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2-8 Inverse of a sum and simplifying

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Inverse of a Sum What happens when we multiply a rational number by -1? The product is the additive inverse of the number. The Property of -1 For any rational number a, (negative one times a is the additive inverse of a) the property of -1 enables us to find an equivalent expression for the additive inverse of a sum

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Examples -(3 + x) = -1(3 + x) Using the distributive property = -3 – x -(3x + 2y + 4) = -1 (3x + 2y + 4) Using the distributive property = -3x – 2y – 4 Try this: a)-(x+2) b) - (5x +2y +8) c) -(a – 7) d) -(3c – 4d + 1)

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Inverse of a Sum Inverse of a sum property For any rational numbers a and b, -(a + b) = -a + -b (The additive inverse of a sum is the sum of the additive inverses) The inverse of a sum property holds for differences as well as sums because any difference can be expressed as a sum. When we apply the inverse of a sum property we sometimes say that we “change the sign of every term” Example: rename each additive inverse without using parentheses -(5 – y) = -5 + y -(2a – 7b – 6) = -2a + 7b +6

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Try this -(6 – t) -(-4a + 3t – 10) -(18 – m – 2n + 4t)

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Simplifying Expressions Involving Parentheses When an expression inside parentheses is added to another expression as in 5x + (2x + 3), the associative property allows us to move the parentheses and simplify the expression to 7x + 3. When an expression inside parentheses is subtracted from another expression as in 3x – (4x + 2), We can subtract by adding the inverse. Example: Simplify 3x – (4x + 2) = 3x + ( - (4x + 2)) = 3x + (-4x – 2) = -x – 2

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Example 5y – (3y + 4) = 5y -3y – 4 = 2y – 4 3y – 2 – ( 2y – 4) = 3y – 2 – 2y + 4 = (3y – 2y) + ( -2 + 4) = y + 2

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Example x – 3(x + y) = x – 3x + (-3y) = x – 3x – 3y = -2x – 3y 3y – 2(4y – 5) = 3y – 8y + 10 = -5y + 10 h) 5x – (3x + 9) i) 5x – 2y – (2y – 3x – 4) j.) y – 9(x + y) k) 5a – 3(7a – 6)

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Grouping Symbols Some expressions contain more than one grouping symbol. Parentheses (), Brackets [ ], and Braces { } are all grouping symbols we use in algebra. When an expression contains more than one grouping symbol, we start with the inner more parentheses and work our way out. [3 – (7+3) = [ 3 – 10] [8 – [9 – (12 + 5)]] = -7 = [ 8 – [ 9 – (17)] = [ 8 – [9 – 17] = [ 8 – [-8] = [8 - -8] = 16

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Try this [ 9 – (6+ 4)] 3(4 + 2) – [ 7 – [ 4 – (6 + 5)]] [3(4 + 2) + 2x] – [4 (y + 2) – 3 (y – 2)]

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