EXPRESSIONS & EQUATIONS 20120803 www.njctl.org 1 Table of - PDF document

Advanced Unit 3: EXPRESSIONS & EQUATIONS Name: _____________________ Advanced Unit 3 EXPRESSIONS & EQUATIONS 2012­08­03 www.njctl.org 1 Table of Contents Commutative and Associative Properties Combining Like Terms Click on a topic to The Distributive Property go to that section. Simplifying Algebraic Expressions Inverse Operations One Step Equations Two Step Equations Multi­Step Equations Distributing Fractions in Equations Translating Between Words and Equations Using Numerical and Algebraic Expressions and Equations Graphing & Writing Inequalities with One Variable Simple Inequalities involving Addition & Subtraction Simple Inequalities involving Multiplication & Division Common Core Standards: 7.EE.1, 7.EE.3, 7.EE.4 2 Commutative and Associative Properties Return to Table of Contents 3

Advanced Unit 3: EXPRESSIONS & EQUATIONS Commutative Property of Addition: The order in which the terms of a sum are added does not change the sum. a + b = b + a Pull Pull 5 + 7 = 7 + 5 12= 12 Commutative Property of Multiplication: The order in which the terms of a product are added does not change the product. ab = ba 4(5) = 5(4) 4 Associative Property of Addition: The order in which the terms of a sum are grouped does not change the sum. (a + b) + c = a + (b + c) (2 + 3) + 4 = 2 + (3 + 4) Pull Pull 5 + 4 = 2 + 7 9 = 9 5 The Associative Property is particularly useful when you are combining integers. Example: ­15 + 9 + (­4)= ­15 + (­4) + 9= Changing it this way allows for the ­19 + 9 = negatives to be added together first. ­10 6

Advanced Unit 3: EXPRESSIONS & EQUATIONS Associative Property of Multiplication: The order in which the terms of a product are grouped does not change the product. 7 1 Identify the property of ­5 + 3 = 3 + (­5) Commutative Property of Addition A Commutative Property of Multiplication B Associative Property of Addition C Associative Property of Multiplication D 8 2 Identify the property of a + (b + c) = (a + c) + b Commutative Property of Addition A Commutative Property of Multiplication B Associative Property of Addition C Associative Property of Multiplication D 9

Advanced Unit 3: EXPRESSIONS & EQUATIONS 3 Identify the property of (3 x ­4) x 8 = 3 x (­4 x 8) Commutative Property of Addition A Commutative Property of Multiplication B Associative Property of Addition C Asociative Property of Multiplication D 10 Discuss why using the associative property would be useful with the following problems: 1. 4 + 3 + (­4) 2. ­9 x 3 x 0 3. ­5 x 7 x ­2 4. ­8 + 1 + (­6) 11 Combining Like Terms Return to Table of Contents 12

Advanced Unit 3: EXPRESSIONS & EQUATIONS An Expression ­ contains numbers, variables and at least one operation. 13 Like terms: terms in an expression that have the same variable raised to the same power Examples: LIKE TERMS NOT LIKE TERMS 6x and 2x 6x 2 and 2x 5y and 8y 5x and 8y 4x 2 and 7x 2 4x 2 y and 7xy 2 14 4 Identify all of the terms like 2x A 5 3x 2 B C 5y D 12y E ­7y 15

Advanced Unit 3: EXPRESSIONS & EQUATIONS 5 Identify all of the terms like 8y A 9y 4y 2 B C 7y D 8 E ­18x 16 6 Identify all of the terms like 8xy A 8x 3x 2 y B C 39xy D 4y E ­8xy 17 7 Identify all of the terms like 2y A 51w B 2x C 3y D 2w E ­10y 18

Advanced Unit 3: EXPRESSIONS & EQUATIONS Identify all of the terms like 14x 2 8 A ­5x 8x 2 B 13y 2 C D x ­x 2 E 19 If two or more like terms are being added or subtracted, they can be combined. To combine like terms add/subtract the coefficient but leave the variable alone. 7x +8x =15x 9v­2v = 7v 20 Sometimes there are constant terms that can be combined. 9 + 2f + 6= 9 + 2f + 6 = 2f + 15 Sometimes there will be both coeffients and constants to be combined. 3g +7 + 8g ­2 11g + 5 Notice that the sign before a given term goes with the number. 21

Advanced Unit 3: EXPRESSIONS & EQUATIONS Try These: 8x + 9x 7y + 5y 2b +6g 3 + 4f + 9f 9j + 3 + 2 4h + 6 + 7h + 3 7a + 4 + 2a ­1 9 + 8c ­12 + 5c 22 9 8x + 3x = 11x True False 23 10 7x + 7y = 14xy True False 24

Advanced Unit 3: EXPRESSIONS & EQUATIONS 11 2x + 3x = 5x True False 25 12 9x + 5y = 14xy True False 26 13 6x + 2x = 8x 2 True False 27

Advanced Unit 3: EXPRESSIONS & EQUATIONS 14 ­15y + 7y = ­8y True False 28 15 ­6 + y + 8 = 2y True False 29 16 ­7y + 9y = 2y True False 30

Advanced Unit 3: EXPRESSIONS & EQUATIONS 17 9x + 4 + 2x = 15x A B 11x + 4 13x + 2x C 9x + 6x D 31 18 12x + 3x + 7 ­ 5 15x + 7 ­ 5 A 13x B 17x C 15x + 2 D 32 19 ­4x ­ 6 + 2x ­ 14 ­22x A ­2x ­ 20 B ­6x +20 C 22x D 33

Advanced Unit 3: EXPRESSIONS & EQUATIONS The Distributive Property Return to Table of Contents 34 An Area Model Imagine that you have two rooms next to each other. Both are 4 feet long. One is 7 feet wide and the other is 3 feet wide . How could you express 4 the area of those two rooms together? 3 7 35 4 7 + 3 You could multiply 4 by 7, then 4 You could add 7 + 3 and then by 3 and add them multiply by 4 OR 4(7) + 4(3) = 4(7+3)= 28 + 12 = 4(10)= 40 40 Either way, the area is 40 feet 2 : 36

Advanced Unit 3: EXPRESSIONS & EQUATIONS An Area Model Imagine that you have two rooms next to each other. Both are 4 yards long. One is 3 yards wide and you don't know how wide the other is. How could you express 4 the area of those two rooms together? x 3 37 You cannot add x and 3 because they aren't like terms, so you can 4 only do it by multiplying 4 by x and 4 by 3 and adding 4(x) + 4(3)= 4x + 12 x + 3 The area of the two rooms is 4x + 12 (Note: 4x cannot be combined with 12) 38 The Distributive Property Finding the area of the rectangles demonstrates the distributive property. Use the distributive property when expressions are written like so: a(b + c) 4(x + 2) 4(x) + 4(2) 4x + 8 The 4 is distributed to each term of the sum (x + 2) 39

Advanced Unit 3: EXPRESSIONS & EQUATIONS Write an expression equivalent to: 5(y + 4) Remember to distribute the 5 to the x and the 3 5(y) + 5(4) 5y + 20 6(x + 2) 3 (x + 4) 4(x ­ 5) 7 (x ­ 1) 40 The Distributive Property is often used to eliminate the parentheses in expressions like 4(x + 2). This makes it possible to combine like terms in more complicated expressions. EXAMPLE: Be careful ­2(x + 3) = ­2(x) + ­2(3) = ­2x + ­6 or ­2x ­ 6 with your 3(4x ­ 6) = 3(4x) ­ 3(6) = 12x ­ 18 signs! ­2 (x ­ 3) = ­2(x) ­ (­2)(­3) = ­2x ­ 6 TRY THESE: 3(4x + 2) = ­1(6m + 4) ­3(2x ­ 5) = 41 Keep in mind that when there is a negative sign on the outside of the parenthesis it really is a ­1. For example: ­(2x + 7) = ­1(2x + 7) = ­1(2x) + ­1(7) = ­2x ­ 7 What do you notice about the original problem and its answer? Remove to see answer. The numbers are turned to their opposites. Try these: ­(9x + 3) = ­(­5x + 1) = ­(2x ­ 4) = ­(­x ­ 6) = 42

Advanced Unit 3: EXPRESSIONS & EQUATIONS 20 4(2 + 5) = 4(2) + 5 True False 43 21 8(x + 9) = 8(x) + 8(9) True False 44 22 ­4(x + 6) = ­4 + 4(6) True False 45

Advanced Unit 3: EXPRESSIONS & EQUATIONS 23 3(x ­ 4) = 3(x) ­ 3(4) True False 46 24 Use the distributive property to rewrite the expression without parentheses 3(x + 4) A 3x + 4 B 3x + 12 C x + 12 D 7x 47 25 Use the distributive property to rewrite the expression without parentheses 5(x + 7) x + 35 A B 5x + 7 5x + 35 C 40x D 48

Advanced Unit 3: EXPRESSIONS & EQUATIONS 26 Use the distributive property to rewrite the expression without parentheses (x + 5)2 A 2x + 5 B 2x + 10 C x + 10 D 12x 49 27 Use the distributive property to rewrite the expression without parentheses 3(x ­ 4) A 3x ­ 4 B x ­ 12 C 3x ­ 12 D 9x 50 28 Use the distributive property to rewrite the expression without parentheses 2(w ­ 6) A 2w ­ 6 B w ­ 12 C 2w ­ 12 D 10w 51

Advanced Unit 3: EXPRESSIONS & EQUATIONS 29 Use the distributive property to rewrite the expression without parentheses ­4(x ­ 9) A ­4x ­ 36 B x ­ 36 C 4x ­ 36 D ­4x + 36 52 30 Use the distributive property to rewrite the expression without parentheses 5.2(x ­ 9.3) A ­5.2x ­ 48.36 B 5.2x ­ 48.36 C ­5.2x + 48.36 D ­48.36x 53 31 Use the distributive property to rewrite the expression without parentheses A B C D 54

Advanced Unit 3: EXPRESSIONS & EQUATIONS We can also use the Distributive Property in reverse. This is called Factoring . When we factor an expression, we find all numbers or variables that divide into all of the parts of an expression. Example: 7x + 35 Both the 7x and 35 are divisible by 7 7(x + 5) By removing the 7 we have factored the problem We can check our work by using the distributive property to see that the two expressions are equal. 55 We can factor with numbers, variables, or both. 2x + 4y = 2(x + 2y) 9b + 3 = 3(3b + 1) ­5j ­ 10k + 25m = ­5(j + 2k ­ 5m) *Careful of your signs 4a + 6a + 8ab = 2a(2 + 3 + 4b) 56 Try these: Factor the following expressions: 6b + 9c = ­2h ­ 10j = 4a + 20ab + 12abc = 57