Name: _____________________________________________________ Date: __________________________ Period: ____________ 6.1B KEY Solve Equations and Justify You have discussed many algebraic properties and are now ready to logically justify your reasons for solving equations. Justify each step of the following solution of an equation. 1.) a.) b.) c.) d.) e.) f.) g.) h.) i.) j.) k.) l.) (𝟐𝒙 + 𝟑) + 𝟕 = 𝟓𝒙 + 𝟐 2𝑥 + (3 + 7) = 4𝑥 + 2 2𝑥 + 12 = 4𝑥 + 2 – 2x – 2x 0 + 12 = 2𝑥 + 2 12 = 2𝑥 + 2 –2 –2 10 = 2x + 0 10 = 2x 2 2 5 = 1𝑥 5=𝑥 𝑥=5 𝟔𝒙+𝟒 2.) 𝟒 6𝑥+4 a.) b.) c.) d.) e.) f.) g.) h.) i.) Associative Prop Addition Property of Equality Additive Inverse Additive Identity Property of Equality Additive Inverse Additive Identity Property of Equality Multiplicative Inverse Multiplicative Identity Symmetric a.) b.) c.) d.) e.) f.) g.) h.) i.) j.) k.) l.) m.) Property of Equality Multiplicative Inverse Distributive Prop Property of Equality Additive Inverse Additive Identity Property of Equality Additive Inverse Additive Identity Property of Equality Multiplicative Inverse Multiplicative Identity Symmetric a.) b.) c.) d.) e.) f.) g.) h.) i.) Distributive Prop Commutative Prop Addition Property of Equality Additive Inverse Additive Identity Property of Equality Multiplicative Inverse Multiplicative Identity = 𝟐𝒙 − 𝟓 a.) 4 ( 4 ) = 4(2𝑥 − 5) b.) 6𝑥 + 4 = 4(2𝑥 − 5) c.) 6𝑥 + 4 = 8𝑥 − 20 d.) −6𝑥 −6𝑥 e.) 0 + 4 = 2𝑥 − 20 f.) 4 = 2𝑥 − 20 g.) + 20 + 20 h.) 24 = 2𝑥 + 0 i.) 24 = 2x j.) 2 2 k.) 12 = 1𝑥 l.) 12 = 𝑥 m.) 𝑥 = 12 3.) a.) b.) c.) d.) e.) f.) g.) h.) i.) j.) k.) l.) 𝟑𝒙 + 𝟐(𝟒 + 𝒙) = 𝟏𝟑 3𝑥 + 8 + 2𝑥 = 13 3𝑥 + 2𝑥 + 8 = 13 5𝑥 + 8 = 13 –8 –8 5𝑥 + 0 = 5 5x = 5 5 5 1x = 1 𝑥=1 Fill in the missing step for solving each equation and the missing justifications. 4.) a.) b.) c.) d.) e.) f.) g.) 𝟏𝟐(𝒕 − 𝟓) + 𝟓 = 𝟎 12𝑡 − 60 + 5 = 0 12𝑡 − 55 = 0 + 55 +55 12𝑡 + 0 = 55 12t = 55 12 12 55 1𝑡 = 12 h.) 5.) a.) b.) c.) d.) e.) f.) g.) h.) i.) 𝑡= a.) b.) c.) d.) e.) f.) g.) h.) i.) j.) 12 𝟗𝒙 − 𝟕 = 𝟐(𝟓𝒙 + 𝟏𝟐) 9𝑥 − 7 = 10𝑥 + 24 -9x -9x 0 − 7 = 1𝑥 + 24 −7 = 1𝑥 + 24 – 24 – 24 −31 = 1𝑥 + 0 −31 = 1𝑥 −31 = 𝑥 𝑥 = −31 𝒙−𝟑 6.) 55 = a.) b.) c.) d.) e.) f.) g.) Distributive Prop. Addition Property of Equality Additive Inverse Additive Identity Property of Equality Multiplicative Inverse h.) Multiplicative Identity a.) b.) c. ) d.) e.) f.) g.) h.) i.) Distributive Prop Property of Equality Additive Inverse Additive Identity Property of Equality Additive Inverse Additive Identity Multiplicative Identity Symmetric a.) b.) c.) d.) e.) f.) g.) h.) i.) j.) Property of Equality Multiplication Inverse Distributive Prop Multiplication Property of Equality Additive Inverse Additive Identity Property of Equality Multiplicative Inverse Multiplicative Identity 𝟐 𝟖 𝟓 𝑥−3 2 5 • 8 ( ) = 40 ( ) 5 • 8 8 5(𝑥 − 3) = 8(2) 5𝑥 − 15 = 8(2) 5𝑥 − 15 = 16 + 15 +15 5𝑥 + 0 = 15 5x = 15 5 5 1𝑥 = 3 𝑥=3 5 Solve each equation and justify each step. Answers may vary. 7.) 3( x 4) 2(3x 2) 1 8.) 2x 25 5x 10 7 2 9.) 𝑥 = 6