PreAlgebra 2 Chapter 2 Notes The Distributive Property

1. PreAlgebra 2 Chapter 2 Notes The Distributive Property

2. 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

3. 2.2 The Distributive Property

4. 2.2 The Distributive Property Answer: The total cost of the geodes is $17.85

5. 2.2 The Distributive Property

6. 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.

7. 2.2 The Distributive Property Answer: The total cost of the geodes is $17.85

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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.

16. 2.5 Subtraction Property of Equality

17. 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

18. 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

19. 2.6 Solving Equations using Multiplication or Division

20. 2.6 Solving Equations using Multiplication or Division

21. 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

22. 2.7 Decimal Operations and Equations with Decimals

23. 2.7 Decimal Operations and Equations with Decimals

24. 2.7 Decimal Operations and Equations with Decimals

25. 2.7 Decimal Operations and Equations with Decimals