Grade 8 Expressions, Equations, and Inequalities Name 1
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UNIT SELF-TEST QUESTIONS The Unit Organizer #2 4 BIGGER PICTURE NAME DATE 2 LAST UNIT /Experience 1 CURRENT CURRENT UNIT UNIT 3 NEXT UNIT /Experience 8 UNIT SCHEDULE 5 UNIT MAP UO Launch equations & inequalities that represent word problems 8.8A& B equations with variables on both sides of the equal sign 8.8C the values of x and y that simultaneously satisfy two linear equations 8.9A 7 Unit 6 Test Words to know: Identity simultaneous linear equations Inequality system of equations Coefficients two-step equations Variables Properties 1. What is the process for solving multi-step equations & inequalities? How does the process for solving equations and inequalities differ? 1 2 3 4 2. How do you model an equation with variables on both sides? 1 2 3 4 3. How do you write, model and solve problems that require variables on both sides? 1 2 3 4 4. How do you identify the solution that satisfies two linear equations from a graph? 1 2 3 4 5. How do you algebraically verify a solution that satisfies two linear equations? 1 2 3 4 RELATIONSHIPS 6 UNIT 3
NEW UNIT SELF-TEST QUESTIONS TheUnit Organizer 9 Expanded Unit Map NAME DATE the values of x and y that simultaneously satisfy two linear equations equations & inequalities that represent word problems equations with variables on both sides of the equal sign The cost for a large cheese pizza at Pizza Palace is $10 plus $1.25 for each additional topping. The cost for a large pizza at Luigi s Pizzeria is $8 plus $1.75 for each additional topping. What is value for x that satisfies the equation: y = -x 4 y = 3x + 4 Write and inequality for the scenario and identify the coefficients & variables. Model and solve the inequality. 10 4
The FRAME Routine Key Topic Expressions, Equations, and Inequalities is about Unit 1 Main idea Expressions Main idea Equations Main idea Inequalities Essential details Essential details Essential details So What? (What s important to understand about this?) 5
Expressions, Equations, and Inequalities 3x + 5 4x 2 + 3x 5 = 12 3x > 2x 1 Word Definition Example Nonexample Variable Coefficient Constant Terms Base Exponent Operator Expression Equation Inequality Like Terms Terms can be combined (added or subtracted) ONLY IF they have the same and since that makes them 6
PRACTICE EXAMPLES Expression Like Term or Not? Justification 4x and 3 4x and 3y 4x and 3x 2 4x and 3x Simplify 5x + 7x These are like terms since they have the same and so we can combine them 5x + 7x = Simplify 2x 2 + 3x -4 -x 2 + x + 9 It is often best to group like terms together first, and then simplify: Notes: if no coefficient is shown it is understood to be "1" If no exponent is shown it is understood to be 1 7
Combining Like T erms Practice 8
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Combining Like Terms The three sides of a scalene triangle are 2x-1, 3x, and 4x+2. What is the perimeter of the triangle? The length of the side of an equilateral triangle is 3x-1. What is the perimeter of the triangle? The length of rectangle is 2x+1 and the width is x+4. What is the perimeter? The width of a rectangle is 2 and the length is x+3. What is the PERIMETER and AREA? 10
3. Known Information Unit: 2. Known Concept 4. Characteristics Anchoring Table 1. New Concept Name: Date: Solving Equations 6. Characteristics Shared 5. Characteristics of New Concept 7. Understanding of the New Concept: 11
One Step Equations The goal of solving an equation is to isolate the variable (get the variable by itself). This is done by performing inverse operations. remember, what you do to, you must do exactly the same to the. http://www.mathsisfun.com/flash.php?path=%2falgebra/images/algebrascales.swf&w=948&h=432&col=%23ffffff&title=balance+when+adding+and+su btracting http://www.pbslearningmedia.org/asset/mgbh_int_balance/ Inverse Operations: Inverse Operations are Practice: 1. I have 3 identical boxes that weigh a total of 9 pounds. How much does each box weigh? 2. I have a small box and a large box that weigh a total of 8 pounds. If the small box weighs 2 pounds how much does the large box weigh? 3. I have 2 large, identical boxes plus a smaller box that have a combined weight of 10 pounds. If the small box weighs 2 pounds what is the weight of each of the large boxes? Method 1: Represent each situation with an algebraic expression then find for each situation using algebraic methods. Method 2: Use the graphing calculator to enter each expression and find the solution using a graph and a table Method 3: Create a model and solve using objects or manipulatives 12
One-Step Equations Solve the following equations, using multiple representations to check your answer 1. 1x + 4 = 5 2. -9 = x - 4 3. 2x = 10 4. x 2 = 5 13
One-Step Equations Solve each equation algebraically, showing all steps. Check your answers using the graphing calculator for add numbers and a visual model for even numbered problems. 14
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One Step Equation Word Problems Formulate an equation for the word problems. Solve the equation and check your answer. 1. A number decreased by five is seven. What is the number? 2. A number subtracted from ten is eight. What is the number? 3. The product of four and a number is 25. What is the number? 4. Three-fourths of a number is eight. What is the number? 5. 150% of number is -90. What is the number? 6. The quotient of a number and 7 is 4. What is the number? 16
Algebraic Expressions Write the matching algebraic expression: the product of a number and 7 is 9 more than 54 Write the matching algebraic expression: the sum of 3 times a number and 11 is 14 My cell phone bill is $32 per month plus $0.10 per text message. Write the equation that matches this statement. The height of a woman in centimeters can be estimated by multiplying the length of her radius bone by 3.34 and adding 81. Write the equation that matches this statement, 17
Basic Equations 1 One box and two bags weigh 10 pounds. If a bag weighs 3 pounds how much does 1 box weigh? A bag that weighs 10 pounds plus 3 boxes weighs a total of 25 pounds. How much does each box weigh? Seven identical boxes plus two bottles weigh a total of 53 pounds. If each bottle weighs 2 pounds how much does each box weigh? I bought 4 oranges and 3 apples for $6.00 total. I purchased 3 oranges and 3 apples for $5.25 total. How much does an orange cost and how much does an apple cost? 18
Provide 2 examples each of: Variable Constant Like Term Basic Equations 2 The cost of salmon is $12 per pound. If I spend $42 on salmon how many pounds did I buy? I took 8 trips on the train and spent a total of $18. At this rate, how much does 1 trip on the train cost? Solve for x: 3x = 27 19
Two-Step Equations Two-step Equations Two-step equations combine the use of addition/subtraction and multiplication/division properties to isolate the variable. Use the first to move the numeric value or constant. The value on the same side of the equation and farthest away from the variable. Then, use to isolate the variable. Multi-step Equations Distribute to clear grouping symbols. Multiply by to clear fractions. Combine Use to move variable terms to one side and numeric values to the other side. Use to isolate the variable. Practice Problems Solve the following equations, using multiple representations to check your answer 1. 4x -5 = 7 2. -3x 5 = 10 3. x 3 + 4 = 19 20
Two-Step Equations: Solve each equation algebraically; show all steps. For odd numbered problems check your answer with the graphing calculator and for even numbered problems check your answer with a visual model. 1. 2 x + 20 = 4 2. 2 x + 4 = 20 3. 4 x + 6 = 18 4. 2 x + 10 = 18 5. 2 x + 18 = 2 6. 4 x + 19 = 7 7. 19 x + 19 = 0 8. 6 x + 4 = 16 9. 3 x + 19 = 1 10. 3 x + 20 = 2 21
11. x 5 1 = 0 12. x 2 2 = 10 13. x 4 3 = 0 14. x 2 1 = 0 15. x 2 6 = 7 16. x 4 2 = 8 17. x 5 6 = 0 18. x 2 10 = 10 22
Two Step Equations Word Problems Formulate an equation for the word problems. Solve the equation algebraically showing all steps. Check your answer with the graphing calculator. 1. Twice a number increased by four is fourteen. What is the number? 2. Five less than six times a number is seven. What is the number? 3. At the local HUB's grocery store, the cost of a carton of tamales, c, depends on the number of tamales, t. The cost of the carton is $0.25 and each tamale is $0.50. a. What equation could be used to represent this situation? b. If a carton contained one dozen tamales, what would be the cost? c. If the cost of a carton of tamales is $15.25, how many tamales are in the carton? 4. Surf City charges a base fee of $10 and $25 per hour to rent a surf board. a. What equation could be used to represent the situation? b. If Toni rented a surf board for 5 hours, how much would she have to pay? c. Steven paid $185 for renting a surf board from Surf City. For how many hours did Steven rent the surf board? 23
Algebraic Properties Equations are composed of two expressions set equal to one another. The properties of algebra can be used to simplify algebraic expressions much more efficiently and applied to solve multiple-step equations. Properties of Addition and Multiplication Property Rule Example Commutative of Addition Commutative of Multiplication Associative of Addition a + b = b + a ab = ba (a+b)+c = a+(b+c) Associative of Multiplication (a b)c = a(b c) Distributive a(b+c) = ab + ac or ab + ac = a(b+c) The three properties are used to simplify algebraic expressions. The ---------------------- is used to expand groups and remove parentheses from expressions. The ---------------------- is used to change the order of the numbers so that like terms are together. The -----------------------is used to associate or group like terms. Sample Problems: Simplify using the algebraic properties. 1. 3(x - 4) - 2(8 - x) 1. 4(m + n) + 3(2m-5n) 2. 5(2a + b) - 2(3a - 5b) 3. -2(3p - q) + 5(2p + q) 24
Using the Distributive Property 25
Multi-Step Equations 26
Multi-Step Equation Word Problems For each word problem, formulate an equation, solve the equation, and justify the solution. 1. Thirty plus twice a larger number equals 100. Find the number. 2. Two coats and four dresses cost $420. If each dress cost $50, how much do the coats cost each? 3. In his science class, Andrew is required to convert temperatures he collected in degrees Fahrenheit into degrees Celsius. The teacher has said that degrees Celsius are equal to five-ninths of the difference between degrees Fahrenheit and 32. a. What is the general equation that represents degrees Celsius as a function of degrees Fahrenheit? b. If Andrew determined the temperature of his solution is 113 F, what would be the temperature of his solution in degree Celsius? c. If Jaylee measured the temperature of her solution to be 55 C, what would be the temperature of her solution in degrees Fahrenheit? 27
Formulate an equation for the word problems. Solve the equation. Justify the solution. 4. The larger of two numbers is 23. If three times the larger is 5 more than eight times the smaller, find the smaller number. 5. If four times Catherine's age is subtracted from 5 times Jose's age, the difference is 32 years. Jose is sixteen. Find Catherine's age. 6. Charles has been asked to insert a trapezoid in the floor tiling of the rotunda of the courthouse. One base must be 8 feet and the height must be 10 feet. a. What general equation can be used to represent the area of the trapezoid as a function of the second base? b. If Charles wants the area of the trapezoid to be 65 square feet, what should be the measure of the second base? 28
7. Two times the difference of a number decreased by five is fourteen. What is the number? 8. Five times the difference of a number subtracted from ten is forty. What is the number? 9. Four times the sum of three times a number increased by two is equal to three times the difference of two times the number decreased by eight. What is the number? 10. A box contains red and blue chips. There are 2 more blue chips than red chips. If the number of red chips is tripled and the number of blue chips is doubled, there will be an equal number of red and blue chips. How many blue chips does the original box contain? 29
Evaluating Expressions Evaluate the expressions below by substituting the given values for the variables. Assume: a = 3, b = -2, c = 4 for problems #1 through #5 1. 6 a 6 b + 9 c 2. 8 a 8 b + 2 c 3. 10 a 2 b + 4 c 4. 4 a 4 b + 3 c 5. 2 a 5 b + 10 c 4. 3m 2n + 6p for m = ½,n =3, p =1/4 5. 6w + 2v 3u for w = -2, v = ½, u = 4 6. -2x + 3y -4z for x =-1/2, y = 2/3, z = 2 30
Evaluating Expressions 31
Multi-Step Equations 1 Solve for x: 6x + 4 = 3x 5 One half of a number plus 3 is 4. What is the number? 2 more than 3 times a number is 14. What is the number? One third of a number minus 7 is 9. What is the number? 32
Multi-Step Equations 2 Solve for x: 5( 3 x) = 2x + 6 Solve for x: 2 5 (10x 15) = x + 6 Solve for x: 3(x 5) = 2x + 10 Solve for x: 4(2 x) = 6x + 18 33
Equations and the Distributive Property Solve for x: 2(x + 4) = 13 Solve for x: 4 2(x 3) = 30 Solve for x: 4(3 x) = 18 + 2x Solve for x: x 2(x + 4) = 6 3x 34
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Inequality Symbols > means > means < means < means Properties of Inequalities Graphing Inequality Symbols: Practice: For each of the graphs below create the matching inequality statement. 36
Properties of Inequalities Addition Property: Subtraction Property: Multiplication Property: Division Property: Transitive Property: Special Cases: Pick 2 different single, digit positive numbers: (a) and (b) Pick a third different single digit positive number (c) Inequality Symbol a > b a+c > b+c What did you notice when we multiplied or divided by a negative number? 37
Seeing Inequalities Compare the three statements below. What is different about each? What is the same? Write an algebraic expression in terms of x to represent each verbal expression in the box below the statement. Seven less than five times a number Seven is five times a number Seven is less than five times a number Using the algebraic expression for "seven less than five times a number'': A. Evaluate the expression for x = 6. B. Evaluate the expression for x = -2. C. Solve for x if the expression is equal to 7. D. Solve for x if the expression is more than 8. E. Solve for x if the expression is at most 17. F. Solve for x if the expression is at least 18. G. Solve for x if the expression is less than -22. 38
Solving Inequalities Inequalities are solved for x using the same steps as equations EXCEPT if we Solutions can be written verbally, symbolically, and graphically. Justify solutions algebraically by testing a value in the selected interval solution Practice: x is less than two Verbal Svmbolic Graphic x is less than or equal to two x is greater than two x is greater than or equal to two Symbols used in Special Case examples W means all real numbers work, shown graphically as 0 means no solution, no graph since no numbers work 39
Solve these inequalities algebraically 1. 6(x + 2) - 4x > 8 2. 5x 3(x -2) < 7 3. 5(9 -x) > 4(x +18) 4. 4x >-7 +4x Special Case 5.. 5x +7 > 5x -8 Special Case What do both special cases have in common? How do you know if the answer is empty set or all real numbers? 40
One-Step Inequalities Solve each inequality and graph its solution. 1. x 1 <= 8 2. x 8 <= 1 3. x 3 <= 2 4. x 4 >= 4 5. x 1 >= 3 6. x 2 < 0 7. x 7 >= 2 8. x 3 < 4 41
9. x 11 <= 7 10. x 5 <= 7 11. x 13 < 1 12. x 8 < 7 13. ( 10)x < 20 14. 4 x < ( 12) 15. ( 3)x >= ( 15) 16. ( 2)x <= 16 42
Two-Step Inequalities Solve each inequality and graph its solution. 1. 10 x + 4 < 10 2. 7 x + 2 >= 8 3. 8 x + 7 > 1 4. 7 x + 9 >= 8 5. x 5 1 >= 11 6. x 5 3 < 2 7. x 4 9 >= 11 8. x 2 2 > 0 9. ( 12)x 19 <= ( 15) 10. 20 x + 5 > 17 11. 8 x + 6 >= ( 19) 12. 3 x + 8 <= 11 43
Seeing Inequalities Practice Problems Directions: Write an inequality for each of the problem situations below. Solve each inequality, giving the solution in both symbolic and number line form. 1. A number decreased by -4 is at least 9. 2. The sum of twice a number and 5 is less than 17. 3. Five times a number decreased by 10 is greater than -5. 4. Four times a number is at least -48. 5. A number divided by negative six is no less than 5. 6. Charlotte must have at least 320 points in her science class to get a B or better. She needs at least a B so she can play on the softball team. Charlotte currently has 168 points from four tests and 79 points from her quizzes. Charlotte will be taking her final exam worth 100 points. What must Charlotte score on the final exam to reach her goal? 7. What is the width of a rectangle whose length is 4 feet and perimeter is at most 20 feet? 8. Triangle FGH is an obtuse triangle. The greatest angle in the triangle has a measure of (6d) 0 What are the possible values of "d'? 9. Susie has a budget of $92 to spend on clothes. The shorts she wants to buy are on sale for $14 each. The shirts are on sale for $12 each. If Susie purchases four shorts, what is the maximum number of shirts she can buy to go with the shorts? 10. For what values of x will the angle be acute? 44
Practice Problems Solving Inequalities Directions: Solve each inequality as indicated, giving the solution in both symbolic and number line form. Show all work and solutions on paper. 1. 3(x + 4) - 5(x - 1) > 8 2. -4(x 1) + 3x < 4 4. 5x < 5x + 11 5. 13 < 17 -x 6. 5(x + 3) - 2x <-21 7. 3(3x + 1) - (x - 1) > 6(x + 10) 8. 4(2x + 1) < 2(x - 1) + 6(x + 2) 45
Equations and Inequalities with Consecutive Numbers Within a problem situation, verbal phrases identify if the problem is an equation or inequality. Fill in the symbol used to represent the verbal phrases below. Verbal Phrase is is equal to is less than is greater than is at most is at least is no more than is no less than Symbol Because consecutive integers follow specific patterns, they can be used to solve problems. Type Numeric Representation Consecutive integers {5, 6, 7, 8} Consecutive even integers {2, 4, 6, 8} Consecutive odd integers {3, 5, 7, 9} Algebraic Generalization What do you notice about consecutive even or odd integers? Write an equation or inequality for each of the problem situations below. Solve each equation or inequality and give the final solution in set form. 1. The sum of three consecutive integers is 24. What are the integers? 2. The sum of four consecutive integers is at least 126. What is the set of the smallest four consecutive integers that fits this situation? 46
Practice Problems Equations and Inequalities with Consecutive Numbers Directions: Write an equation or inequality for each of the problem situations below. Use the equation or inequality to solve the problem situation. 1. The sum of three consecutive integers is 48. What is the set of the three consecutive integers that fits this situation? 2. The sum of three consecutive even integers is greater than 54. What is the set of the smallest three consecutive even integers that fits this situation? 3. Five decreased by eleven times a number is greater than 71. Evaluate this problem situation for values of the number. 4. Negative four times the difference of six and a number is at least seven times the number. Evaluate this problem situation for values of the number. 5. Four times a number increased by seven must be at least 31 but at most 55. Which interval represents the values of the number in this situation? 6. Change the math sentence, 25 + 27+ 29 2 81, into a verbal phrase using the terminology of consecutive integers. Is there more than one way to write the phrase? If so, give another example. 47
Inequality Word Problems and Multiple Representations 1. Jason has $50 and saves $8 per week. If he needs $200 to buy a Golf club for at least how many weeks does He need to save? Create the inequality that matches this situation and solve algebraically and graphically. 48
2.. Stephen needs to keep his monthly cell bill below $75. If his bill is $25 per month plus $0.15 per text what are the fewest and most texts he can make in a month? Write the inequality that matches this situation and solve it algebraically and create a matching graph. 49
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Inequalities 1 Ryan needs 45 minutes to mow a lawn. He can work for up to 7 hours in a day; write the inequality that shows how many lawns he can mow in a day. Solve for n: 2n 4 < 5n + 2 4 times a number plus 2 is AT MOST 18. Solve the inequality and display it graphically. Write the two inequality statements that match the diagrams above. 51
Inequality Statements Write the algebraic equation or inequality to match the statement: seven less than 5 times a number Write the algebraic equation or inequality to match the statement: seven is 5 times a number Write the algebraic equation or inequality to match the statement: seven is less than 5 times a number Write the algebraic equation or inequality to match the statement: seven is at least 5 times a number 52
Equation Word Problems To rent the theater it costs $100 plus $45 per hour. If the total cost was $280 for how many hours was it rented? Jackie is 5 years older than twice Stephen s age. Stephen is 9 years older than three times Erin s age. If Erin is 4, how old is Jackie? The sum of three consecutive integers is 81. What are the three integers? To rent the limo it costs $40 plus $18 per person. If the total cost was $148 how many people were in the limo? 53
Inequalities 2 Jake has 4 acres of land to fertilize. If a bag can fertilize at most 1/3 of an acre write the inequality statement that represents how many bags she will need. Ariel needs at least $300 to buy the TV she wants. She has $25 already and saves $18 per week. Write the inequality statement that shows for at least how many weeks she needs to reach her goal. Sierra has to put 50 books into boxes. If a box can hold UP TO 6 books, write the inequality statement that shows how many boxes she will need and determine how many boxes she will need. Jason has to make 100 cupcakes. If he can make half a dozen cupcakes in 16 minutes write the inequality that shows the number of minutes it will take him to complete the job (assume he has to make the cupcakes in batches of half a dozen). 54
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