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1.4 Properties of Real Numbers and Algebraic Expressions

Algebraic equation is a statement that two expressions have equal value. Algebraic Equations Equality is denoted by the phrases equals gives is/was/should be yields amounts to represents is the same as

Write each sentence as an equation. The difference of 7 and a number is – x = 42 The quotient of y and twice x is the same as the product of 4 and z Example

Equality and Inequality Symbols SymbolMeaning a = b a  b a < b a > b a  b a  b a is equal to b. a is not equal to b. a is less than b. a is greater than b. a is less then or equal to b. a is greater than or equal to b.

Example Tell whether each mathematical statement is true or false. a. 4 < 5 True b. 27 ≥ 27 True c. 0 > 5 False d. 16 ≤ 9 False

Example Insert, or = between the pairs of numbers to form true statements. a. 4.7 > b = c. – 4 > – 7 d. <

Example Translate each sentence into a mathematical statement. a. Thirteen is less than or equal to nineteen. 13 ≤ 19 b. Five is greater than two. 5 > 2 c. Seven is not equal to eight. 7 ≠ 8

Addition 0 is the identity since a + 0 = a and 0 + a = a. Multiplication 1 is the identity since a · 1 = a and 1 · a = a. Identities

Additive and Multiplicative Inverses The numbers a and –a are additive inverses or opposites of each other because their sum is 0; that is a + ( – a) = 0. The numbers b and (for b ≠0) are reciprocals or multiplicative inverses of each other because their product is 1; that is Inverses

Example Write the additive inverse, or opposite, of each. a. 5 b. c.  3.7

Example Write the multiplicative inverse, or reciprocal, of each. a. 5 b. c.  3

Commutative and Associative Property Associative property Addition: (a + b) + c = a + (b + c) Multiplication: (a · b) · c = a · (b · c) Commutative property Addition: a + b = b + a Multiplication: a · b = b · a

Example Use the commutative or associative property to complete. a. x + 8 = ______ 8 + x b. 7 · x = ______ x · 7 c. 3 + (8 + 1) = _________ (3 + 8) + 1 d. ( ‒ 5 ·4) · 2 = _________ ‒ 5(4 · 2) e. (xy) ·18 = ___________ x · (y ·18)

For real numbers, a, b, and c. a(b + c) = ab + ac Also, a(b  c) = ab  ac Distributive Property

Example Use the distributive property to remove the parentheses. 7(4 + 2) = (7)(4) = = 42 +(7)(2) 7(4 + 2) =

Example Use the distributive property to write each expression without parentheses. Then simplify the result. a. 7(x + 4y) = 7x + 28y b. 3( ‒ 5 + 9z) = 3( ‒ 5) + (3)(9z) = ‒ z c. ‒ (8 + x ‒ w) = ( ‒ 1)(8) + ( ‒ 1)(x) ‒ ( ‒ 1)(w) = ‒ 8 ‒ x + w

Example Write each as an algebraic expression. a. A vending machine contains x quarters. Write an expression for the value of the quarters. 0.25x b. The cost of y tables if each tables costs $ y

Example Write each as an algebraic expression. a. Two numbers have a sum of 40. If one number is a, represent the other number as an expression in a. 40 – a b. Two angles are supplementary if the sum of their measures is 180 degrees. If the measure of one angle is x degrees represent the other angle as an expression in x. 180 – y

Terms of an expression are the addends of the expression. Like terms contain the same variables raised to the same powers. Like Terms

Simplify by combining like terms. a. b. Example

Simplify each expression. a. b. Example

Simplify by using the distributive property to multiply and then combining like terms.

Example Simplify by using the distributive property to multiply and then combining like terms.