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Expressions & Equations
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Setting the PowerPoint View
Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. On the View tab, click Slide Master | Page Setup. Select On-screen Show (4:3) under Slide sized for and click Close Master View. On the Slide Show menu, confirm that Resolution is set to 1024x768. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 10 for an example.)
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Click on a topic to go to that section.
Table of Contents Commutative and Associative Properties Combining Like Terms Click on a topic to go to that section. The Distributive Property and Factoring Simplifying Algebraic Expressions Inverse Operations One Step Equations Two Step Equations Multi-Step 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
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Commutative and Associative Properties
Return to table of contents
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Commutative Property of Addition: The order in which the terms of a sum are added does not change the sum. a + b = b + a 5 + 7 = 7 + 5 12= 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) Teacher’s instructions: When you commute to work you are going back and forth but it is the same route. So it is the same backwards and forwards.
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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) 5 + 4 = 2 + 7 9 = 9 Teacher’s instructions: You are at a party. You associate with Alan and Barbara and then with Chris. This is the same as associating with Barbara and Chris and then Alan, just in a different order.
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The Commutative Property is particularly useful when you are combining integers.
Example: (-4)= -15 + (-4) + 9= Changing it this way allows for the = negatives to be added together first. -10
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Associative Property of Multiplication: The order in which the terms of a product are grouped does not change the product.
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Identify the property of -5 + 3 = 3 + (-5)
1 Identify the property of = 3 + (-5) A Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Associative Property of Multiplication Answer: A
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Identify the property of a + (b + c) = (a + c) + b
2 Identify the property of a + (b + c) = (a + c) + b A Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Associative Property of Multiplication Answer: C
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Identify the property of (3 * (-4)) * 8 = 3 * ((-4) * 8)
A Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Asociative Property of Multiplication Answer: D
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Discuss why using the Commutative Property would be useful with the following problems:
(-4) x 3 x 0 x 7 x -2 (-6)
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Combining Like Terms Return to table of contents
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An Algebraic Expression - contains numbers, variables and at least one operation.
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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 x2 and 2x 5y and 8y x and 8y 4x2 and 7x x2y and 7xy2
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Identify all of the terms like 2x
4 Identify all of the terms like 2x A 5x B 3x2 C 5y D 12y E 2 Answer: A
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Identify all of the terms like 8y
5 Identify all of the terms like 8y A 9y B 4y2 C 7y D 8 E -18x Answer: A, C
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Identify all of the terms like 8xy
6 Identify all of the terms like 8xy A 8x B 3x2y C 39xy D 4y E -8xy Answer: C, E
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Identify all of the terms like 2y
7 Identify all of the terms like 2y A 51w B 2x C 3y D 2w E -10y Answer: C, E
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Identify all of the terms like 14x2
8 Identify all of the terms like 14x2 A -5x B 8x2 C 13y2 D x E -x2 Answer: B, E
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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 9v-2v = 7v
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Sometimes there are constant terms that can be combined. 9 + 2f + 6 =
Sometimes there will be both coeffients and constants to be combined. 3g g - 2 11g + 5 Notice that the sign before a given term goes with the number.
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Try These: 1.) 2b +6g(3) + 4f + 9f 2.) 9j + 3 + 24h + 6 + 7h + 3
3.) 7a a c c 4.) 8x + 56xy + 5y Answers: 1. 2b+18g +13f 2. 9j+31h+12 3. 9a-27+13c 4. 8x +56xy+5y
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9 8x + 3x = 11x A True B False Answer: A True
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10 7x + 7y = 14xy A True B False Answer: B False
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11 2x + 3x = 5x A True B False Answer: A True
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12 9x + 5y = 14xy A True B False Answer: B False
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13 6x + 2x = 8x2 A True B False Answer: B False
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14 -15y + 7y = -8y A True B False Answer: A True
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15 -6 + y + 8 = 2y A True B False Answer: B False
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16 -7y + 9y = 2y A True B False Answer: A True
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17 9x x = A 15x B 11x + 4 C 13x + 2x D 9x + 6x Answer: B
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18 12x + 3x A 15x B 13x C 17x D 15x + 2 Answer: D
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19 -4x x - 14 A -22x B -2x - 20 C -6x +20 D 22x Answer: B
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The Distributive Property
and Factoring Return to table of contents
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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 the area of those two rooms together? 4 7 3
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Either way, the area is 40 feet2:
7 + 3 4 4 7 3 You could multiply 4 by 7, then 4 by 3 and add them 4(7) + 4(3) = = 40 You could add and then multiply by 4 4(7+3)= 4(10)= 40 OR Either way, the area is 40 feet2:
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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 the area of those two rooms together? 4 x 3
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The area of the two rooms is
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 area of the two rooms is (Note: 4x cannot be combined with 12) 4 x + 3
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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)
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Write an expression equivalent to: 5(y + 4) 5(y) + 5(4) 5y + 20
6(x + 2) 3(x + 4) 4(x - 5) 7(x - 1) Remember to distribute the 5 to the y and the 4
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-2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6
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: -2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6 3(4x - 6) = 3(4x) - 3(6) = 12x - 18 -2 (x - 3) = -2(x) - (-2)(3) = -2x + 6 TRY THESE: 3(4x + 2) = -1(6m + 4) = -3(2x - 5) = Be careful with your signs!
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-(2x + 7) = -1(2x + 7) = -1(2x) + -1(7) = -2x - 7
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) =
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20 4(2 + 5) = 4(2) + 5 A True B False Answer: B False
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21 8(x + 9) = 8(x) + 8(9) A True B False Answer: A True
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22 -4(x + 6) = (6) A True B False Answer: B False
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23 3(x - 4) = 3(x) - 3(4) A True B False Answer: A True
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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 Answer: B
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25 Use the distributive property to rewrite the expression without parentheses 5(x + 7) A x + 35 B 5x + 7 C 5x + 35 D 40x Answer: C
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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 Answer: B
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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 Answer: C
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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 Answer: C
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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 Answer: D
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30 Use the distributive property to rewrite the expression without parentheses 5.2(x - 9.3) A -5.2x B 5.2x C -5.2x D -48.36x Answer: B
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31 Use the distributive property to rewrite the expression without parentheses A B C D Answer: D
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We can also use the Distributive Property in reverse
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.
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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( b)
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Factor the following expressions: 1.) 6b + 9c = 2.) -2h - 10j =
Try these: Factor the following expressions: 1.) 6b + 9c = 2.) -2h - 10j = 3.) 4a + 20ab + 12abc = Answers: 1. 3(2b+3c) 2. -2(h+5j) 3. 4a(1+5b+3bc)
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Factor the following: 4p + 24q
32 Factor the following: 4p + 24q A 4 (p + 24q) B 2 (2p + 12q) C 4(p + 6q) D 2 (2p + 24q) Answer: C
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Factor the following: 5g + 15h
33 Factor the following: 5g + 15h A 3(g + 5h) B 5(g + 3h) C 5(g + 15h) D 5g (1 + 3h) Answer: B
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Factor the following: 3r + 9rt + 15rx
34 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) Answer: B
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Factor the following: 2v+7v+14v
35 Factor the following: 2v+7v+14v A 7(2v + v + 2v) B 7v( ) C 7v (1 + 2) D v( ) Answer: D
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Factor the following: -6a - 15ab - 18abc
36 Factor the following: -6a - 15ab - 18abc A -3a(2 + 5b + 6bc) B 3a(2+ 5b + 6bc) C -3(2a - 5b - 6bc) D -3a (2 -5b - 6bc) Answer: A
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What divides into the expression: -5n - 20mn - 10np
Answer: -5n
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- If a regular pentagon has a perimeter of 10x + 25, what does each side equal? Answer: 2x + 5
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Simplifying Algebraic Expressions
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Now we will use what we know about combining like terms and the distributive property to simplify algebraic expressions. Remember, like terms have the same variable and same exponent.
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To simplify: 4 + 5(x + 3) First Distribute 4 + 5(x) + 5(3) 4 + 5x Then combine Like Terms 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.
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37 7x +3(x - 4) = 10x - 4 A True B False Answer: B False
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38 8 +(x + 3)5 = 5x + 11 A True B False Answer: B False
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39 4 +(x - 3)6 = 6x -14 A True B False Answer: A True
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40 2x + 3y + 5x + 12 = 10xy + 12 A True B False Answer: B False
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41 5x2 + 2x + 7(x + 1) + x2 = 6x2 + 9x + 7 A True B False
Answer: A True
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42 2x3 + 4x2 + 6(x2 + 3x) + x = 2x3 + 10x2 + 4x A True B False
Answer: B False
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Which expression represents the perimeter of the figure?
43 The lengths of the sides of home plate in a baseball field are represented by the expressions in the accompanying figure. yz y y Which expression represents the perimeter of the figure? x x A 5xyz B x2 + y3z Answer: D C 2x + 3yz 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.
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44 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 Answer: B
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Inverse Operations Return to table of contents
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What is an equation? An equation is a mathematical statement containing an equal sign to show that two expressions are equal. 2 + 3 = 5 9 – 2 = 7 5 + 3 = 1 + 7 An algebraic equation is just an equation that has algebraic symbols in one or both of the expressions. 4x = 24 9 + h = 15
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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 any given value of x it is true that 4x + 1 = 13 x = 3 because 4(3) + 1 = 13 = 13 13 = 13
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An equation can be compared to a balanced scale.
Both sides need to contain the same quantity in order for it to be "balanced".
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For example, 9+ 11 = 6 + 14 represents an equation because both sides simplify to 20.
= 20 = 20 Any of the numerical values in the equation can be represented by a variable. Examples: 15 + c = 25 x + 10 = 25 = y
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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).
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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 Multiplication Division Square Square Root
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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.)?
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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… to keep the equation in balance.
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To solve for "x" in the following equation...
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 = 25 In the original equation, replace x with 25 and see if it makes the equation true. x + 7 = 32 = 32 32 = 32
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a.) y +7 = 14 subtract 7 b.) a - 21 = 10 add 21
For each equation, write the inverse operation needed to solve for the variable. a.) y +7 = 14 subtract b.) a - 21 = add 21 c.) 5s = divide by 5 d.) x = multiply by 12 12 move move move move
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Which method is better? Why?
Think about this... To solve c - 3 = 12 Which method is better? Why? Kendra Added 3 to each side of the equation c - 3 = 12 c = 15 Ted Subtracted 12 from each side, then added 15. c - 3 = 12 c - 15 = 0 c = 15
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What is the inverse operation needed to solve this equation?
45 What is the inverse operation needed to solve this equation? 2x = 14 A Addition B Subtraction C Multiplication D Division Answer: D
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What is the inverse operation needed to solve this equation?
46 What is the inverse operation needed to solve this equation? x - 3 = -12 A Addition B Subtraction C Multiplication D Division Answer: A
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What is the inverse operation needed to solve this problem? -2 + x = 9
47 What is the inverse operation needed to solve this problem? -2 + x = 9 A Addition B Subtraction C Multiplication D Division Answer: A, B Add 2 Subtract -2
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One Step Equations Return to table of contents
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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!
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Examples: y + 3 = 13 The inverse of adding 3 is subtracting 3 y = 10 4m = 32 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!!!
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Solve each equation then click the box to see work & solution.
One Step Equations Solve each equation then click the box to see work & solution. x - 5 = 2 x = 7 2 = x - 4 6 = x click to show inverse operation click to show inverse operation x + 5 = -14 x = -19 6 = x + 1 5 = x click to show inverse operation click to show inverse operation x + 9 = 5 x = -4 12 = x + 17 -5 = x click to show inverse operation click to show inverse operation
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One Step Equations 4x = 16 4 4 x = 4 x (2) (2) = 9 2 x = 18 -2x = -12
x = 4 -2x = -12 x = 6 -20 = 5x -4 = x x click to show inverse operation (2) = 9 (2) 2 x = 18 click to show inverse operation x click to show inverse operation = 36 (-6) click to show inverse operation (-6) -6 x = -216 click to show inverse operation
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48 Solve. x - 7 = 19 Answer: 26
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49 Solve. j + 15 = 17 Answer: 2
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50 Solve. 42 = 6y Answer: 7
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51 Solve. -115 = -5x Answer: 23
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52 Solve. = 12 x 9 Answer: 108
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53 Solve. w - 17 = 37 Answer: 54
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54 Solve. -3 = x 7 Answer: -21
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55 Solve. 23 + t = 11 Answer: -12
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56 Solve. 108 = 12r Answer: 9
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Sometimes the operation can be confusing.
For example: x = 7 This looks like you should use subtraction to undo the problem. However, -2 + 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. -2 + x = 7 x - 2 = 7 +2 +2 x = 9 OR OR -2 + x = x = 7 - (-2) -(-2) x = x = 9
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-2 + x = 7 = -2 -4 + x = 5 This did not cancel out anything. +2 +2 x = 9 This did cancel out to find the answer. x - 2 = 7 This is the same as the middle problem
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Try these: 1.) -4 + b = 7 2.) -2 + r = 4 3.) -3 + w = 6 4.) -5 + c = 9
Answers: 1. b=11 2. r=6 3. w=9 4. c=14
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To which does the "-" belong?
Think about this... In the expression To which does the "-" belong? 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:
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57 Solve. Answer: 15
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58 Solve. -5 + q = 15 Answer: 20
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59 Solve. Answer: -96
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60 Solve Answer: -63
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61 Solve. Answer: -1.4
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62 Solve. Answer: -19.1
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63 Solve. Answer: 7.5
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Sometimes you will have an equation where you are multiplying a variable by a fraction.
𝟑 𝟒𝒙 =𝟗
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To undo the fraction you: 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
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1 times any number is itself so this is why it can cancel out.
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64 Solve. Answer: 15
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65 Solve Answer: 12
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66 Solve. Answer: .4
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Two-Step Equations Return to table of contents
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Sometimes it takes more than one step to solve an equation
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!
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4 4 Undo multiplication second x = 2 -2y - 9 = -13
Examples: 4x + 2 = 10 Undo addition first 4x = 8 Undo multiplication second x = 2 -2y - 9 = -13 Undo subtraction first -2y = -4 Undo multiplication second y = 2 Remember - whatever you do to one side of an equation, you MUST do to the other!!!
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Solve each equation then click the box to see
Two Step Equations Solve each equation then click the box to see work & solution. 5b + 3 = 18 5b = 15 b = 3 w + 6 = 10 2 w 2 = w = 8 3j - 4 = 14 3j = 18 j = 6 -2x + 3 = -1 -2x = -4 x = 2 -2m - 4 = 22 -2m = 26 m = -13 +5 = +5 t = 15
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67 Solve the equation. 5x - 6 = -56 Answer: -10
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68 Solve the equation. 14 = 3c + 2 Answer: 4
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69 Solve the equation. x 5 - 4 = 24 Answer: 140
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70 Solve the equation. 5r - 2 = -12 Answer: -2
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71 Solve the equation. 14 = -2n - 6 Answer: -10
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72 Solve the equation. + 7 = 13 x 5 Answer: 30
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73 Solve the equation. + 2 = -10 x 3 - Answer: 36
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74 Solve the equation. Answer: 30
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75 Solve the equation. Answer: 1.5
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76 Solve the equation. Answer: -3
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77 Solve the equation. Answer: -3
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78 Solve -3 5 1 2 1 10 x = Answer: 2/3
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79 Solve the equation. Answer: -16.8
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80 Solve the equation. Answer: 2
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Multi-Step Equations Return to table of contents
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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!
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Examples: 5x + 7x + 4 = 28 12x + 4 = 28 Combine Like Terms Undo Addition 12x = 24 Undo Multiplication x = 2 -1 = 2r - 7r +19 -1 = -5r Combine Like Terms -19 = Undo Subtraction -20 = -5r Undo Multiplication 4 = r
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Try these. 12h - 10h + 7 = 25 -17q + 7q -13 = 27 17 - 9f + 6 = 140 h = 9 q = - 4 f = -13
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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) -2(4x - 7) = -2(4x) - (-2)(7)
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Examples: 2(b - 8) = 28 2b - 16 = Distribute the 2 through (b - 8) Undo subtraction 2b = 44 Undo multiplication b = 22 3r + 4(r - 2) = 13 3r + 4r - 8 = 13 Distribute the 4 through (r - 2) 7r - 8 = Combine Like Terms Undo subtraction 7r = 21 Undo multiplication r = 3
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Try these. 3(w - 2) = 9 4(2d + 5) = 92 6m + 2(2m + 7) = 54 w = 5 d = 9 m = 4
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81 Solve. 9 + 3x + x = 25 Answer: 4
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82 Solve -8e e = -13 Answer: 4
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83 Solve. -27 = 8x x - 11 Answer: -2
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84 Solve. n n - 5 = 13 Answer: 4
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85 Solve. 32 = f - 3f + 6f Answer: 8
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86 Solve. 6g - 15g = -38 Answer: 3
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87 Solve. 3(a - 5) = -21 Answer: -2
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88 Solve. 4(x + 3) = 20 Answer: 2
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89 Solve. 3 = 7(k - 2) + 17 Answer: 0
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90 Solve. 2(p + 7) -7 = 5 Answer: -1
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91 Solve. 3m -1m + 3(m-2) = 19.75 Answer: 5.15
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92 Solve. Answer: 2.1
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93 Solve. Answer: 4
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94 Solve. Answer: 7
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Distributing Fractions in Equations
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Remember... 1. 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!
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There is more than one way to solve an equation with a fraction coefficient. While you can, you don't need to distribute. Multiply by the reciprocal Multiply by the LCD 3 5 72 5 (-3 + 3x) = (-3 + 3x) = 3 5 72 (-3 + 3x) = 3(-3 + 3x) = 72 -9 + 9x = 72 9x = 81 x = 9 (-3 + 3x) = 3 5 72 (-3 + 3x) = -3 + 3x = 24 3x = 27 x = 9
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Some problems work better when you multiply by the reciprocal and some work better multiplying by the LCM. Which strategy would you use for the following? Why?
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95 Solve. Answer: 13
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96 Solve. Answer: -7 5/6
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97 Solve. (8 - 3c) = 2 3 16 Answer: 0
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98 Solve. Answer: 3
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99 Solve. Answer: 3/4
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Translating Between Words and Expressions
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List words that indicate
addition Answers: Sum More than Total All together Add Plus increased by
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List words that indicate
subtraction Answers: Minus Decreased by Difference Less Take away Fewer Than Less than Subtracted From Subtract
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List words that indicate
multiplication Answers: Product Multiplied by Times Of Twice... Double...
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List words that indicate
division Answers: Divided by Divisible by Quotient of Divisibility Half... Fractions
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List words that indicate
equals Answers: Total The same value as Is The same as
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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: Three less than eight: When you see "less than", you need to switch the order of the numbers.
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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 - 8 ·3 more than x means x + 3 ·x less than 2 means x _______________ click to reveal
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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. The following are not allowed: 3xa ... The multiplication sign looks like another variable a3 ... The number is always written in front of the
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Representation of division...
How do you represent "b divided by 12"? b ÷ 12 b ∕ 12 b 12
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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)
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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 1 + + + 2 3 j - - - - 4 5 6 7 8 9 ÷ ÷ ÷ ÷
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Write the expression for each statement.
Then check your answer. The sum of twenty-three and m 23 + m
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Write the expression for each statement.
Then check your answer. Twenty-four less than d d - 24
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Write the expression for each statement.
Remember, sometimes you need to use parentheses for a quantity. Four times the difference of eight and j 4(8-j)
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Write the expression for each statement. Then check your answer.
The product of seven and w, divided by 12 7w 12
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Write the expression for each statement.
Then check your answer. The square of the sum of six and p (6+p)2
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The quotient of 200 and the quantity of p times 7
100 The quotient of 200 and the quantity of p times 7 A 200 7p B 200 - (7p) C 200 ÷ 7p D 7p 200 Answer: A
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35 multiplied by the quantity r less 45
101 35 multiplied by the quantity r less 45 A 35r - 45 B 35(45) - r C 35(45 - r) D 35(r - 45) Answer: D
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Mary had 5 jellybeans for each of 4 friends.
102 Mary had 5 jellybeans for each of 4 friends. A 5 + 4 B 5 - 4 C 5 x 4 D 5 ÷ 4 Answer: C
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103 If n + 4 represents an odd integer, the next larger odd integer is represented by A n + 2 B n + 3 C n + 5 D n + 6 Answer: D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.
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104 a less than 27 A 27 - a B a 27 C a - 27 D 27 + a Answer: A
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If h represents a number, which equation is a correct translation of:
105 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 Answer: B From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.
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Using Numerical and Algebraic Expressions and Equations
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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.
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The cafeteria sold 150 grilled chicken sandwiches and 75 tenders.
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? 2c + c = 225 c + 2c = 225 3c = 225 c = 75 The cafeteria sold 150 grilled chicken sandwiches and 75 tenders. chicken sandwiches chicken tenders total meals
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Julie needs 1 inches on each side.
Julie is matting a picture in a frame. Her frame is 9 inches wide and her picture is 7 inches wide. How much matting should she put on either side? 2m + 7 = 1 2 9 1 2 9 2m + 7 = 2m = 2 m = 1 Julie needs 1 inches on each side. 1 4 2 both sides of the mat size of picture size of frame
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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?
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George is buying video games online. The cost of the video is $30
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 cost of one video game number of games
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George is buying video games online. The cost of the video is $30
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 cost of one video game number of games the cost of the shipping
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George is buying video games online. The cost of the video is $30
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 = 127 cost of one video game number of games the total amount the cost of the shipping
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George is buying video games online. The cost of the video is $30
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 = 127 30g = 120 g = 4 George bought 4 video games.
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106 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 feet, how many feet should be on either side of the fence? 1 2 Answer: 9 ¾ 2x+41/2= 24 x= 93/4 feet
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107 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 Answer: C
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108 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? Answer: 4 Careful with this type of problem. It asks how many people are going with Lewis. This means that you have to find the answer and then subtract one because Lewis is one of the people going and the problem asks how many are going with him. 4 people went with Lewis
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109 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 Answer: B C 68d + 9 = 239 D 77d = 239
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110 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? Answer: 19 weeks
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Be prepared to show your equation!
111 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! Answer: 14 15[x+(x-17)]=675 14 shirts today 31 shirts yesterday
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Be prepared to show your equation!
112 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! Answer: 3 2.49p = 20 p=3 packs
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Write and solve an equation to determine the width of the rectangle.
113 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! Answer: 16 2w+2(w+9)=82 w= 16cm
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The product of -4 and the sum of 7 more than a number is -96.
114 The product of -4 and the sum of 7 more than a number is -96. Write and solve an equation to determine the number. Be prepared to show your equation! Answer: 17 -4(7+x)= -96 x=17
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Be prepared to show your equation! How many subscribers last year?
115 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? Answer: 182x+ 182(x+2100)=2,566,200 x=6000 6000 subscribers last year 8100 subscribers this year
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Be prepared to show your equation! How many subscribers this year?
116 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 this year? Answer: 182x+ 182(x+2100)=2,566,200 x=6000 6000 subscribers last year 8100 subscribers this year
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The perimeter of a hexagon is 13.2 cm.
117 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! Answer: 2.2 6x=13.2 x=2.2 cm
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Graphing and Writing Inequalities with One Variable
Return to table of contents
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When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. Important Words Sample Sentence Equivalent Translation is more than Trenton is more than 10 miles away. t > 10 is greater than A is greater than B. A > B must exceed The speed must exceed 25 mph. The speed is greater than 25 mph. s > 25
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Important Words Sample Sentence Equivalent Translation cannot exceed
When you need to use an inequality to solve a word problem, you may encounter one of the phrases below. Important Words Sample Sentence Equivalent Translation cannot exceed Time cannot exceed 60 minutes. Time must be less than or equal to 60 minutes. t < 60 is at most At most, 7 students were late for class. Seven or fewer students were late for class. n < 7 is at least Bob is at least 14 years old. Bob's age is greater than or equal to 14. B > 14
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How are these inequalities read?
2 + 2 > Two plus two is greater than 3 2 + 2 > Two plus two is greater than or equal to 3 2 + 2 ≥ Two plus two is greater than or equal to 4 2 + 2 < Two plus two is less than 5 2 + 2 ≤ Two plus two is less than or equal to 5 2 + 2 ≤ Two plus two is less than or equal to 4
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Writing inequalities Let's translate each statement into an inequality. words x is less than 10 translate to inequality statement x 10 < 20 is greater than or equal to y 20 > y
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You try a few: 1. 14 is greater than a 2. b is less than or equal to 8
3. 6 is less than the product of f and 20 4. The sum of t and 9 is greater than or equal to 36 5. 7 more than w is less than or equal to 10 decreased by p is greater than or equal to 2 7. Fewer than 12 items 8. No more than 50 students 9. At least 275 people attended the play Answers: > a 2. b ≤ 8 3. 6 < 20f 4. t + 9 ≥ 36 w ≤ 10 p ≥ 2 7. n < 12 8. s < 50 9. p > 275
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Do you speak math? Change the following expressions from English into math. Answer Double a number is at most four. 2x ≤ 4 Answer Three plus a number is at least six. 3 + x ≥ 6
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Five less than a number is less than twice that number.
Answer x - 5 < 2x The sum of two consecutive numbers is at least thirteen. Answer x + (x + 1) ≥ 13 Three times a number plus seven is at least nine. Answer 3x + 7 > 9
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Let e represent an employee's wages.
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. An employee earns e at least > $7.50 7.5 7.5
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Try this: The speed limit on a road is 55 miles per hour. Define a variable and write an inequality. Answer: Let s = speed 0 < s < 55
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118 You have $200 to spend on clothes. You already spent $140 and shirts cost $12. Which equation shows this scenario? A 200 < 12x + 140 B x + 140 ≤ C 200 > 12x + 140 D x + 140 ≥ Answer: D
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119 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 125 < 37 + x x ≤ B C 125 > 37 + x D x ≥ Answer: D
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120 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 + 6 25 ≤ C 4a + 6 > 25 D 4a + 6 ≥ 25 Answer: D
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121 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 Answer: B
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Solution Sets A solution to an inequality is NOT a single number. It will have more than one value. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 This would be read as the solution set is all numbers greater than or equal to negative 5.
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Let's name the numbers that are solutions of the given inequality.
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 Answer: {15, 20} are solutions of the inequality r > 10
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Let's try another one. 30 ≥ 4d; {3, 4, 5, 6, 7, 8} 30 ≥ 4d 30 ≥ (4)3
30 ≥ 12 30 ≥ 4d 30 ≥ (4)4 30 ≥ 16 30 ≥ 4d 30 ≥ (4)5 30 ≥ 20 click to reveal click to reveal click to reveal 30 ≥ 4d 30 ≥ (4) 6 30 ≥ 24 30 ≥ 4d 30 ≥ (4)7 30 ≥ 28 30 ≥ 4d 30 ≥ (4)8 30 ≥ 32 click to reveal click to reveal click to reveal
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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". < >
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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.
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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. -1 -2 -3 -4 -5 1 2 3 4 5
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Graphing Inequalities
What is the number in the inequality? What kind of circle should be used? In what direction does the line go?
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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. -1 -2 -3 -4 -5 1 2 3 4 5
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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. -1 -2 -3 -4 -5 1 2 3 4 5 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. -1 -2 -3 -4 -5 1 2 3 4 5
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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. -1 -2 -3 -4 -5 1 2 3 4 5
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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. -1 -2 -3 -4 -5 1 2 3 4 5 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. -1 -2 -3 -4 -5 1 2 3 4 5
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You try Graph the inequality x > 2 Graph the inequality -3 > x
1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 click 2 on the number line for answer click -3 on the number line for answer .05.05 Graph the inequality -3 > x 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Teacher’s instructions: Notice that the inequality symbol is pointing right but the arrow is pointing left. This is because the variable is on the right side.
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Graph the inequalities.
Try these. Graph the inequalities. 1. x > -3 -1 -2 -3 -4 -5 1 2 3 4 5 .05. 2. x < 4 -1 -2 -3 -4 -5 1 2 3 4 5
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State the inequality shown.
Try these. State the inequality shown. 1. -1 -2 -3 -4 -5 1 2 3 4 5 2. -1 -2 -3 -4 -5 1 2 3 4 5 Answers: 1. x 5 2. x -1
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This solution set would be x > -4.
122 This solution set would be x > -4. A True B False 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: B False
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State the inequality shown.
123 State the inequality shown. -1 -2 -3 -4 -5 1 2 3 4 5 A x > 3 B x < 3 C x < 3 Answer: D D x > 3
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State the inequality shown.
124 State the inequality shown. A 11 < x B 11 > x C 11 > x D 11 < x Answer: A
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State the inequality shown.
125 State the inequality shown. -1 -2 -3 -4 -5 1 2 3 4 5 A x > -1 B x < -1 C x < -1 D x > -1 Answer: C
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State the inequality shown.
126 State the inequality shown. -1 -5 1 5 -2 -3 -4 2 3 4 A -4 < x B -4 > x C -4 < x D -4 > x Answer: C
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State the inequality shown.
127 State the inequality shown. -1 -2 -3 -4 -5 1 2 3 4 5 A x > 0 B x < 0 C x < 0 D x > 0 Answer: B
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Simple Inequalities Involving Addition
and Subtraction Return to table of contents
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Remembers how to solve an algebraic equation?
x + 3 = 13 x = 10 Use the inverse of addition Be sure to check your answer! Does = 13 13 = 13
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· 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.
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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 + 5 Subtract to undo addition 7 > x
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7 > x The symbol is > so it is an open circle and it is numbers less than 7 so it goes to the left. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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Solve and graph. A. j + 7 > -2 A. j + 7 > -2 -7 -7 j > -9
j > -9 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -9 is not included in solution set; therefore we graph with an open circle.
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Solve and graph. B. r - 2 > 4 r - 2 > 4 +2 +2 r > 6 11 10 12
r > 6 11 10 12 13 14 9 8 7 6 5 4 3 2 1
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Solve and graph. 9 > w + 4 C. 9 > w + 4 - 4 5 > w w < 5 1
2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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128 Solve the inequality. 3 < s + 4 ____ < s Answer: -1
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Solve the inequality and graph the solution. -4 + b < -2
129 Solve the inequality and graph the solution. -4 + b < -2 A 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: B b < 2 D 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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Solve the inequality and graph the solution. -8 > b - 5
130 Solve the inequality and graph the solution. -8 > b - 5 A 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: C -3 > b D 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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131 Solve the inequality. m + 6.4 < 9.6 m < ______ Answer: 3.2
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Simple Inequalities Involving Multiplication
and Division Return to table of contents
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Multiplying or Dividing by a Positive Number
3x > -36 x > -12 Since x is multiplied by 3, divide both sides by 3 for the inverse operation.
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( ) ( ) Solve the inequality. 2 3 r < 4
Since r is multiplied by 2/3, multiply both sides by the reciprocal of 2/3. ( ) ( ) 3 2 2 3 3 2 r < 4 r < 6
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132 3k > 18 A 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 D 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: B k > 6
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133 -30 > 3q A 10 > q B -10 < q C -10 > q D 10 < q
Answer: C
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134 X 2 < -3 A B C D Answer: D x < -6 3 4 5 6 7 8 9 -3 -4 -5 -6
2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: D x < -6 D
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135 3 4 g > 27 A g > 36 B g > 108 C g > 36 D g > 108
Answer: A
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136 -21 > 3d A d > -7 B d > -7 C d < -7 D d < -7
Answer: D
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Now let's see what happens when we multiply or divide by negative numbers.
·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.
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Write down two numbers and put the appropriate
inequality (< or >) between them. 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
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What happened with the inequality symbol in your
results? 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.
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Let's see what happens when we multiply this inequality
by -1. 5 > -1 -1 • 5 ? -1 • -1 -5 ? 1 -5 < 1 We know 5 is greater than -1 Multiply both sides by -1 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?
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Helpful Hint The direction of the inequality changes only if the number you are using to multiply or divide by is negative.
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Dividing each side by -3 changes the > to <.
Solve and graph. A. -3y > 18 -3y < 18 y < -6 Dividing each side by -3 changes the > to <. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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Flip the sign because you divided by a negative.
Solve and graph. B. -7m > -28 -7m < -28 m < 4 Divide each side by -7 Flip the sign because you divided by a negative. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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The sign does NOT change because you did not divide by a negative.
Solve and graph. C. 5m > -25 m > -5 Divide each side by 5. The sign does NOT change because you did not divide by a negative. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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Solve and graph. D. -8y > 32 E. -9f > -54 1 2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 E. -9f > -54 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answers: y -4 f < 6
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-r 2 < 5 ( ) ( ) -r 2 Multiply both sides by the reciprocal of -1/2. -2 > 5 -2 Why did the inequality change? r > -10 You multiplied by a negative. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10
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Solve and graph each inequality.
Try these. Solve and graph each inequality. h < 42 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 2. 4x > -20 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answers: h > -7 x > -5
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Solve and graph each inequality.
Try these. Solve and graph each inequality. 3. 5m < 30 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 a -2 > -3 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answers: m < 6 4. a 6
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137 Solve and graph. 3y < -6 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6
2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: y < -2
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Solve and graph. 138 x < -2 -4 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5
2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: x > 8
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139 Solve and graph. -5y ≤ -25 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6
2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: y 5
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140 Solve and graph. n > 2 -2 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5
2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Answer: n < -4
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