Unit 3. Linear Equations & Inequalities. Created by: M. Signore & G. Garcia

Transcription

1 Unit 3 Linear Equations & Inequalities Created by: M. Signore & G. Garcia 1

2 Lesson #13: Solving One Step Equations Do Now: 1. Which sentence illustrates the distributive property? a) xy = yx b) x(yz) = (xy)z c) x(y + z) = xy + xz d) 1(xy) = xy 2. What is the additive inverse of -4a? a) 4 a b) 4a c) - 4 a 1 d) - 4a 3] What is the reciprocal of 5? 4] What is the reciprocal of 7 2? 5] In the step-by-step simplification of the expression below, which property is not used? 3(1 + x) 3(x + 1) 3 x a) Commutative b) Distributive c) Associative d) Identity 2

3 Simplify the following shown below: 6] x + x = 7] x x = 8] (3xy 3 z 2 ) 3 9] (x 4) 2 Solving Equations An equation is a mathematical statement that two expressions have the same value, which are separated by an equal sign, =. Algebra is the art of reducing and solving equations. - Al-Khwarizmi Algebra has been used for over 4500 years dating back to ancient Babylon, where assignments were written on clay tablets using the ends of little sticks to make wedge-shaped marks. About 1000 years later, Egyptian students wrote their assignments on papyrus, a parchment-like material that was easier to write on than a clay tablet. The Greek mathematician, Diophantus, introduced his style of writing equations. 3

4 Steps for Solving ALL Equations: 1 st Distribute to get rid of the parentheses, if necessary. 2 nd Combine like terms on the same side of the =. 3 rd Add & subtract to get all constants to one side of the = and all variables to the other side of the =. 4 th Multiply & Divide to get the variable by itself. Simply, undo 5 th Check using substitution. what has been done to the For Solving ALL EQUATIONS, you need to add, subtract, multiply, and/or divide from BOTH sides of the equation in order to ISOLATE the given variable. One-Step Equations: Adding/Subtracting on Both Sides: x + 14 = 21 Check: -13 = -4 + h Check: To Check, substitute your SOLUTION into the ORIGINAL equation! 4

5 QUICK CHECK: Solve the following equations k + 11 = -21 Check: h 26 = -29 Check: 12 + z = -36 Check: 23 = n Check: Multiplying/Dividing on Both Sides: 5y = 30 Check: x 4 Check: 8 To Check, substitute your SOLUTION into the ORIGINAL equation! 5

6 QUICK CHECK: Solve the following equations 4r = -28 Check: 11 = 5 x Check: 14 = a Check: 5x = -45 Check: Independent Practice: Solve and check the following equations x + 2 = 10 Check: 3x = -15 Check: x 3 = -6 Check: -7 = k Check: X Check: -8 = x 7 Check: 6

7 Homework #13: Solving One Step Equations Directions: Solve and check each of the following equations. Show all of your work! 1) Solve y 4 = 11 or 11 = y 4 Check 2) n + 40 = 25 or 40 + n = 25 3) 3q = 10 7

8 Solve Check 4) b = 35 or = 35 5) d = ) 3 k =

9 Lesson #14: Solving Two Step Equations DO NOW: 9

10 Two-Step Equations: Your turn! Be sure to show all of your steps 2x 10 = X = 45 10

11 Independent Practice: 4x + 5 = -27 Check it! x - 20 = k + 15 = -1 3 x 10 =

12 HW #15: Solving Two Step Equations 1) Directions: Solve for x and check. Show all work! Solve 4x + 5 = 29 Check 2) 7 5x = 22 3) 30 x = 40 x 6 4)

13 Solving Two Step Equations Regents Test 13

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16 Lesson #16: Solving Multi-Step Equations DO NOW: Solve and check the following equations: y 2) 61 7 y 26 1) Solving Multi-step equations: A multi-step equation requires you to combine like terms first. 16

17 Guided Example: A) n + 4n - 11 = 19 B) 9 - y + 6y = -6 C) 60-12b + 12 = 0 You Try D) 8x 4x +15 = -5 E) 4a a = 18 17

18 F) -10c + 5-8c = 59 G) y +21 = 53 H) p +3 = 63 I) 3 7a +6a -4 = 8 18

19 Homework #15: Solving Multi-Step Equations 1) 7a +3a +2 a = 20 2) 5m 18-6m = 4 3) -5y -3y +4 = -4 4) 8x 3x -10 = 20 5) 9d -2d +4 = 32 6) 9 = 7z -13z

20 Lesson #16: Solving Equations with the Distributive Property 1) 8 3x x (3x ) 8(1) 56 24x x x 2 SOLVE * Distribute * Keep + sign Replace variable with your answer CHECK (2) x Rewrite Replace Recalculate 2) 3x ) 30 2(10 y) 20

21 SOLVE CHECK 4) 7x ) y ) 1 (8x 6)

22 Homework #16: Solving Equations with the Distributive Property 1) Solve and check each equation! SOLVE 2x CHECK 2) x ) x

23 Lesson #17: Solving Equations with Variables on Both Sides 1) 9x 44 2x 9x 44 2x 2x 2x 11x x 4 SOLVE Steps: 1) Variables on both sides 2) Perform inverse operation 3) Now you only have your variable on one side of the equation CHECK 9x 44 2x 9(4) 44 2(4) Rewrite original Replace solution Recalculate 2) 5 c 28 c 23

24 3) 9 x 72 x 4) 7r 10 3r 50 5) 6 d 12 d 9d 53 d 24

25 Homework #17:Solving Equations with Variables on Both Sides 1) 7n - 3 = n + 27 SOLVE CHECK 2) y y 14 3) 7x 4 5x x 35 25

26 Solving Equations with Variables on Both Sides Regents Test = = = = = = = =

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28 Lesson #18: Solving Literal Equations Do Now: 1] Evaluate x 2 4y 2, when, x = -2 and y = -8. 2] Simplify: (x 2) 2 28

29 A literal equation is an equation that contains two or more variables. You can use inverse operations to solve for one variable in terms of others (aka transforming formulas). What is it being solved for? y = 2x + 3 Is y 2x = 3 equivalent? In Science, m D Formula or Literal equation? v What is it being solved for? Example: Given 2y + 5x = 16, solve for y in terms of x. Isolate 29

30 Example: Given ax + b= c, express a in terms of b and c Isolate Distance Formula: Distance = rate time Example: Given D = r t, solve for the following shown below, Solve for r: Solve for t: Example: Given A = 1 bh, solve for h. 2 Formula for Area Steps: Divide by b A = 1 2 bh 30

31 Temperature: The formula, F= 9 C + 32 gives the temperature F in degrees 5 Fahrenheit in terms of a given temperature C in degrees Celsius. Solve for C in terms of F. January 2011 Regents Questions: ey If k t, what is y in terms of e, n, k, and t? n You Try: 1 3 1) A p prt solve for p 2) V 2 r h solve for r 3) P 2l 2w solve for w cx 4) f b solve for x d 31

32 HW #18: Solving Literal Equations Solve the following for the indicated variable shown below: 1] Solve for x: 2] Solve for p: 3ax + b = c 2m + 2p = 16 3] Solve for p: 4] Solve for y: A = 2 1 p + rt -4x + 2y = 12 5] The formula for potential energy is P = mgh, where P is potential energy, m is mass, g is gravity, and h is height. Express g in terms of P, m, and h 32

33 Literal Equations Regents Test 33

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36 Lesson #19: Translating Algebraic Expressions Do Now: Place the following words in the appropriate boxes Word Bank: more than decreased by take away increased by Diminished total plus minus exceeds Greater than sum difference less than Added to Addition Subtraction 3 more than x the sum of 10 and a number c a number n increased by 4.5 a number t decreased by 4 the difference between 10 and a number y 6 less than a number z Examples Highlight the words that indicate what operation to use and UNDERLINE the numbers. 1. The sum of a number and The difference of a number and less than an number 4. A number increased by 7 36

37 Word Bank: quotient times product divided by multiplied by Twice double of divide triple half Multiplication Division the product of 3 and a number t twice the number x 4.2 times a number e the quotient of 25 and a number b the number y divided by 2 Examples Highlight the words that indicate what operation to use and UNDERLINE the numbers 1. the quotient of a number and 5 3. The product of a number and twice a number plus 8 4. Half of a number minus 1 37

38 You Try: Write an algebraic expression for each verbal expression 1. eight less than a number 2. a number increased by seven 3. the quotient of m and n 4. a number squared 5. nine times a number 6. a number decreased by three 7. seven more than the cube of a number 8. one-half the product of x and y 9. the product of twice a and b *10. twice the product of a and b 11. two less than five times a number 12. twice a number increased by 3 times that number *13. the sum of 3 times a and b *14. three times the sum of a and b Write a verbal expression for each algebraic expression 15. 3x x x y 18. ½ (x + y) 19. x y 2 38

39 Homework #19: Translating Algebraic Expression 39

40 Lesson #20: Coin Word Problems Do Now: 40

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48 Homework #20: Coin Word Problems 48

49 Lesson #21: Consecutive Integer Word Problems Do Now: 1] Simplify: (x 7) 2 2] Solve: 6y (6 + 4y) = 26 3] The sum of two numbers is 84, and one of them is 12 more than the other. What are the two numbers? 49

50 Consecutive Integers Find the pattern and fill in the blank: Consecutive Integer 2, 3,, 5 Expression: What are consecutive numbers?!? Add 2 or Add 1? Consecutive Even Integer 2, 4,, 8 Expression: Add 2 or Add 1? Consecutive Odd Integer 9, 11, 13, Expression:, -10,-8, -6 QUICK CHECK: 1] If x +3 represents an integer, then the next consecutive integer in terms of x is (1) x (2) x +2 (3) x +4 (4) x +5 2] If x - 4 represents an odd integer, then the next consecutive odd integer in terms of x is (1) x - 6 (2) x - 4 (3) x - 3 (4) x

51 Steps for Solving ALL Word Problems: 1 st Assign the variable 2 nd Write an equation 3 rd Solve 4 th Check your solution Consecutive Integer Example: The sum of two consecutive integers is 15. Find the integers Consecutive Integer Example: The sum of three consecutive integers is 99. Find the three integers. Consecutive Integer Example w/ Translation: Find three consecutive numbers whose sum is 9 more than twice the largest number. 51

52 Consecutive Even & Odd Integer Examples: Remember! Consecutive Even Example: Find three consecutive even integers such that their sum is 42 n, n+2, n+4,... Consecutive Odd Example: Find three consecutive odd integers such that their sum is 57. Consecutive Even Example w/ Translation: Find three consecutive odd integers such that twice the sum of the second and the third is 43 more than three times the first. 52

53 Tie-The-KnoTT: Three sisters have ages that are consecutive odd integers. Find the ages if the sum of the age of the youngest and three times the age of the oldest is five less than five times the middle sister s age. U-Try: 1] Find two consecutive integers such that their sum is 89. Only an algebraic solution will be accepted 2] Find three consecutive integers with sum ] Find three consecutive odd integers such that the sum of the first and third equals the sum of the second and

54 HW #21: Consecutive Integer Word Problems 1] If x +2 represents an odd integer, then the next consecutive odd integer in terms of x is (1) x +1 (2) x +3 (3) x +4 (4) x +5 2] Find three consecutive integers with sum 168. Which is the greatest of the three? [A] 59 [B] 57 [C] 19 [D] 53 3] Find three consecutive integers with sum 99. 4] Find three consecutive odd integers such that the sum of the first and third equals the sum of the second and

55 5] Find two consecutive integers such that the larger is nine more than twice the smaller. 6] Three brothers have ages that are consecutive odd integers. The difference between the age of the oldest and twice the age of the youngest is twenty-four less than the middle brother s age. Find their ages 55

56 Lesson #23: Solving and Graphing Inequalities on a Number Line Do Now: 1] Simplify: 2a 3b (b 2 4c +2a) 2] Solve: 2x + 3x 4 = 11 3] Solve: 1 = y + 3(y 9) 4] Simplify: (x 3) 2 5] Solve for x: 3x ] Simplify: (3x 3 y 2 z 4 ) 2 56

57 Solving Linear Inequalities An inequality is a statement that consists of two mathematical expressions joined by an inequality symbol. Fill in the blanks with the appropriate symbols: ( >,,, <, = ) 1] 7 5 2] ] ] ] ]

58 Graphing Inequalities on a Number Line: Exclusive Vs. Inclusive Graph: Graph: Open Circle: x < 2 Graph: x > 3 Graph: Solving Linear Inequalities: Closed Circle: x 2 x > 3 All the rules for solving equations apply to inequalities To solve inequalities, treat them like EQUATIONS! When an inequality is multiplied or divided by ANY NEGATIVE number, then the DIRECTION of the inequality sign CHANGES 58

59 The SOLUTION of an inequality includes ALL values that make the Inequality TRUE = SOLUTION SET Solve and Graph solution set on a number line for the following: b w < 20 3r r (2x + 4) > 4x + 10 Symbol How to Read It Circle s Appearance greater than open (not a solution) less than open (not a solution) greater than or equal to closed (is a solution) less than or equal to closed (is a solution) 59

60 YOU TRY 1) 11 > 4x ) 12 ⅔ x > 6 3) 3(5x + 7) > 81 4) 3(4x + 1) <

61 Writing An Inequality Based On A Graph

62 HW #22: Solving and Graphing Inequalities on a Number Line Directions: Solve and graph the solution set for each of the following: 1] x (-12) < 4 2] 2m + 7 > 17 3] 5(2h 6) 7(h + 7) > 4h 4] 12x 17 > 19 62

63 Solving Inequalities Practice Problems 63

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65 Solving Inequalities Regents Test 65

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67 Lesson #23: Solving and Graphing Compund Inequalities Do Now: Algebraic Symbols Math Symbols Symbol In WORDS Graphed on # Line Open vs. Closed = > Greater Than < Less most 67

68 Compound Inequalities: A compound inequality is two simple inequalities joined by "and" or "or". Compound Inequalities: Solving with AND Solving with OR Graph: Graph: Solution: Solution: Graph the solution set of each compound inequality: 1] x > -3 and x < 4 2] n -5 or n -1 68

69 Solving Compound Inequalities: Solve and graph solution set for the following: 1] -3 < x ] -4 < -6x + 2 < 8 3] In the set of positive integers, what is the solution set of the inequality 2x - 3 < 5? A) {0, 1, 2, 3} B) {1, 2, 3} C) {0, 1, 2, 3, 4} D) {1, 2, 3, 4} REMEMBER: The SOLUTION(S) of an INEQUALITY are ALL the values that make it TRUE! 4] Which of the following is NOT a solution of - 4 < -6x + 2 < 8? A) 0 B) 1 C) 2 D) -1 69

70 Homework #24: Solving and Graphing Compound Inequalities Directions: Solve and graph the solution set for each of the following: 1] x (-12) < 4 2] 2m + 7 > 17 3] -7 < 2x 5 < 7 4] 5(2h 6) 7(h + 7) > 4h 70

71 Inequality Word Problem Practice Use each scenario to write an inequality that can be used to solve each situation: 1. R & G Catering specializes in catering wedding receptions. They charge $550 for setting up the buffet and an additional $6.50 per guest. Mr. and Mrs. Henderson want to spend no more than $1200 on their daughter s wedding reception. Write an inequality in terms of the number of guests, g that they can invite to the wedding reception. 2. Anthony is carpeting several rooms in his home. The carpet costs $14.95 per square yard plus $200 for installation. He can afford to spend no more than $3000. Write an inequality to represent how many square yards of carpet Anthony can afford. 71

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73 Inequality Regents Test 73

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