7th Grade Math. Expressions & Equations. Table of Contents. 1 Vocab Word. Slide 1 / 301. Slide 2 / 301. Slide 4 / 301. Slide 3 / 301.

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1 Slide 1 / 301 Slide 2 / 301 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. 7th Grade Math Expressions & Equations Click to go to website: Inverse Operations One Step Equations Two Step Equations Multi-Step Equations Distributing Fractions in Equations Slide 3 / 301 Table of Contents Commutative and Associative Properties Combining Like Terms The Distributive Property and Factoring Simplifying Algebraic Expressions 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 Glossary Common Core Standards: 7.EE.1, 7.EE.3, 7.EE.4 1 Vocab Word Slide 5 / 301 The charts have 4 parts. Factor A whole number that can divide into another number with no remainder A whole number that multiplies with another number to make a third number. 3 Click on a topic to go to that section. 2 Its meaning 5 R.1 16 (As it is used in the lesson.) Slide 4 / 301 Vocabulary words are identified with a dotted underline. Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. How many thirds are in 1 whole? How many fifths are in 1 whole? How many ninths are in 1 whole? (Click on the dotted underline.) The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall. Slide 6 / 301 Commutative and Associative Properties Return to Table of Contents 3 is a factor of 15 3 Examples/ Counterexamples 3 x 5 = 15 3 and 5 are factors of 15 3 is not a factor of 16 4 Back to Instruction Link to return to the instructional page.

2 Slide 7 / 301 Commutative Property of Addition: The order in which the terms of a sum are added does not change the sum. a + b = b + a = = 12 Commutative Property of Multiplication: The order in which the terms of a product are multiplied does not change the product. ab = ba 4(5) = 5(4) Slide 8 / 301 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) = = 9 Slide 9 / 301 Slide 10 / 301 The Associative Property is particularly useful when you are combining integers. Example: (-4)= 5 + (-4) + 9= Changing it this way allows for the = negatives to be added together first. Associative Property of Multiplication: The order in which the terms of a product are grouped does not change the product. Slide 11 / Identify the property of = 3 + (-5) Slide 12 / Identify the property of a + (b + c) = (a + c) + b A B C D Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication A B C D Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication

3 Slide 13 / Identify the property of (3 x -4) x 8 = 3 x (-4 x 8) A Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Asociative Property of Multiplication Slide 14 / 301 Discuss why using the associative property would be useful with the following problems: (-4) 2. x 3 x x 7 x () Slide 15 / 301 Slide 16 / 301 Combining Like Terms An Expression - contains numbers, variables and at least one operation. Return to Table of Contents Slide 17 / 301 Slide 18 / 301 Like terms: terms in an expression that have the same variable raised to the same power 4 Identify all of the terms like 2x A 5x Examples: LIKE TERMS 6x and 2x NOT LIKE TERMS 6x 2 and 2x B 3x 2 C 5y D 12y E 2 5y and 8y 5x and 8y 4x 2 and 7x 2 4x 2 y and 7xy 2

4 Slide 19 / 301 Slide 20 / Identify all of the terms like 8y 6 Identify all of the terms like 8xy A 9y A 8x B 4y 2 B 3x 2 y C 7y C 39xy D 8 D 4y E 8x E xy Slide 21 / 301 Slide 22 / Identify all of the terms like 2y 8 Identify all of the terms like 14x 2 A 51w A -5x B 2x B 8x 2 C 3y C 13y 2 D 2w D x E y E -x 2 Slide 23 / 301 Slide 24 / 301 If two or more like terms are being added or subtracted, they can be combined. Sometimes there are constant terms that can be combined f + 6= 9 + 2f + 6= To combine like terms add/subtract the coefficient but leave the variable alone. 7x +8x =15x 9vv = 7v 2f + 15 Sometimes there will be both coeffients and constants to be combined. 3g g g + 5 Notice that the sign before a given term goes with the number.

5 Slide 25 / 301 Slide 26 / 301 Try These: 1.) 2b +6g(3) + 4f + 9f 9 8x + 3x = 11x True False 2.) 9j h h ) 7a a 9 + 8c 2 + 5c 4.) 8x + 56xy + 5y Slide 27 / 301 Slide 28 / x + 7y = 14xy 11 2x + 3x = 5x True False True False Slide 29 / 301 Slide 30 / x + 5y = 14xy 13 6x + 2x = 8x 2 True False True False

6 Slide 31 / 301 Slide 32 / y + 7y = y True False 15 + y + 8 = 2y True False Slide 33 / 301 Slide 34 / y + 9y = 2y True False 17 9x x = A 15x B 11x + 4 C 13x + 2x D 9x + 6x Slide 35 / 301 Slide 36 / x + 3x A 15x B 13x C 17x D 15x x x - 14 A 2x B x - 20 C x +20 D 22x

7 Slide 37 / 301 The Distributive Property and Factoring Return to Table of Contents 4 Slide 38 / 301 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. 7 3 How could you express the area of those two rooms together? 4 Slide 39 / 301 Slide 40 / 301 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. You could add and then multiply by 4 4(7+3)= 4(10)= OR Either way, the area is 40 feet 2 : You could multiply 4 by 7, then 4 by 3 and add them 4(7) + 4(3) = = 40 4 x 3 How could you express the area of those two rooms together? Slide 41 / 301 Slide 42 / You cannot add x and 3 because they aren't like terms, so you can only do it by multiplying 4 by x and 4 by 3 and adding 4(x) + 4(3)= 4x + 12 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) x + 3 The area of the two rooms is 4x + 12 (Note: 4x cannot be combined with 12) 4(x + 2) 4(x) + 4(2) 4x + 8 The 4 is distributed to each term of the sum (x + 2)

8 Slide 43 / 301 Write an expression equivalent to: 5(y + 4) 5(y) + 5(4) 5y + 20 Remember to distribute the 5 to the y and the 4 6(x + 2) 3(x + 4) 4(x - 5) 7(x - 1) Slide 44 / 301 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: (x + 3) = (x) + (3) = x + or x - 6 3(4x - 6) = 3(4x) - 3(6) = 12x - 18 (x - 3) = (x) - ()(3) = x + 6 TRY THESE: 3(4x + 2) = (6m + 4) = Be careful with your signs! (2x - 5) = Slide 45 / 301 Slide 46 / 301 Keep in mind that when there is a negative sign on the outside of the parenthesis it really is a. For example: -(2x + 7) = (2x + 7) = (2x) + (7) = x - 7 What do you notice about the original problem and its answer? Try these: -(9x + 3) = -(-5x + 1) = -(2x - 4) = -(-x - 6) = Remove to see answer. The numbers are turned to their opposites. 20 4(2 + 5) = 4(2) + 5 True False Slide 47 / 301 Slide 48 / (x + 9) = 8(x) + 8(9) True False 22-4(x + 6) = (6) True False

9 Slide 49 / 301 Slide 50 / (x - 4) = 3(x) - 3(4) True False 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 Slide 51 / 301 Slide 52 / Use the distributive property to rewrite the expression without parentheses 5(x + 7) 26 Use the distributive property to rewrite the expression without parentheses (x + 5)2 A x + 35 B 5x + 7 C 5x + 35 D 40x A 2x + 5 B 2x + 10 C x + 10 D 12x Slide 53 / 301 Slide 54 / Use the distributive property to rewrite the expression without parentheses 3(x - 4) 28 Use the distributive property to rewrite the expression without parentheses 2(w - 6) A 2w - 6 A 3x - 4 B x - 12 C 3x - 12 D 9x B w - 12 C 2w - 12 D 10w

10 Slide 55 / 301 Slide 56 / Use the distributive property to rewrite the expression without parentheses -4(x - 9) 30 Use the distributive property to rewrite the expression without parentheses 5.2(x - 9.3) A -4x - 36 B x - 36 C 4x - 36 D -4x + 36 A -5.2x B 5.2x C -5.2x D x Slide 57 / 301 Slide 58 / Use the distributive property to rewrite the expression without parentheses A B 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 C D 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. Slide 59 / 301 Slide 60 / 301 We can factor with numbers, variables, or both. Try these: 2x + 4y = 2(x + 2y) Factor the following 9b + 3 = 3(3b + 1) expressions: -5j - 10k + 25m = -5(j + 2k - 5m) *Careful of your signs 4a + 6a + 8ab = 2a( b) 1.) 6b + 9c = 2.) h - 10j = 3.) 4a + 20ab + 12abc =

11 Slide 61 / Factor the following: 4p + 24q A 4 (p + 24q) B 2 (2p + 12q) C 4(p + 6q) D 2 (2p + 24q) Slide 62 / Factor the following: 5g + 15h A 3(g + 5h) B 5(g + 3h) C 5(g + 15h) D 5g (1 + 3h) Slide 63 / Factor the following: 3r + 9rt + 15rx A 3(r+ 3rt + 5rx) B 3r(1 + 3t + 5x) C 3r (3t + 5x) D 3 (r + 9rt + 15rx) Slide 64 / Factor the following: 2v+7v+14v A 7(2v + v + 2v) B 7v( ) C 7v (1 + 2) D v( ) Slide 65 / Factor the following: a - 15ab - 18abc Slide 66 / What divides into the expression: -5n - 20mn - 10np A B C D a(2 + 5b + 6bc) 3a(2+ 5b + 6bc) (2a - 5b - 6bc) a (2-5b - 6bc)

12 Slide 67 / 301 Slide 68 / If a regular pentagon has a perimeter of 10x + 25, what does each side equal? Simplifying Algebraic Expressions Return to Table of Contents Slide 69 / 301 Slide 70 / 301 Now we will use what we know about combining like terms and the distributive property to simplify algebraic expressions. To simplify: 4 + 5(x + 3) First Distribute 4 + 5(x) + 5(3) 4 + 5x + 15 Then combine Like Terms Remember, like terms have the same variable and same exponent. 5x + 19 Notice that when combining like terms, you add/subtract the coefficients but the variable remains the same. Remember that you can combine coefficient or constant terms. Slide 71 / 301 Slide 72 / x +3(x - 4) = 10x - 4 True False (x + 3)5 = 5x + 11 True False

13 Slide 73 / 301 Slide 74 / (x - 3)6 = 6x 4 True False 42 2x + 3y + 5x + 12 = 10xy + 12 True False Slide 75 / x 2 + 2x + 7(x + 1) + x 2 = 6x 2 + 9x + 7 True False Slide 76 / x 3 + 4x 2 + 6(x 2 + 3x) + x = 2x x 2 + 4x True False Slide 77 / 301 Slide 78 / The lengths of the sides of home plate in a baseball field are represented by the expressions in the accompanying figure. yz A B C 5xyz y x x 2 + y 3 z 2x + 3yz x y Which expression represents the perimeter of the figure? 46 A rectangle has a width of x and a length that is double that. What is the perimeter of the rectangle? A 4x B 6x C 8x D 10x D 2x + 2y + yz From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

14 Slide 79 / 301 Slide 80 / 301 What is an equation? An equation is a mathematical statement containing an equal sign to show that two expressions are equal. Inverse Operations Return to Table of Contents 2+3=5 9= = An algebraic equation is just an equation that has algebraic symbols in one or both of the expressions. 4x = h = 15 Slide 81 / 301 Slide 82 / 301 Equations can also be used to state the equality of two expressions containing one or more variables. In real numbers we can say, for example, that for the given value of x it is true that: 4x + 1 = 13 x = 3 because 4(3) + 1 = = = 13 An equation can be compared to a balanced scale. Both sides need to contain the same quantity in order for it to be "balanced". Slide 83 / 301 Slide 84 / 301 For example, = represents an equation because both sides simplify to = = 20 Any of the numerical values in the equation can be represented by a variable. Examples: 15 + c = 25 x + 10 = = y When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).

15 Slide 85 / 301 In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides of the equation. Let's review the inverses of each operation: Addition Subtraction Slide 86 / 301 There are two questions to ask when solving an equation: *What operation is in the equation? *What is the inverse of that operation (This will be the operation you use to solve the equation.)? Multiplication Division Slide 87 / 301 A good phrase to remember when doing equations is: Whatever you do to one side of the equation, you do to the other. For example, if you add three on one side of the equal sign you must add three to the other side as well. Slide 88 / 301 To solve for "x" in the following equation... x + 7 = 32 Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides (in this case, it is subtraction). x + 7 = 32-7 x = 25 In the original equation, replace x with 25 and see if it makes the equation true. x + 7 = = = 32 Slide 89 / 301 Slide 90 / 301 Think about this... For each equation, write the inverse operation needed to solve for the variable. a.) y +7 = 14 subtract move 7 b.) a - 21 = 10 add 21 move c.) 5s = 25 divide move by 5 d.) x = 5 multiply moveby To solve c - 3 = 12 Which method is better? Why? Kendra Added 3 to each side of the equation c - 3 = c = 15 Ted Subtracted 12 from each side, then added 15. c - 3 = c - 15 = c = 15

16 Slide 91 / 301 Slide 92 / What is the inverse operation needed to solve this equation? 2x = What is the inverse operation needed to solve this equation? x - 3 = 2 A B C D Addition Subtraction Multiplication Division A B C D Addition Subtraction Multiplication Division Slide 93 / 301 Slide 94 / What is the inverse operation needed to solve this problem? + x = 9 A B C D Addition Subtraction Multiplication Division One Step Equations Return to Table of Contents Slide 95 / 301 Slide 96 / 301 Examples: To solve equations, you must work backwards through the order of operations to find the value of the variable. Remember to use inverse operations in order to isolate the variable on one side of the equation. Whatever you do to one side of an equation, you MUST do to the other side! y + 3 = 13-3 The inverse of adding 3 is subtracting 3 y = 10 4m = The inverse of multiplying by 4 is dividing by 4 m = 8 Remember - whatever you do to one side of an equation, you MUST do to the other!!!

17 Slide 97 / 301 One Step Equations Solve each equation then click the box to see work & solution. x - 5 = 2 2 = x click to show click to show inverse operation inverse operation x = 7 6 = x 6 = x + 1 x + 5 = 4 click to show -5-5 click to show inverse operation inverse operation 5 = x x = 9 x + 9 = 5 12 = x click to show click to show inverse operation inverse operation x = -4-5 = x 4x = x = 4 click to show inverse operation x = 2 x = 6 click to show inverse operation 0 = 5x 5 5 click to show -4 = x inverse operation Slide 98 / 301 One Step Equations () x (2) = 9 (2) 2 x = 18 click to show inverse operation x = 36 x = 16 click to show inverse operation () Slide 99 / 301 Slide 100 / Solve. x - 7 = Solve. j + 15 = 17 Slide 101 / 301 Slide 102 / Solve. 42 = 6y 53 Solve. 15 = -5x

18 Slide 103 / 301 Slide 104 / Solve. x 9 = Solve. w - 17 = 37 Slide 105 / 301 Slide 106 / Solve. = x 7 57 Solve t = 11 Slide 107 / 301 Slide 108 / Solve. 108 = 12r Sometimes the operation can be confusing. For example: + x = 7 This looks like you should use subtraction to undo the problem. However, + x = 7 is the same as x - 2 = 7 so while it appears to be addition, it is really subtraction. In order to undo this we can add. + x = 7 x - 2 = x = 9 OR + x = 7 - () -() x = 9

19 Slide 109 / 301 Slide 110 / x = 7 = -4 + x = 5 This did not cancel out anything. + x = x = 9 This did cancel out to find the answer. + x = 7 x - 2 = x = 9 This is the same as the middle problem Try these: 1.) -4 + b = 7 2.) + r = 4 3.) + w = 6 4.) -5 + c = 9 Slide 111 / 301 Slide 112 / 301 Think about this... In the expression To which does the "-" belong? 59 Solve Does it belong to the x? The 3? Both? The answer is that there is one negative so it is used once with either the variable or the 3. Generally, we assign it to the 3 to avoid creating a negative variable. So: Slide 113 / 301 Slide 114 / Solve. 61 Solve q = 15

20 Slide 115 / 301 Slide 116 / Solve 63 Solve. Slide 117 / 301 Slide 118 / Solve. 65 Solve. Slide 119 / 301 Sometimes you will have an equation where you are multiplying a variable by a fraction. To undo the fraction you: Slide 120 / 301 Multiply by the reciprocal of the coefficent This means that you will flip the fraction and then multiply **Dividing by a fraction is the same as multiplying by its reciprocal

21 Slide 121 / 301 Slide 122 / Solve. 1 times any number is itself so this is why it can cancel out. Slide 123 / 301 Slide 124 / Solve 68 Solve. Slide 125 / 301 Slide 126 / 301 Two-Step Equations Return to Table of Contents Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable. This means that you undo in the opposite order (PEMDAS): 1st: Addition & Subtraction 2nd: Multiplication & Division 3rd: Exponents 4th: Parentheses Whatever you do to one side of an equation, you MUST do to the other side!

22 Examples: Slide 127 / 301 4x + 2 = Undo addition first 4x = Undo multiplication second x = 2 y - 9 = Undo subtraction first y = -4 Undo multiplication second y = 2 Remember - whatever you do to one side of an equation, you MUST do to the other!!! 5b + 3 = 18 5b = b = 3 x + 3 = - 3 x = -4 x = 2 Slide 128 / 301 Two Step Equations Solve each equation then click the box to see work & solution. w + 6 = 10 2 w 2 = w = 8 m - 4 = m = 26 m = 3 3j - 4 = j = j = 6 +5 = +5 t = Solve the equation. Slide 129 / Solve the equation. Slide 130 / 301 5x - 6 = = 3c + 2 Slide 131 / 301 Slide 132 / Solve the equation. x 5-4 = Solve the equation. 5r - 2 = 2

23 Slide 133 / 301 Slide 134 / Solve the equation. 14 = n Solve the equation. x = 13 Slide 135 / 301 Slide 136 / Solve the equation. 76 Solve the equation. - x = Slide 137 / 301 Slide 138 / Solve the equation. 78 Solve the equation.

24 Slide 139 / 301 Slide 140 / Solve the equation. 80 Solve 5 x + 1 = Slide 141 / 301 Slide 142 / Solve the equation. 82 Solve the equation. Slide 143 / 301 Slide 144 / 301 Multi-Step Equations Return to Table of Contents Steps for Solving Multiple Step Equations As equations become more complex, you should: 1. Simplify each side of the equation. (Combining like terms and the distributive property) 2. Use inverse operations to solve the equation. Remember, whatever you do to one side of an equation, you MUST do to the other side!

25 Examples: Slide 145 / 301 5x + 7x + 4 = 28 12x + 4 = 28 Combine Like Terms -4-4 Undo Addition 12x = Undo Multiplication x = 2 = 2r - 7r +19 = -5r + 19 Combine Like Terms 9 = - 19 Undo Subtraction 0 = -5r -5-5 Undo Multiplication 4 = r 12h - 10h + 7 = 25 h = 9 7q + 7q 3 = 27 q = f + 6 = 140 f = 3 Slide 146 / 301 Try these. Slide 147 / 301 Slide 148 / 301 Always check to see that both sides of the equation are simplified before you begin solving the equation. Sometimes, you need to use the distributive property in order to simplify part of the equation. Remember: The distributive property is a(b + c) = ab + ac Examples 5(20 + 6) = 5(20) + 5(6) 9(30-2) = 9(30) - 9(2) 3(5 + 2x) = 3(5) + 3(2x) Examples: 2(b - 8) = 28 2b - 16 = 28 Distribute the 2 through (b - 8) Undo subtraction 2b = Undo multiplication b = 22 3r + 4(r - 2) = 13 3r + 4r - 8 = 13 Distribute the 4 through (r - 2) 7r - 8 = 13 Combine Like Terms Undo subtraction 7r = Undo multiplication r = 3 (4x - 7) = (4x) - ()(7) Slide 149 / 301 Slide 150 / 301 Try these. 83 Solve. 3(w - 2) = x + x = 25 w = 5 4(2d + 5) = 92 d = 9 6m + 2(2m + 7) = 54 m = 4

26 Slide 151 / 301 Slide 152 / Solve e e = 3 85 Solve. 7 = 8x - 4-2x - 11 Slide 153 / 301 Slide 154 / Solve. n n - 5 = Solve. 32 = f - 3f + 6f Slide 155 / 301 Slide 156 / Solve. 6g - 15g = 8 89 Solve. 3(a - 5) = 1

27 Slide 157 / 301 Slide 158 / Solve. 4(x + 3) = Solve. 3 = 7(k - 2) + 17 Slide 159 / 301 Slide 160 / Solve. 2(p + 7) = 5 93 Solve. 3m m + 3(m) = Slide 161 / 301 Slide 162 / Solve. 95 Solve.

28 Slide 163 / 301 Slide 164 / Solve. Distributing Fractions in Equations Return to Table of Contents Remember... Slide 165 / Simplify each side of the equation. 2. Solve the equation. (Undo addition and subtraction first, multiplication and division second) Remember, whatever you do to one side of an equation, you MUST do to the other side! Slide 166 / 301 There is more than one way to solve an equation with a fraction coefficient. While you can, you don't need to distribute. 3 ( + 3x) = Multiply by the reciprocal Multiply by the LCD ( + 3x) = ( + 3x) = x = x = x = ( + 3x) = ( + 3x) = ( + 3x) = x = x = x = 9 Slide 167 / 301 Some problems work better when you multiply by the reciprocal and some work better multiplying by the LCD. 97 Solve. Slide 168 / 301 Which strategy would you use for the following? Why?

29 Slide 169 / 301 Slide 170 / Solve. 99 Solve. 2 3 (8-3c) = 16 3 Slide 171 / 301 Slide 172 / Solve. 101 Solve. Slide 173 / 301 Slide 174 / 301 List words that indicate addition Translating Between Words and Expressions PULL Return to Table of Contents

30 Slide 175 / 301 List words that indicate subtraction Slide 176 / 301 List words that indicate multiplication PULL PULL Slide 177 / 301 List words that indicate division Slide 178 / 301 List words that indicate equals PULL PULL Slide 179 / 301 Slide 180 / 301 Be aware of the difference between "less" and "less than". For example: "Eight less three" and "Three less than Eight" are equivalent expressions. So what is the difference in wording? Eight less three: 8-3 Three less than eight: 8-3 When you see "less than", you need to switch the order of the numbers. As a rule of thumb, if you see the words "than" or "from" it means you have to reverse the order of the two items on either side of the word. Examples: 8 less than b means b more than x means x + 3 x less than 2 means 2 - x click to reveal

31 Slide 181 / 301 The many ways to represent multiplication... How do you represent "three times a"? (3)(a) 3(a) 3 a 3a The preferred representation is 3a When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable. Slide 182 / 301 Representation of division... How do you represent "b divided by 12"? b 12 b 12 b 12 The following are not allowed: 3xa... The multiplication sign looks like another variable a3... The number is always written in front of the variable Slide 183 / 301 Slide 184 / 301 When choosing a variable, there are some that are often avoided: l, i, t, o, O, s, S Why might these be avoided? It is best to avoid using letters that might be confused for numbers or operations. In the case above (1, +, 0, 5) TRANSLATE THE WORDS INTO AN ALGEBRAIC EXPRESSION Three times j Eight divided by j j less than 7 5 more than j 4 less than j j Slide 185 / 301 Slide 186 / 301 Write the expression for each statement. Then check your answer. Write the expression for each statement. Then check your answer. The sum of twenty-three and m Twenty-four less than d 23 + m d - 24

32 Slide 187 / 301 Slide 188 / 301 Write the expression for each statement. Remember, sometimes you need to use parentheses for a quantity. Write the expression for each statement. Then check your answer. Four times the difference of eight and j The product of seven and w, divided by 12 4(8-j) 7w 12 Slide 189 / 301 Slide 190 / 301 Write the expression for each statement. Then check your answer. 102 The quotient of 200 and the quantity of p times 7 The square of the sum of six and p A 200 7p B (7p) (6+p) 2 C D 200 7p 7p 200 Slide 191 / 301 Slide 192 / multiplied by the quantity r less Mary had 5 jellybeans for each of 4 friends. A 35r - 45 B 35(45) - r C 35(45 - r) D 35(r - 45) A 5+4 B 5-4 C 5 x 4 D 5 4

33 Slide 193 / 301 Slide 194 / If n + 4 represents an odd integer, the next larger odd integer is represented by 106 a less than 27 A n + 2 B n + 3 C n + 5 D n + 6 A B 27 - a a 27 C a - 27 D 27 + a From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, Slide 195 / If h represents a number, which equation is a correct translation of: Sixty more than 9 times a number is 375? A 9h = 375 B 9h + 60 = 375 C 9h - 60 = 375 D 60h + 9 = 375 Slide 196 / 301 Using Numerical and Algebraic Expressions and Equations Return to Table of Contents From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, Slide 197 / 301 Slide 198 / 301 We can use our algebraic translating skills to solve other problems. We can use a variable to show an unknown. A constant will be any fixed amount. If there are two separate unknowns, relate one to the other. The school cafeteria sold 225 chicken meals today. They sold twice the number of grilled chicken sandwiches than chicken tenders. How many of each were sold? chicken sandwiches 2c + c = 225 chicken tenders total meals c + 2c = 225 3c = c = 75 The cafeteria sold 150 grilled chicken sandwiches and 75 tenders.

34 Slide 199 / Julie is matting a picture in a frame. Her frame is 9 2 inches wide and her picture is 7 inches wide. How much matting should she put on either side? both sides of the mat 2m + 7 = 9 size of picture 1 2 size of frame 1 2m + 7 = m = m = 4 1 Julie needs 14inches on each side. Slide 200 / 301 Many times with equations there will be one number that will be the same no matter what (constant) and one that can be changed based on the problem (variable and coefficient). Example: George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all? Slide 201 / 301 George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all? Notice that the video games are "per game" so that means there could be many different amounts of games and therefore many different prices. This is shown by writing the amount for one game next to a variable to indicate any number of games. 30g Slide 202 / 301 George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all? Notice also that there is a specific amount that is charged no matter what, the flat fee. This will not change so it is the constant and it will be added (or subtracted) from the other part of the problem. 30g + 7 cost of one video game number of games cost of one video game number of games the cost of the shipping Slide 203 / 301 George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all? "Total" means equal so here is how to write the rest of the equation. 30g + 7 = 127 cost of one video game number of games the cost of the shipping the total amount Slide 204 / 301 George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all? Now we can solve it. 30g + 7 = g = g = 4 George bought 4 video games.

35 Slide 205 / Lorena has a garden and wants to put a gate to her fence directly in the middle of one side. The whole length of the fence is 24 feet. If the gate is 4 1feet, 2 how many feet should be on either side of the fence? Slide 206 / Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. Which equation shows this problem? A 12p + 27 = 147 B 12p + 27p = 147 C 27p + 12 = 147 D 39p = 147 Slide 207 / Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. How many people went to the amusement park WITH Lewis? Slide 208 / Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle? Which equation represents the situation? A = 239 B 9d + 68 = 239 C 68d + 9 = 239 D 77d = 239 Slide 209 / 301 Slide 210 / Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle? 113 You are selling t-shirts for $15 each as a fundraiser. You sold 17 less today then you did yesterday. Altogether you have raised $675. Write and solve an equation to determine the number of t-shirts you sold today. Be prepared to show your equation!

36 Slide 211 / 301 Slide 212 / Rachel bought $12.53 worth of school supplies. She still needs to buy pens which are $2.49 per pack. She has a total of $20.00 to spend on school supplies. How many packs of pens can she buy? Write and solve an equation to determine the number of packs of pens Rachel can buy. Be prepared to show your equation! 115 The length of a rectangle is 9 cm greater than its width and its perimeter is 82 cm. Write and solve an equation to determine the width of the rectangle. Be prepared to show your equation! Slide 213 / 301 Slide 214 / The product of -4 and the sum of 7 more than a number is 6. Write and solve an equation to determine the number. Be prepared to show your equation! 117 A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200. Write and solve an equation to determine the number of subscribers they had each year. Be prepared to show your equation! How many subscribers last year? Slide 215 / 301 Slide 216 / A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200. Write and solve an equation to determine the number of subscribers they had each year. Be prepared to show your equation! 119 The perimeter of a hexagon is 13.2 cm. Write and solve an equation to determine the length of a side of the hexagon. Be prepared to show your equation! How many subscribers this year?

37 Slide 217 / 301 Slide 218 / 301 Graphing and Writing Inequalities with One Variable Return to Table of Contents When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. Important Words is more than Sample Sentence Trenton is more than 10 miles away. Equivalent Translation t > 10 is greater than must exceed A is greater than B. The speed must exceed 25 mph. The speed is greater than 25 mph. A > B s > 25 Slide 219 / 301 When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. How are these inequalities read? Slide 220 / > 3 Two plus two is greater than 3 Important Words Sample Sentence Equivalent Translation > 3 Two plus two is greater than or equal to 3 cannot exceed Time cannot exceed 60 minutes. Time must be less than or equal to 60 minutes. t < Two plus two is greater than or equal to < 5 Two plus two is less than 5 is at most At most, 7 students were late for class. Seven or fewer students were late for class. n < Two plus two is less than or equal to Two plus two is less than or equal to 4 is at least Bob is at least 14 years old. Bob's age is greater than or equal to 14. B > 14 Slide 221 / 301 Slide 222 / 301 Writing inequalities Let's translate each statement into an inequality. x is less than 10 words translate to You try a few: is greater than a 2. b is less than or equal to is less than the product of f and The sum of t and 9 is greater than or equal to 36 Answers x < 10 inequality statement 5. 7 more than w is less than or equal to decreased by p is greater than or equal to 2 20 is greater than or equal to y 20 > y 7. Fewer than 12 items 8. No more than 50 students 9. At least 275 people attended the play

38 Slide 223 / 301 Do you speak math? Change the following expressions from English into math. Slide 224 / 301 Five less than a number is less than twice number. x - Answer 5 < 2x that Double a number is at most four. Three plus a number is at least six. 2x Answer Answer x 6 The sum of two consecutive numbers is at least thirteen. x + (x Answer + 1) 13 Three times a number plus seven is at least nine. 3x + Answer 7 > 9 Slide 225 / 301 A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Let e represent an employee's wages. Try this: Slide 226 / 301 The speed limit on a road is 55 miles per hour. Define a variable and write an inequality for the speed of all cars on the road. An employee earns at least $7.50 Answer e > 7.5 Slide 227 / 301 Slide 228 / You have $200 to spend on clothes. You already spent $140 and shirts cost $12. Which equation shows this scenario? 121 A sea turtle can live up to 125 years. If one is already 37 years old, which scenario shows how many more years could it live? A 200 < 12x B x C 200 > 12x D x A 125 < 37 + x B x C 125 > 37 + x D x

39 Slide 229 / 301 Slide 230 / The width of a rectangle is 3 in longer than the length. The perimeter is no less than 25 inches. A 4a + 6 < 25 B 4a C 4a + 6 > 25 D 4a The absolute value of the sum of two numbers is less than or equal to the sum of the absolute values of the same two numbers. A B C D Slide 231 / 301 Slide 232 / 301 Solution Sets Let's name the numbers that are solutions of the given inequality. A solution to an inequality is NOT a single number. It will have more than one value. r > 10 Which of the following are solutions? {5, 10, 15, 20} 5 > 10 is not true So, not a solution 10 > 10 is not true So, not a solution 15 > 10 is true So, 15 is a solution 20 > 10 is true So, 20 is a solution This would be read as the solution set is all numbers greater than or equal to negative 5. Answer: {15, 20} are solutions of the inequality r > d 30 (4) d 30 (4) Slide 233 / 301 Let's try another one. 30 4d; {3, 4, 5, 6, 7, 8} 30 4d 30 (4) d 30 (4) d 30 (4) click to reveal click to reveal click to reveal click to reveal click to reveal 30 4d 30 (4) click to reveal Slide 234 / 301 Graphing Inequalities - The Circle An open circle on a number shows that the number is not part of the solution. It is used with "greater than" and "less than". The word equal is not included. < > A closed circle on a number shows that the number is part of the solution. It is used with "greater than or equal to" and "less than or equal to". < >

40 Slide 235 / 301 Graphing Inequalities - The Arrow The arrow should always point in the direction of those numbers who satisfy the inequality. *If the variable is on the left side of the inequality, then < and will show an arrow pointing left. *If the variable is on the left side of the inequality, then > and will show an arrow pointing right. Slide 236 / 301 Notice that < and look like an arrow pointing left and that > and look like an arrow pointing right. But what if the variable isn't on the left? Do the opposite of where the inequality symbol points Slide 237 / 301 Slide 238 / 301 Graphing Inequalities What is the number in the inequality? What kind of circle should be used? In what direction does the line go? Graphing Inequalities x is less than 5 Step 1: Rewrite this as x < 5. Step 2: What kind of circle? Because it is less than, it does not include the number 5 and so it is an open circle Slide 239 / 301 x < 5 Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality Slide 240 / 301 Graphing Inequalities x is less than or equal to 5 Step 1: Rewrite this as x 5. Step 2: What kind of circle? Because it is less than or equal to, it does include the number 5 and so it is a closed circle

41 Slide 241 / 301 Slide 242 / 301 x 5 You try Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality. Graph the inequality x > 2 Graph the inequality > x click 2 on the number line for answer click on the number line for answer Slide 243 / 301 Slide 244 / 301 Try these. Graph the inequalities. 1. x > Try these. State the inequality shown x < Slide 245 / 301 Slide 246 / This solution set would be x > State the inequality shown. True False A x > 3 B x < 3 C x < 3 D x > 3

42 Slide 247 / 301 Slide 248 / State the inequality shown. 127 State the inequality shown. A B C D < x > x 11 > x 11 < x A x > B x < C x < D x > Slide 249 / 301 Slide 250 / State the inequality shown. 129 State the inequality shown A -4 < x B -4 > x C -4 < x A x > 0 B x < 0 D -4 > x C x < 0 D x > 0 Slide 251 / 301 Slide 252 / 301 Who remembers how to solve an algebraic equation? Simple Inequalities Involving Addition and Subtraction Return to Table of Contents x + 3 = x = 10 Use the inverse of addition Does Be sure 10 + to 3 check = 13 your answer! 13 = 13

43 Slide 253 / 301 Slide 254 / 301 Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations. Remember, whatever you do to one side, you do to the other. To find the solution, isolate the variable x. Remember, it is isolated when it appears by itself on one side of the equation. 12 > x Subtract to undo addition 7 > x Slide 255 / 301 Slide 256 / > x Solve and graph. The symbol is > so it is an open circle and it is numbers less than 7 so it goes to the left. A. j + 7 > A. j + 7 > j > is not included in solution set; therefore we graph with an open circle. Slide 257 / 301 Slide 258 / 301 Solve and graph. Solve and graph. B. r - 2 > 4 r - 2 > r > 6 C. 9 > w > w > w w <

44 Slide 259 / 301 Slide 260 / Solve the inequality. 3 < s Solve the inequality and graph the solution b < < s A B C D Slide 261 / 301 Slide 262 / Solve the inequality and graph the solution. > b Solve the inequality. m < 9.6 m < A B C D Slide 263 / 301 Slide 264 / 301 Multiplying or Dividing by a Positive Number Simple Inequalities Involving Multiplication and Division Return to Table of Contents 3x > 6 3x > x > 2 Since x is multiplied by 3, divide both sides by 3 for the inverse operation.

45 Slide 265 / 301 Slide 266 / 301 Solve the inequality k > 18 3 ( 2) 2 3 r < r < r < 6 ( ) Since r is multiplied by 2/3, multiply both sides by the reciprocal of 2/3. A B C D Slide 267 / 301 Slide 268 / > 3q 136 X 2 < A 10 > q A B C D < q > q 10 < q B C D Slide 269 / 301 Slide 270 / g > > 3d A g > 36 A d > B g > 108 B d > C g > 36 C d < D g > 108 D d <

46 Slide 271 / 301 Slide 272 / 301 Now let's see what happens when we multiply or divide by negative numbers. 1. Write down two numbers and put the appropriate inequality (< or >) between them. Sometimes you must multiply or divide to isolate the variable. Multiplying or dividing both sides of an inequality by a negative number gives a surprising result. 2. Apply each rule to your original two numbers from step 1 and simplify. Write the correct inequality(< or >) between the answers. A. Add 4 B. Subtract 4 C. Multiply by 4 D. Multiply by -5 E. Divide by 4 F. Divide by -4 Slide 273 / 301 Slide 274 / What happened with the inequality symbol in your results? Let's see what happens when we multiply this inequality by. 4. Compare your results with the rest of the class. 5. What pattern(s) do you notice in the inequalities? How do different operations affect inequalities? Write a rule for inequalities. 5 > 5? -5? 1-5 < 1 We know 5 is greater than Multiply both sides by Is -5 less than or greater than 1? You know -5 is less than 1, so you should use < What happened to the inequality symbol to keep the inequality statement true? Slide 275 / 301 Slide 276 / 301 Helpful Hint Solve and graph. A. y > 18 The direction of the inequality changes only if the number you are using to multiply or divide by is negative. y < 18 y < Dividing each side by changes the > to <.

47 Slide 277 / 301 Slide 278 / 301 Solve and graph. Solve and graph. B. m > 8 m < - 28 m < 4 Divide each side by Flip the sign because you divided by a negative. C. 5m > 5 5m > m > -5 Divide each side by 5. The sign does NOT change because you did not divide by a negative. Slide 279 / 301 Slide 280 / 301 Solve and graph. D. y > 32 () -r 2 < 5 ( ) -r 2 > 5 Multiply both sides by the reciprocal of /2. E. f > -54 r > Why did the inequality change? You multiplied by a negative. Try these. Solve and graph each inequality. Slide 281 / 301 Try these. Solve and graph each inequality. Slide 282 / h < m < x > 0 a 4. >

48 Slide 283 / 301 Slide 284 / Solve and graph. 3y < If you can't put the inequality in your responder, just put the number. 140 Solve and graph. x -4 < If you can't put the inequality in your responder, just put the number. Slide 285 / 301 Slide 286 / Solve and graph. -5y 5 If you can't put the inequality in your responder, just put the number. 142 Solve and graph. n > 2 If you can't put the inequality in your responder, just put the number. Slide 287 / 301 Slide 288 / 301 Associate Property of Addition Glossary The order in which the terms of a sum are grouped does not change the sum. Return to Table of Contents (a + b) + c = a + (b + c) (2 + 3) + 4 = = (3 + 4) = = 9 (2 + 3) + 4 = 2 + (3 + 4) = = 9 Back to Instruction

49 Slide 289 / 301 Associate Property of Multiplication The order in which the terms of a product are grouped does not change the product. Slide 290 / 301 Commutative Property of Addition The order in which the terms of a sum are added does not change the sum. (a b) c = a (b c) (3 4) 5 = 12 5 = 60 3 (4 5) = 3 20 = 60 (3 4) 5 = 3 (4 5) 12 5 = = 60 a + b = b + a = = = = 5 Back to Back to Instruction Instruction Slide 291 / 301 Commutative Property of Multiplication The order in which the terms of a product are multiplied does not change the product. Slide 292 / 301 Distributive Property For all real numbers a, b, c, a(b+c) = ab + ac and a(b - c) = ab - ac. ab = ba 2 3 = = = = 6 Back to Instruction a(b + c) = ab + ac a(b - c) = ab - ac 3(2 + 4) = (3)(2) + (3)(4) = = 18 3(2-4) = (3)(2) - (3)(4) = 6-12 = 3(x + 4) = 48 (3)(x) + (3)(4) = 48 3x + 12 = 48 3x = 36 x = 12 Back to Instruction Slide 293 / 301 Equation A mathematical statement containing an equal sign to show that two expressions are equal. Slide 294 / 301 Expression Numbers, symbols and operators grouped together that show the value of something = 3 22 = = 13 7x = 21 (where x = 3) 3y + 2 = 11 (where y = 3) 11-1 = 3z + 1 (where z = 3) = 90 5 = x + 6 = 11 (where x = 3) Back to 7x 3y b -0.5a 7 x 6 7x = = 3y = 3z + 1 Remember! 7x "7 times x" "7 divided by x" Back to Instruction Instruction

50 The distributive property in reverse. 7x + 35 = = 1 7 x Slide 295 / 301 Factoring To find all numbers or variables that divide into all of the parts of an expression. 2x + 4y = = 7(x + 5) 1 2 x y 2(x + 2y) Slide 297 / x + 8z cannot be factored (terms have nothing in common). PRIME Back to Instruction Addition + _ Subtraction Multiplication x Division Slide 296 / 301 Inverse The operation that reverses the effect of another operation x = x = 10 Slide 298 / = 3y = 3y = y Back to Instruction Like Terms Terms in an expression that have the same variable and raised to the same power. 3x x 1/2x 5x 15.7x.3x x 3 x 3 27x 3 1/4x 3 5 5x 3 NOT LIKE TERMS! 5x 5x 2 5x 4-5x 3 2.7x 3 Back to Back to Instruction Instruction Slide 299 / 301 Slide 300 / 301 Back to Instruction Back to Instruction

51 Slide 301 / 301 Back to Instruction