 Real Numbers: Properties MathBitsNotebook.com  Decide upon your answer and then check the "Explanation" drop down. 1. Explain why the following statements are equivalent: 4(x + y) = 4x + 4y Explanation The 4 is being multiplied across the parentheses, which demonstrates the Distributive Property for Real Numbers. 2. Rewrite this expression using the Distributive Property: 6x + 12 Explanation Use the Distributive Property in reverse. Factor out a 6 to create a parentheses situation: 6x + 12 = 6(x + 2). 3. Using the Distributive Property, write an expression equivalent to 5(x - 6). Explanation Since (x - 6) = (x + (-6)), distribute across the parentheses, being careful of the negative sign: 5x + (- 30) or 5x - 30. 4. Rewrite this expression using the Distributive Property: -5x - 25 Explanation Working in reverse, factoring out 5 will give 5(-x -5) or factoring out -5 will give -5(x + 5). While both solutions use the Distributive Property, the solution -5(x + 5) is the more accepted answer. 5. Explain why the following statements are equivalent: a + (b + 4) = a + (4 + b) Explanation Only the elements inside the parentheses changed places, so this is the Commutative Property for Addition at work. 6. Using the Commutative Property, write an equivalent expression for 5•(7x). Explanation Do not simplify. Simply have the elements exchange places with respect to the multiplication symbol: (7x)•5 or 5•(x•7) or (x•7)•5. 7. Explain why the following equation is true: (x + 4) + (-x + (- 4)) = 0. Explanation An expression (x + 4) plus its negation (-x + (-4)) yields the additive identity element of 0. This is the Additive Inverse Property. 8. Simplify this expression and justify your steps: 12 + 3(a + 2b) - 2a - 3b - 9 + b (The Explanation shown below is one possible solution.) 12 + 3(a + 2b) - 2a - 3b - 9 + b Reason GIVEN 12 + 3a + 6b - 2a - 3b - 9 + b Reason Distributive Property 12 - 9 + 3a - 2a + 6b - 3b + b Reason Commutative Property of Addition aligns similar terms (12 - 9) + (3a - 2a) + (6b - 3b + b) Reason Associative Property of Addition groups similar terms (3) + (3a - 2a) + (6b - 3b + b) Reason Numerical subtraction 3 + a(3 - 2) + b(6 - 3+ 1) Reason Distributive Property (in reverse) 3 + a(1) + b(4) Reason Numerical subtraction and addition 3 + a + 4b Reason Commutative Property of Multiplication to get 4b and 1a. Multiplicative Identity gives 1a = a. 