PreAlgebra 2 Chapter 2 Notes The Distributive Property

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## PreAlgebra 2 Chapter 2 Notes The Distributive Property

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**PreAlgebra 2**Chapter 2 Notes The Distributive Property**2.2**The Distributive Property Camping:You and a friend are going on a camping trip. You each buy a backpack that costs $90 and a sleeping bag that cost $60. What is the total cost of the camping equipment? Answer: The total cost of the camping equipment is $300**2.2**The Distributive Property**2.2**The Distributive Property Answer: The total cost of the geodes is $17.85**2.2**The Distributive Property**2.2**The Distributive Property 8 – 3 y 7 12 2 x + 5 Answer: The area is ( 14 x + 35 ) square units. Answer: The area is ( 48 – 18 y ) square units.**2.2**The Distributive Property Answer: The total cost of the geodes is $17.85**2.3**Simplifying Variable Expressions Vocabulary: Term (values separated by ADDor SUBTRACToperation) Coefficient (number value that precede a variable, 5x ) Constant Term ( a number value with no variable, 5) Like Terms ( values that has the same variable with the same exponent, 3x + 4x) Example 1 Identify the terms, like terms, coefficients, and constant terms of the expression y + 8 – 5y – 3 Solution Write the expression as a sum: y + 8 + (– 5y) + (– 3) Identify the parts of the expression. Note that because y = 1y, the coefficient of y is 1 Terms: y, 8, – 5y. – 3 Like terms; y and – 5y; 8 and – 3 Coefficients: 1, – 5 Constant terms: 8, – 3**2.3**Simplifying Variable Expressions Example 2 : Simplifying and Expression 4n – 7 – n + 9 = 4n + (– 7) + (– n ) + 9 Write as a sum = 4n + (– n) + (– 7 ) + 9 Commutative property = 4n + (– 1n) + (– 7 ) + 9 Coefficient of – n is – 1 = [4 + (– 1) ] n + (– 7 ) + 9 Distributive property = 3n + 2 Simplify A quick way to combine like terms containing variables is to add their coefficients mentally. For example, 4n + (– n) = 3n Because 4 – 1 = 3 Example 3 : Simplifying Expressions with parentheses a.) 2 (x – 4) + 9 x+ 1 = 2x – 8 + 9x + 1 Distributive property = 2x – 8 + 9x + 1 Group like terms = 11x – 7 Combine like terms b) 3k – 8 (k + 2) = 3k – 8k – 16 Distributive property = – 5 k – 16 Combine like terms 4a – ( 4a – 3 ) = 4a – 1 (4a – 3 ) Identity property = 4a – 4a + 3 Distributive property = 0 + 3 Combine like terms = 3 Simplify**2.3**Simplifying Variable Expressions X The error in this problem is – • – = +: 5a – (3a – 7 ) = 5a – 3a + 7 = 2a+7 Correct the error in this problem: 5a – (3a – 7 ) = 5a – 3a – 7 = 2a – 7 Guided Practice What are terms that have a number but no variable called? What is the coefficient of y in the expression 8 – 3 y + 1 ? 6x + x + 2 + 4 – 4 k – 12 + 3k 5n + 1 – n – 8 5x + 2 + 3(x – 1) – 7 (2r + 3) + 11 r p + 6 – 6 (p – 2 ) 1)______________ 2)______________ 3) 7x + 6 4) – 1 k – 12 5) 4 n – 7 6) 8 x – 1 7) – 3 r – 21 8) – 5p + 18**2.4**Variables and Equations An Equation is a mathematical sentence formed by placing an equal sign, =, between two expressions. A Solution of an equation with a variable is a number that produces a true statement when it is substituted for the variable. Numerical Expression is an expression, with no equation or inequality, that has no variables, just numbers. Ex. 3 ( 6 + 2 ) Variable Expression is an expression, with no equation or inequality, that has at lease one variable. Ex. 3 ( x + 2 ) • Example 1: Writing Verbal Sentences as Equations • The sum (addition) of x and 6 is 9. • The difference (subtraction) of 12 and y is 15. • The product (multiplication) of – 4 and p is 32. • The quotient (division) of n and 2 is 9. • Equation • x + 6 = 9 • 12 – y = 15 • – 4 p = 32 • n = 9 • 2**2.4**Variables and Equations Example 2: Checking Possible Solutions Tell whether 9 or 7 is a solution of x – 5 = 2. Substitute 9 for x b. Substitute 7 for x x – 5 = 2 x – 5 = 2 9 – 5 = 2 7 – 5 = 2 4 ≠ 2 2 = 2 Answer 9 is not a solution Answer 7 is a solution**2.4**Variables and Equations Solve the equation using mental math: 4) x – 10 = 7 5) 2 + n = - 6 6) 3 w = - 15 7) 4 = 36 s**2.4**Variables and Equations Guided Practice 1. A(n) ___________? Of an equation is a number that produces a true statement when it is substituted for a variable. The sum (addition) of x and 10 is 15. The product (multiplication) of – 6 and x is 54 The difference (subtraction) of 3 and x is 2. The quotient (division) of – 40 and x is – 8. Solution x + 10 - 15 – 6 x = 54 3 x = 2 – 40 = – 8 x**2.5**Solving Equations using Addition or Subtraction Inverse operations are two operations that undo each other, such as addition and subtraction. Equivalent equations have the same solution.**2.5**Subtraction Property of Equality**2.5**Solving Equations using Addition or Subtraction Example 2 : Solving an Equation Using Addition Solve 23 = y – 11 Write original equation 23 + 11 = y – 11 + 11 Add 11 to each side 34 = y Simplify**2.5**Solving Equations using Addition or Subtraction X The error in this problem is – 8 to one side of equation and + 8 to other side of equation. The value to both sides MUST be equal, including sign. Correct the error in this problem: x + 8 = 10 x + 8 – 8 = 10 + 8 x = 18 Guided Practice Addition and subtraction are ___________ operations? Why are x – 5 = 7 and x = 12 are equivalent equations? x + 4 = 10 t + 9 = – 5 u – 3 = 6 y – 7 = – 2 16 = a + 25 – 70 = b – 30 1) Inverse / opposite 2) Both = 12 3) x = 6 4) t = – 14 5) u = 9 6) y = 5 7) a = – 9 8) b = – 40**2.6**Solving Equations using Multiplication or Division**2.6**Solving Equations using Multiplication or Division**2.6**Solving Equations using Multiplication or Division Guided Practice Multiplication and __________ are inverse operations? Which property of equality would you use to solve are x = 12 ? 5 3) 5 c = – 15 54 = 9 x 5) 6 = u 4 6) y = 7 – 10 1) Division 2) Multiplication 3) c = – 3 4) x = 6 u = 24 y = – 70**2.7**Decimal Operations and Equations with Decimals**2.7**Decimal Operations and Equations with Decimals**2.7**Decimal Operations and Equations with Decimals**2.7**Decimal Operations and Equations with Decimals