she ended up with an untrue inequality. Explain where Jessica made a mistake. Then solve the problem correctly and check your work. Incorrect solution: 4x x x 4 4x x 1 Check: 4(2) Description of mistake: Corrected solution (with check step):
21 Topic 15: Solving linear inequalities 85 STAYING SHARP On the coordinate plane, sketch a line that has a slope of zero and that passes through the point (- 3,5). Then, write an equation for the line. 2. On the coordinate plane, sketch a line that passes through the points (3, 5) and (3,3). Then write an equation for the line. Practicing algebra skills & concepts Equation of line: Equation of line: Preparing for upcoming lessons Complete these tables to answer questions 3 and 4. Table A Table B y = 2 + 3x y = 4 + 3x a. As x increases by 1, by how much does y increase in Table A? b. As x increases by 1, by how much does y increase in Table B? 4. Will there be a common (x,y) pair in the two tables if the tables are extended? Explain. Use this graph to help you answer questions 5 and Write an equation of the line in slope- intercept form. Focus skill: Slope and geometric connections 6. On the line, there is a point with an x- coordinate of 7. Find that point s y- coordinate. Explain how you found the answer.
22 86 Unit 5 Linear equations and inequalities
23 Topic 15: Solving linear inequalities 87 Lesson 15.5 Inequalities in the plane 15.5 OPENER Match each point with a description of its coordinate. 1. Point A 2. Point B 3. Point C 4. Point D 15.5 CORE ACTIVITY Suppose you and some friends go to the movies and buy some snacks. The snack bar charges $2 for a box of candy and $6 for the combo. The combo is a medium drink and popcorn. 1. If x = the number of boxes of candy purchased and y = the number of combos purchased, write an expression that represents the total amount you could spend at the snack bar on candy and combos. 2. After buying the movie tickets, you have $12 left to spend for snacks. Use your cost expression from question 1 to write an inequality that indicates that you would get change from your snack bar purchase. 3. a. Use substitution to determine whether (5,1) is a solution to the inequality you wrote in question 2. b. What does the coordinate pair (5,1) mean in the context of the problem? 4. Is (2,1) a solution to the inequality 2x + 6y < 12? What does the ordered pair (2,1) mean in the context of the problem?
24 88 Unit 5 Linear equations and inequalities 5. a. Which of the given ordered pairs are solutions to the inequality you wrote in question 2? Explain how you know. (3,2) (0,1) (9,- 1) (6,- 2) (- 4,2) (5,0) b. What do the solutions mean in the context of the problem? 6. Graph the line 2x + 6y = 12 and plot all of the points listed in question 5. Label points that are solutions to the inequality 2x + 6y < 12 with the letter T and the non- solutions with the letter F. Where do all the solutions lie? 7. How would you shade your graph to show ALL the ordered pairs that make the inequality true?
25 Topic 15: Solving linear inequalities Match each of the inequalities with its corresponding graph. Write the inequality in the blank space beneath the graph it represents. y > - 5 5x 4y > 20 x < y x < - 5 5x 4y 20 5x 4y < 20 Inequality: Inequality: Inequality: Inequality: Inequality: Inequality: 15.5 ONLINE ASSESSMENT Today you will take an online assessment.
26 90 Unit 5 Linear equations and inequalities HOMEWORK 15.5 Notes or additional instructions based on whole- class discussion of homework assignment: A snack bar charges $3 for a box of candy and $5 for the combo. The combo is a medium drink and popcorn. 1. Write an expression that represents the total amount you could spend at the snack bar on candy and combos. 2. Suppose you can spend at most $12. Find at least three ordered- pair solutions representing the number of boxes of candy and combos you can buy. 3. Suppose you want to spend at least $12. Find at least three ordered- pair solutions representing the number of boxes of candy and combos you can buy. 4. Suppose you want to spend exactly $12. Find an ordered pair solution representing the number of boxes of candy and combos you can buy. 5. Using the expression you wrote in question 1, write an equation to represent the situation in question 4, in which you spend exactly $12. Graph the equation. Then plot all seven ordered- pair solutions to questions 2-4 on the graph. a. Describe the region of the graph where you find the solutions to question 2. b. Describe the region of the graph where you find the solutions to question 3. c. Describe the region of the graph where you find the solution to question 4.
27 Topic 15: Solving linear inequalities Tanya s mother will pay for Tanya s entire cell phone bill as long as the usage charge is less than $35 for the month. On Tanya s cell phone plan, the usage charge is $0.15 per text message and $0.10 per minute for calls. a. Write an inequality that expresses the number of text messages Tanya may send, t, and the number of minutes Tanya may talk, m, so that her mother will pay the entire cell phone bill for the month. b. Create a graph showing all of the possible combinations of texts and minutes so that Tanya s mother will pay the entire bill. m t c. Use the graph you constructed to find five possible combinations of texts and minutes that would keep Tanya s usage charge under $35.
28 92 Unit 5 Linear equations and inequalities 7. For each inequality, create a graph showing all of the coordinate pairs that make the inequality true. a. y 1 2 x 3 b. 5x 6y < 30 c. x > - 3
29 Topic 15: Solving linear inequalities 93 STAYING SHARP 15.5 Practicing algebra skills & concepts 1. Solve the following inequality for x: 8x Answer with supporting work: 2. Write an equation for a line that has a slope of 4 and passes through the point (1,6). Then, name the coordinates of one other point that the line passes through. Answer with supporting work: Preparing for upcoming lessons 3. Report the point of intersection for the graph. Include the context of the situation represented in the graph. Intersection point: 4. The functions y = 2x + 4 and y = 2x + 1 are graphed here. How can you tell from their equations that the two lines will never intersect each other? Answer: Focus skill: Slope and geometric connections 5. Graph the line with a y- intercept of (0, 2) and a slope of 4 on the coordinate plane. (Consider using slope 3 triangles to graph efficiently.) 6. Write an equation of the line you graphed in question 5.
30 94 Unit 5 Linear equations and inequalities
31 Topic 15: Solving linear inequalities 95 Lesson 15.6 Compound inequalities in the plane 15.6 OPENER 1. Graph the following compound inequality on a number line: x < - 1 or x > 4 2. Graph the following inequalities on the plane: a. x < - 1 b. x > 4 3. Using the graphs you constructed in questions 1 and 2, predict what the graph of x < - 1 or x > 4 would look like in the plane. Sketch your prediction. Explain why you think your graph is correct.
32 96 Unit 5 Linear equations and inequalities 15.6 CORE ACTIVITY 1. How close was your prediction of the graph of x < - 1 or x > 4 in the plane to the correct graph? Explain. 2. Graph the inequality x > - 2 and x < 7 on the number line. 3. Graph the inequality x > - 2 and x < 7 on the plane. How is the graph related to the graph you constructed in question 2? 4. For each inequality, create a graph showing all of the coordinate pairs that make the inequality true. a. y - 4 b. x Graph the following compound inequalities in the plane: a. y - 4 or x - 2 b. y - 4 and x REVIEW ONLINE ASSESSMENT
33 Topic 15: Solving linear inequalities 97 You will work with your class to review the online assessment questions. Problems we did well on: Skills and/or concepts that are addressed in these problems: Problems we did not do well on: Skills and/or concepts that are addressed in these problems: Addressing areas of incomplete understanding Use this page and notebook paper to take notes and re- work particular online assessment problems that your class identifies. Problem # Work for problem: Problem # Work for problem: Problem # Work for problem:
34 98 Unit 5 Linear equations and inequalities HOMEWORK 15.6 Notes or additional instructions based on whole- class discussion of homework assignment: Next class period, you will take an end- of- unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts, especially by reviewing related course materials. Important skills and ideas you have learned in the unit Linear equations and inequalities: Analyze situations involving linear functions and formulate linear equations and inequalities to solve problems Use various methods to solve linear equations and inequalities: inspection, tables, graphs, and use of algebraic operations in connection with the properties of equality Interpret and determine the reasonableness of solutions to linear equations for given contexts Apply techniques for solving equations in one variable to solve literal equations Graph solutions to linear inequalities in one variable on a number line Graph solutions to linear inequalities in two variables on a coordinate plane Graph solutions of compound linear inequalities in two variables on a coordinate plane Homework Assignment Part I: Study for the end- of- unit assessment by reviewing the key ideas listed above. Part II: Complete the online More practice in the topic Solving linear inequalities. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study. Part III: Complete Staying Sharp 15.6
35 Topic 15: Solving linear inequalities 99 STAYING SHARP a. Find a number that satisfies the inequality x < Write an equation for a line that passes through the points (3,5) and (6,3). Practicing algebra skills & concepts b. Check if the number you found in part a also satisfies the inequality 3x < 9. c. Will any number that satisfies x < 3 also satisfy 3x < 9? Explain. Answer with supporting work: Preparing for upcoming lessons 3. Complete this input- output table for both function rules. x y = 4x + 18 y = 2x A classmate has been away and missed several lessons. Explain to that classmate how to use the table you created in question 3 to find the intersection point of the two function rules, and how to check that the coordinates of that intersection point are correct. Focus skill: Slope and geometric connections 5. A line passes through the point (2,3) and has a slope of 2. Graph the line on the coordinate grid. (Consider using slope triangles to graph efficiently.) 6. Write an equation of the line you graphed in question 5.
36 100 Unit 5 Linear equations and inequalities
37 Topic 15: Solving linear inequalities 101 Lesson 15.7 Checking for understanding 15.7 OPENER Bianca and Joe are starting their own pet grooming business called House of Groom. They figure that they can spend no more than $50 a month on pet shampoo. A local dealer of pet shampoo, The Pet Pantry, sells quart bottles of shampoo for $4.00 a bottle plus a $5.00 handling fee per order. 1. Write an expression that represents the amount of money charged by The Pet Pantry for an order of shampoo. 2. Write an inequality that represents the amount House of Groom is willing to pay per month for The Pet Pantry's shampoo. 3. Solve the inequality you wrote in question 2 using any method END-OF-UNIT ASSESSMENT Today you will take the end- of- unit assessment.
38 102 Unit 5 Linear equations and inequalities 15.7 CONSOLIDATION ACTIVITY 1. For each numbered card, find a match using the lettered cards. Record the matches here. 2. Answer the following questions to reflect on your performance and effort this unit. a. Summarize your thoughts on your performance and effort in math class over the course of this unit of study. Which areas were strong? Which areas need improvement? What are the reasons that you did well or did not do as well as you would have liked? b. Set a new goal for the next unit of instruction. Make your goal SMART. Description of goal: Description of enabling goals that will help you achieve your goal:
39 Topic 15: Solving linear inequalities 103 HOMEWORK 15.7 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Explain the phrase compound inequality in the plane in your own words. 2. Graph each compound inequality on the coordinate plane provided. a. x < 5 and x > - 3 b. y < 4 and y 0 c. x - 2 and y < 3 d. y < - 5 or y > 5 e. x 7 or x >2 f. y - 3 or x - 1
40 104 Unit 5 Linear equations and inequalities STAYING SHARP 15.7 Practicing algebra skills & concepts 1. A line with a slope of 1 passes through the point 3 (3,6). Write an equation for this line. Answer with supporting work: 2. Which of the following points are on the line with equation y = 1 2 x + 7? (4,5) ( 4,3) (4,3) ( 4,5) Preparing for upcoming lessons 3. Graph the function rules y = 4x 3 and y = 2x + 9 on the coordinate plane. Then write the coordinates of their point of intersection. 4. Complete this input- output table for the function rules. x y = 4x 3 y = 2x a. For what x value are the y- values of the two functions equal? b. How does your answer to question 4a relate to your answer to question 3? Focus skill: Slope and geometric connections Intersection point: 5. Use slope triangles to graph each function rule on the coordinate grid. a. y = 1 2 x + 2 b. y = 2 3 x In which quadrant do the two lines intersect?
Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and
Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and
Topic 18: Other methods for solving systems 175 OTHER METHODS FOR SOLVING SYSTEMS Lesson 18.1 The substitution method 18.1 OPENER 1. Evaluate ab + 2c when a = 2, b = 3, and c = 5. 2. Following is a set
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TEST BANK How TO USE THIS TEST BANK In this section you will find one chap-ter test for each of the 14 chapters in Algebra: Themes, Tools, Concepts. For Chapters 3, 5, 6, 7, 9, 10, and 13, you will also
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Overall instructions: Your Name Teacher ALGEBRA FINAL EXAM 2006 There is a mix of easier and harder problems. Don t give up if you see some questions that you don t know how to answer. Try moving on to
SUGGESTED LEARNING STRATEGIES: Marking the Text, Shared Reading, Create Representations, Think/Pair/Share Geri wants to become a commercial airline pilot someday. She found this information while doing
Algebra 1 Spencer Unit 4 Notes: Inequalities and Graphing Linear Equations Unit Calendar Date Topic Homework Nov 5 (A ) 6.1 Solving Linear Inequalities +/- 6.2 Solving Linear Inequalities x/ 6.3 Solving
COMMON CORE MATHEMATICS CURRICULUM Lesson 1 8 4 Lesson 1: Writing Equations Using Symbols Classwork Exercises Write each of the following statements using symbolic language. 1. The sum of four consecutive
7.1 Solving Systems of Equations: Graphing Name Part I - Warm Up with ONE EQUATION: a. Which of the following is a solution to the equation: y 3x 1? a. (-2, -3) b. (1, 3) c. (2, 5) d. (-2, -6) Partt II
3.0 Distributive Property and Expressions Teacher Notes Distributive Property: To multiply a sum or difference by a number, multiply each number in the sum or difference by the number outside of the parentheses.
Systems of Equations Unit Five ONE NONE INFINITE Standards: 8.EE.8 Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables
Math 8 Unit 4: A Study of Linear Equations By the end of this unit, students should be able to: 1) Show that the slope of a line can be calculated as rise/run. Explain why the slope is the same between
MATH 080: Review for the Final Exam In order to prepare for the final exam, you need to understand and be able to work problems involving the following topics: I. Simplifying Expressions: Do you know how
Lesson 8: Representing Proportional Relationships with Equations Student Outcomes Students use the constant of proportionality to represent proportional relationships by equations in real world contexts
Applications 1 p = 7.5n - 55, where n represents the number of car washes and p represents the profit in dollars. 2 t = 0.5 + 2a, where a represents the area of the grass and t represents the time in hours
Patterns and RELATIONS Solving Equations Chapter 2 Big Idea Developing and solving equations can help me solve problems. Learning Goals I can use words to show number relationships. I can use equations
Standards PLUS Flexible Supplemental K-8 ELA & Math Online & Print Grade 8 SAMPLER Mathematics EL Strategies DOK 1-4 RTI Tiers 1-3 15-20 Minute Lessons Assessments Consistent with CA Testing Technology
Unit 5: Moving Straight Ahead Investigation 3: Solving Equations using tables and Graphs Practice Ace Problems Directions: Please complete the necessary problems to earn a maximum of 16 points according
th Grade Math Lesson Plan Unit.A Teacher: Dates: Texas Essential Knowledge and Skills:. Algebraic Reasoning The student applies mathematical process standards to develop concepts of expressions and equations.
CC Math 1A Name Unit 1 Writing and Evaluating Algebraic Expressions Day Date Lesson Assignment Mon 8/25 Lesson 1 Writing and Evaluating Algebraic Expressions Tues 8/26 Lesson 2 Combining Like Terms & Distributive
24 4.6 Taking Sides A Practice Understanding Task Joaquin and Serena work together productively in their math class. They both contribute their thinking and when they disagree, they both give their reasons
, and Absolute Value In this lesson, we first look at literal equations, which are equations that have more than one variable. Many of the formulas we use in everyday life are literal equations. We then
Lesson 1 Lesson 1: Writing Equations Using Symbols Classwork Exercises Write each of the following statements using symbolic language. 1. The sum of four consecutive even integers is 28. 2. A number is
Section 2.3 Objectives Use the inequality symbols to compare two numbers. Determine if a given value is a solution of an inequality. Solve simple inequalities. Graph the solutions to inequalities on the
Part 1: Introduction Solve Systems of Equations Algebraically Develop Skills and Strategies CCSS 8.EE.C.8b You know that solutions to systems of linear equations can be shown in graphs. Now you will learn
Pennsylvania Keystone Exams Algebra I Item and Scoring Sampler 2016 Keystone Algebra I Sampler Table of Contents INFORMATION ABOUT ALGEBRA I Introduction.......................................................................................
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
Name Period Date Equations and Inequalities Student Packet 4: Inequalities EQ4.1 EQ4.2 EQ4.3 Linear Inequalities in One Variable Add, subtract, multiply, and divide integers. Write expressions, equations,
Foundations for Algebra Introduction to Algebra I Variables and Expressions Objective: To write algebraic expressions. Objectives 1. I can write an algebraic expression for addition, subtraction, multiplication,
GRE 501 LESSON/NOTES Period Name CRS SKILL LEVEL DESCRIPTION Level 1 ALL students must GRE 301 Locate points on the number line attain mastery at this level R- XEI 506 Solve first degree inequalities that
Copyright 2015 Edmentum All rights reserved. Linear Equations & Graphs 1. A line has a y intercept of and a slope of. Find the equation of the line. A. B. C. D. Evaluate Functions 2. The graph of the function
Section 2.2 Objectives Solve multi-step equations using algebra properties of equality. Solve equations that have no solution and equations that have infinitely many solutions. Solve equations with rational
Name Algebra 1 Date Midterm Review Period 1) Solve each equation: a) x 2x + 2 = 3 b) 5 5 + 9 = 13 c) 64 = 9x +1 d) x 7 2 = 10 4x e) x + 2 3 = 3x 2) Solve for y: a) 6x 3y = 12 b) 4y 8x = 16 3) Solve and
Section 2.1 Video Guide Linear Equations: The Addition and Multiplication Properties of Equality Objectives: 1. Determine If a Number Is a Solution of an Equation 2. Use the Addition Property of Equality
Pennsylvania Keystone Exams Algebra I Item and Scoring Sampler 2014 Keystone Algebra I Sampler Table of Contents INFORMATION ABOUT ALGEBRA I Introduction.......................................................................................
4.2 Equal Values Method Name: Recall Writing and solving equations involving rate of change and initial value a. In a jumping frog race, your frog received a 6 in. head start and jumps 3 in. every 2 seconds.
Write inequalities to represent the following problem, and then solve to answer the question. 1. The Rent-A-Lemon Car Rental Company charges $60 a day to rent a car and an additional $0.40 per mile. Alex
Name: 2014-15 Math Class: Algebra I Summer Review Packet DUE DATE: About Algebra I Algebra I teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions
UNIT 2 SOLVING EQUATIONS NAME: GRADE: TEACHER: Ms. Schmidt _ Solving One and Two Step Equations The goal of solving equations is to. We do so by using. *Remember, whatever you to do one side of an equation.
Answer: Journal Prompt Cristina was working on a multi step problem but she made a mistake. Find the error and explain how she could fix it. 3 x 6 + 4 x + 1 = 7x 10 3x 6 + 4x + 4 = 7x 10 x 2 = 7x 10 +2
Name Algebra 1 Fall Review 2013-2014 Date 1. Write an inequality to best represent the graph shown at right. (A.1.D.) m: b: inequality: 2. Write an inequality to best describe the graph shown at right.
Name: Teacher: School: Contents: Key Words... 1 Tools and Strategies... 2 Open Response Rubric... 3 EQAO Formula Sheet... 5 Sample Questions from the 2014 Assessment... 7 Sample Questions from the 2013
Name Period Date POLYNOMIALS Student Packet 3: Factoring Polynomials POLY3 STUDENT PAGES POLY3.1 An Introduction to Factoring Polynomials Understand what it means to factor a polynomial Factor polynomials
Be Greater Than Solving Inequalities with Inverse 4 Operations WARM UP Graph each inequality on a number line. 1. x. 5. x $ 1 3. x, 6. 4. x # 9 LEARNING GOALS Solve and graph one- and two-step inequalities.
A Story of Ratios Eureka Math Grade 8, Module 4 Student File_B Contains, and Assessment Materials Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced,
13 (3 1) Chapter 3 Inequalities in One Variable 95. Designer jeans. A pair of ordinary jeans at A-Mart costs $50 less than a pair of designer jeans at Enrico s. In fact, you can buy four pairs of A-Mart
Keystone Exam Concept Review Name: Properties and Order of Operations COMMUTATIVE Property of: Addition ASSOCIATIVE Property of: Addition ( ) ( ) IDENTITY Property of Addition ZERO PRODUCT PROPERTY Let
Lesson ~ Parallel, Intersecting or the Same Line Graph the two linear equations in each system on a single coordinate plane and state whether the lines are intersecting, parallel or the same line.. x 5
Name Period Date LINEAR FUNCTIONS STUDENT PACKET 6: SYSTEMS OF LINEAR EQUATIONS LF6.1 LF6.2 LF6.3 Introduction to Systems of Linear Equations Understand the definition of a system of linear equations Understand
Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you
More with Systems of Equations In 2008, 4.7 million Americans went on a rafting expedition. In Georgia, outfitters run whitewater expeditions for ages 8 and up on the Chattooga River. 12.1 Systems of Equations
Name: Essential Skills Practice for students entering Geometry or Accelerated Geometry Use this document to review the mathematics that you have learned previously. Completion of the Essential Skills Practice
DIRECT VARIATION TABLES AND SLOPE LESSON 3-B When a graph on a coordinate plane is a straight line that goes through the origin it is called a direct variation graph. In this lesson you will investigate
.1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle
SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS 6.EE.6 and 6.EE.7 CONTENTS The types of documents contained in the unit are listed below. Throughout the unit, the documents are arranged by lesson.