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1 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, and inequalities. Graph solutions to inequalities. Determine whether equations and inequalities are true or false. Solving Linear Inequalities in One Variable Solve linear equations and inequalities in one variable and graph their solutions. Inequality Problems Change word statements that involve inequalities to symbols. Use inequalities to solve problems. Graph solutions of inequalities EQ4 STUDENT PACKET EQ4.4 Vocabulary, Skill Builders, and Review Equations and Inequalites (Student Packet) EQ4 SP

2 WORD BANK (EQ4) Word Phrase Definition or Explanation Example or Picture boundary point of a solution set equation greater than inequality is equal to is greater than is less than less than Equations and Inequalites (Student Packet) EQ4 SP0

3 4.1 Linear Inequalities in One Variable LINEAR INEQUALITIES IN ONE VARIABLE Ready (Summary) Set (Goals) We will write linear inequalities in one variable and graph their solutions. Add, subtract, multiply, and divide integers. Write expressions, equations, and inequalities. Graph solutions to inequalities. Determine whether equations and inequalities are true or false. Go (Warmup) Here are five mathematical symbols: = < > Select two different symbols to make each statement true. For example, it is true that 6 > 4, and it is also true that or or or (-6) -3(-8) or -4(-6) -3(-8) (-3) or (-3) (3 + (-1)) 8-2 or 5 3-4(3 + (-1)) 8-2 EQ4 SP1

4 4.1 Linear Inequalities in One Variable NUMBER SENTENCES AND THEIR GRAPHS Match each number sentence with its graph. 1. x = 3 A. 2. x > 3 B. 3. x < 3 C. 4. x 3 D. 5. x 3 E < x < 3 x is an integer 7. 3< x < 3 G. F Describe the difference between the symbol and when graphing on a number line. EQ4 SP2

5 4.1 Linear Inequalities in One Variable EXPLORING INEQUALITIES 1. Complete the table. For the last column, write a new inequality that reflects the change. Be sure that your new inequality is in fact a true statement. Begin each operation with this inequality. 4 < 10 Then do this to both sides: Left Steps Right New inequality Add < 15 Add -5 Subtract 3 Subtract -3 4 (-3) 10 (-3) Multiply by 8 Multiply by -8 Divide by 2 Divide by On the first number line below, graph the numbers 4 and 10. Below it, graph the result of subtracting -3 from 4 and -3 from On the first number line below, graph the numbers 4 and 10. Below it, graph the result of dividing 4 by -2 and 10 by In the table above circle every result where the inequality must change direction. When must the direction of the inequality symbol change? EQ4 SP3

6 4.1 Linear Inequalities in One Variable EXPLORING INEQUALITIES (continued) 5. Complete the table. For the last column, write a new inequality that reflects the change. Be sure that your new inequality is in fact a true statement. Begin each operation with this inequality. -6 > -9 Then do this to both sides: Left Steps Right Add Add -5 Subtract 7 Subtract -7 Multiply by 2 Multiply by -2 Divide by 3 Divide by -3 New inequality 6. On the first number line below, graph the numbers -6 and -9. Below it, graph the results of dividing -6 by 3 and -9 by On the first number line below, graph the numbers -6 and -9. Below it, graph the results of dividing -6 by -3 and -9 by In the table above, circle every result where the inequality must change direction. When must the direction of the inequality symbol change? EQ4 SP4

7 4.1 Linear Inequalities in One Variable WORDS, SYMBOLS, AND GRAPHS Complete the table. For cases in which the x is on the right side of the number sentence, also write an equivalent number sentence with x on the left. Words Symbols Graph the solution. Test a number x is equal to 4 x is greater 2. than or equal to 4 x is an integer 3. that is greater than 2 2 is greater 2 > x 4. than x The opposite of -x > 2 5. x is greater (-1)(-x) (-1)(2) than 2 x takes values from 6. < < -2 to 4 (exclusive) x takes values from to 4 (inclusive) The opposite of 8. x is less than or equal to EQ4 SP5

8 4.1 Linear Inequalities in One Variable WORDS, SYMBOLS, AND GRAPHS (continued) Complete the table. For cases in which the x is on the right side of the number sentence, also write an equivalent number sentence with x on the left Words Symbols Graph the solution. Test a number. x 4 is equal to the opposite of x x is an integer less than or equal to 5 5 is less than or equal to x x takes values from -5 to -1 (inclusive) x takes values from -3 and 0 (exclusive) The opposite of x is greater than or equal to -3-1 is greater than x 2 is greater than the opposite of EQ4 SP6

9 4.1 Linear Inequalities in One Variable TRUE OR FALSE? Test each statement by substituting numbers into the equation or inequality. Then make a conjecture as to whether the statement is true or false. Use numerical examples to support a claim that a statement is true. Recall that only one counterexample is needed to disprove a statement. (The symbol for not equal to is.) Statement If a = b, then b = a. If a b, then b a If a > b, then -a > - b If a = b and b = c, then a = c If a > b and b > c, then a > c If a b, and b c, then a c Supporting examples (if any) Counterexamples (if any) True or False? 7. Which false statements are sometimes true? EQ4 SP7

10 4.2 Solving Linear Inequalities in One Variable SOLVING LINEAR INEQUALITIES IN ONE VARIABLE Ready (Summary) We will solve linear inequalities in one variable and graph their solutions. Set (Goals) Solve linear equations and inequalities in one variable and graph their solutions. Go (Warmup) Solve each equation for x. Check the solution in the original equation 1. 3x 5 = 2x 2. -(2x 4) = -2x 4 Check: 3 ( ) 5 2 ( ) Check: Solve each inequality for x and graph the solution(s). 3. -x > 3 and x is an integer x 5. What is the difference between an equation and an inequality? EQ4 SP8

11 4.2 Solving Linear Inequalities in One Variable SOLVING INEQUALITIES Solve each inequality. Graph the solution. Use substitution to check the boundary point and one other point x Graph: Check the boundary point (x = ) Test another point (x = ) 3. 3( x 5) > -12 Graph: Check the boundary point (x = ) Test another point (x = ) x + 6 Graph: Check the boundary point (x = ) Test another point (x = ): 4. 6 x > 2+ x Graph: Check the boundary point (x = ) Test another point (x = ) EQ4 SP9

12 4.2 Solving Linear Inequalities in One Variable SOLVING INEQUALITIES (continued) Solve each inequality. Graph the solution. Use substitution to check the boundary point and one other point. 5. -( x 1) < -4 Graph: Check the boundary point (x = ) Test another point (x = ) x -x 7 Graph: Check the boundary point (x = ) Test another point (x = ) 6. x is an integer -x < -3x 5 Graph: Check the boundary point (x = ) Test another point (x = ): 8. -3( x 1) 2 x Graph: Check the boundary point (x = ) Test another point (x = ) EQ4 SP10

13 4.2 Solving Linear Inequalities in One Variable SOLVE IT! Roll one number cube three times. Create an equation or inequality using the expressions and symbols below. For example, if you roll a 4 on the first cube, a 3 on the second cube, and a 5 on the third cube, you will create the inequality: 2(x 1) < 2. Solve, graph, and check solutions. Number on cube Roll 1 Roll 2 Roll 3 1 2x x = -4 3 x 4 < ( x 1) > 0 5 -x x = 4 Use these templates to record your equations or inequalities, their graphs, and checks Graph: Check the boundary point (x = ) Test another point (x = ) Graph: Check the boundary point (x = ) Test another point (x = ) EQ4 SP11

14 4.2 Solving Linear Inequalities in One Variable MORE SOLVE IT! TEMPLATES Use number cubes or use cards provided by your teacher to create equations or inequalities Graph: Check the boundary point (x = ) Test another point (x = ) 5. Graph: Check the boundary point (x = ) Test another point (x = ) Graph: Check the boundary point (x = ) Test another point (x = ) 6. Graph: Check the boundary point (x = ) Test another point (x = ) EQ4 SP12

15 4.3 Inequality Problems INEQUALITY PROBLEMS Ready (Summary) Set (Goals) We will solve problems that involve inequalities. Change verbal statements that involve inequalities to symbolic statements. Use inequalities to solve problems. Graph solutions of inequalities Go (Warmup) For each word statement, identify the variable. Then choose an appropriate symbolic representation for the statement from these choices. Some choices may not be used at all, and some may be used more than once. A. x = 25 B. x 25 C. x > 25 D. x 25 E. x < 25 F. x 25 G. None of the above Words Identify the variable Symbols 1. Matt has less than $25 Let x = Matt s money in dollars E. x < Derek has exactly $25 3. Shannon has $25 or more 4. Andrew has no more than $25 5. Ron has at least $25 6. Steve has $25 at the most 7. Lamar has approximately $25 8. Graph: x > 2.5, x is an integer. EQ4 SP13

16 4.3 Inequality Problems THE BICYCLE SHOP At the bicycle shop, you are paid $50 per week plus $3 for each bike sold. 1. How much will you earn if you come to work, but do not sell any bikes during the week? 2. How much will you earn if you sell 5 bikes during the week? 3. Write an equation that expresses the amount you earn (e) in terms of the number of bikes (b) sold. 4. This week, you want your pay to be at least $100. Write an inequality that describes the number of bikes (b) you need to sell. 5. Solve the inequality and graph it. 6. Explain your answer in the context of the problem. EQ4 SP14

17 4.3 Inequality Problems RAISING MONEY FOR SCHOOL ACTIVITIES 1. The high school glee club is organizing the spring dance. a. If they charge $23.00 per person for the dance, and 30 people attend, how much will they collect? b. If they charge $23.00 per person for the dance, and p people attend, how much will they collect? c. The cost of the rental of the hall for the dance is $4000. How many people must attend the dance to cover the rental fee? Write an inequality, solve it, and graph it. Explain your answer in the context of the problem. 2. The high school basketball team needs to raise at least $2,700 to travel to a tournament. A booster gives them $250 and they are selling magazines at $27 per subscription to make the rest. How many subscriptions must they sell? Write an inequality, solve it, and graph it. Explain your answer in the context of the problem. EQ4 SP15

18 4.3 Inequality Problems PRACTICE: KATY S SUMMER For each problem below, write an inequality, solve it, and graph it. Then explain the answer in the context of the problem. 1. Katy has $460 in a checking account at the beginning of summer. She wants to have at least $200 in the account by the end of summer. She withdraws $25 each week for her expenses. How many weeks can Katy withdraw this amount of money from this account? 2. A taxi service charges a $2.25 flat rate in addition to $0.64 per mile. Katy wants to spend no more than $10 on a ride. How many miles can Katy travel without exceeding her limit? 3. Katy goes to the Fun Golf Arcade with her friends. They play golf, have lunch, and then play some video games. A round of golf is $6.20. Lunch is $5.60. Video games are $0.50 each. If Katie wants to spend no more than $15.00, how many games can she play? EQ4 SP16

19 4.4 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (EQ4) Use words from the word bank and the clues below to fill in the crossword puzzle. Across Down 3 A mathematical statement that is 14 (2 words) asserts the equality of two expressions is -25 (2 words) 2 A mathematical statement that asserts the relative order of two objects ( 3 words) (3 words) ( 3 words) 8 There are solutions and nonsolutions around this point of a solution set for an inequality. EQ4 SP17

20 4.4 Vocabulary, Skill Builders, and Review 1. Perform the number trick below. SKILLBUILDER 1 Step Words Numbers Pictures Algebraic Process Choose a 1 V n number. 2 Add 3. 3 Multiply by 2. 4 Subtract Divide by the original number. What is the result? What is the number trick? Does this always work? Explain. 2. Draw a picture to represent the situation below, write an equation, solve, and check the solution. Two trains start traveling in the same direction, from the same place and at the same time. Train A travels at a rate of 125 miles per hour and Train B travels at a rate of 110 miles per hour. After some time has passed, one train is ahead by 75 miles. Which train is it, and how many hours have passed? EQ4 SP18

21 4.4 Vocabulary, Skill Builders, and Review SKILLBUILDER 2 Draw the next step suggested by this pattern. Then complete the table and find a rule for the number of squares at step n. step1 step 2 step 3 step 4 Step # n Arithmetic # of squares 1. Label the horizontal and vertical axes and graph the data points. 2. Recursive Rule: 3. Explicit Rule: 4. How many toothpicks are in step #40? 5. In what step number are there squares? exactly 33 EQ4 SP19

22 4.4 Vocabulary, Skill Builders, and Review Complete the table. SKILL BUILDER 3 Words Write and solve. Graph the solution. Test a number. 1. x is less than is less than or equal to the opposite of x 4 x x is an integer 4. - x < x takes values from -2 and 0 (inclusive) < x < EQ4 SP20

23 4.4 Vocabulary, Skill Builders, and Review Underline what you are trying to find, then answer questions a-d. 1. A scalene triangle has a perimeter of 70 units. The second side is 8 units more than the length of the first side. The third side is 4 times the length of the second side. What is the length of each side? a. Define the variables using words or pictures. b. Write an equation and solve. c. Check the solution. d. Write the solution in words. SKILL BUILDER 4 2. A collection of coins contains dimes, quarters, and nickels. There are 4 less nickels than dimes, and twice as many quarters as dimes. The value of the collection is $4.35. Find the number of coins in the collection. Fill in the missing numbers and name the property of arithmetic illustrated by each equation. 3. ( 4 )(5) = ( )(5)=5 4 Property = (2 + 8). 4 = 40 Property 5. (7)(8) = (8)(7) = 56 Property EQ4 SP21

24 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Write all of the mathematical symbols from the box below that apply for each problem. = 1. Example: 5? 3 Answer(s): >,, 3. (-4)(3)? (4)(3) Answer(s): (-6) 3? -12 -(-6) (-3)? 4 (-3) Answer(s): 4. (-4)(-3)? (4)(-3) Answer(s): Answer(s): Answer(s): Solve each equation x 6 = -3(x 6) 8. 2x 5 = 5x x = -(2x + 10) (-6) -3? x 1 = 4( x 3 ) (-6) -3 EQ4 SP22

25 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Find missing values in each input-output table, and write an explicit rule for the data x y y = Rule: x Rule: y x y y = y = Rule: F or each explicit rule, complete the input-output table x y x y x y Rule: y = 4x Rule: y = -3x Rule: y = 3x 2 Careful! Watch order of operations. EQ4 SP23

26 4.4 Vocabulary, Skill Builders, and Review Match the inequality with its graph x +9 > 6-3x 9 > 6-3( x 3) > 6 SKILL BUILDER 7 Solve each inequality. Then graph the solution and check by testing a number Inequality 6 x > 3-5 -x 9 3x-8 < - x +4 3( x+ 1) 4x + 5 a. b. c. Graph Check the boundary point and test another number EQ4 SP24

27 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 8 For each problem, translate it into an inequality and solve it. Then check the boundary point and another value. 1. The sum of the first number and the second number is less than the first number. Find the possibilities for the first number. Write inequality and solve: Answer in words: Check: Evaluate each expression (1) (5) (6) (3) (-1) (2) (-7) (-3) (3) (-9) (12) (11) 2. A number times -3 is less than or equal to two less than the number. Find the possibilities for the number. Write inequality and solve: Answer in words: Check: ( 3) (2) (3) (6) (-2) (1) (-3) (-6) (-6) (-3) (2) (1) EQ4 SP25

28 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 9 Solve for the variable indicated. 3. Solve for x: 3x 2y = 6 4. Solve for y: 3x 2y = 6 5. Solve for r: I = prt 6. Solve for r: 320 = p + prt 7. Solve for r: C = 2π r 8. Solve for π : C = π D Determine if the set of ordered pairs in each table represents a function in x x y x y x y function? function? function? function? x y EQ4 SP26

29 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER In the figure at the right, three parallel lines are cut by a perpendicular line and another transversal. Find x. Then find the value of every angle in the diagram. 5x Solve for x: y = 3x 6 3. Solve for y: x 2y = Find x if y = 0: y = 3x 6 5. Find y if x = 0: x 2y = 14 3x + 50 EQ4 SP27

30 4.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (EQ4) Show your work on a separate sheet of paper and choose the best answer. 1. Describe the graph of -x > 4 A. Open circle at 4; arrow to the right B. Open circle at 4; arrow to the left 2. Solve for x: 10 2x -4(x 2) C. Open circle at - 4; arrow to the right D. Open circle at - 4; arrow to the left A. x 1 B. x -1 C. x 1 D. x Choose the inequality that represents the following problem: 18 less than a number x is less than 25. A. 18 < x < 25 B. 18 < x 25 C. x 18 < 25 D. 18 x < Which expression below is not equivalent to two groups of (-x 3)? A. (-x 3) + (-x 3) B. 2(-x) 3 C. -2x 6 D. -2(x + 3) 5. If n is an odd integer, which expression below represents the odd integer just larger than n? A. n + 1 B. n + 2 C. 2n + 1 D. n(n + 1) 6. You have 20 coins, all quarters and dimes, for a total of $3.35. If x represents the number of quarters you have, which equation represent the amount of money you have (in cents)? A. 25x + 10(20 x) = 335 B. x + (20 x) = 335 C. 25x + 10x = 335 D. 20x = 335 EQ4 SP28

31 4.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (EQ4) Show your work on a separate sheet of paper and write your answers on this page. 8.1 Linear Inequalities in One Variable Word Symbol Graph the Solution Test a Number is greater than x 2. x is an integer that is less than or equal to Solve Linear Inequalities in One Variable Solve each inequality. Graph the solution. Use substitution to check the boundary point and one other point. 9. 2( x +3) > x 3 + x 8.3 Inequality Problems and Challenges 5. The band members will hold a car wash to raise money to purchase sheet music for the Music Department. The Music Department needs at least $275 to purchase the sheet music. If they earn $6.50 for each car they wash, how many cars must the band members wash in order to raise the money to purchase the sheet music? Write an inequality, solve it, and graph it. Explain your answer in the context of the problem. EQ4 SP29

32 HOME SCHOOL CONNECTION Here are some questions to review with your young mathematician. 1. Graph the inequality x 3. Was your symbol on the graph or? Explain why you choose the symbol that you used? 2. Write the difference of x and 2 is greater than 5 in symbols. Solve and graph your inequality. 3. Rose plans to go shopping. She wants to buy a book for $13.50, a pair of sandals for $7.80 and some DVDs. The store has DVDs on sale for $9.90 each. If Rose wants to spend no more than $60, how many DVDs can she buy? Parent (or Guardian) Signature 6.EE.8 7.EE.4b A-CED-3* A-REI-3 SELECTED COMMON CORE MATHEMATICS STANDARDS Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities: Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. EQ4 SP30