Properties of Numbers and Simplifying Expressions

MATH 017

Intermediate Algebra

S. Rook

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Overview

• Section 1.4, Objectives 3 – 5 in the textbook– Properties of Numbers

• Commutative Property• Associative Property• Distributive Property

– Writing Algebraic Equations– Combining Like Terms

Properties of Numbers

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Commutative Property

• Commutative Property of Addition:a + b = b + a

• Commutative Property of Multiplication:a * b = b * a

• The order of the numbers are switched.• The operation (addition/multiplication) is

what we use to distinguish between the two commutative properties.

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Associative Property

• Associative Property of Addition:(a + b) + c = a + (b + c)

• Associative Property of Multiplication:(a * b) * c = a * (b * c)

• The order of evaluation is altered by using parentheses.

• The operation (addition/multiplication) is what we use to distinguish between the two associative properties.

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Properties of Numbers (Example)

Ex 1: Identify the property: 3 * x = x * 3

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Properties of Numbers (Example)

Ex 2: Identify the property: (7 + y) + z = 7 + (y + z)

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Distributive Property

• Distributive Property: taking multiplication over addition or subtraction.

a(b + c) = a * b + a * c

• Usually no operation next to the number outside of the parentheses.– Implied to be multiplication

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Distributive Property (Example)

Ex 3: Apply the distributive property to

2(x + 5)

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Distributive Property

Ex 4: Apply the distributive property to

-(6 + y)

Writing Algebraic Expressions

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Writing Algebraic Expressions

• Use key words in the problem to determine the operation and order of the operands.

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Writing Algebraic Expressions (Example)

Ex 5: Write an expression for the total cost when x units of brand A are bought at $2.45 each and y units of brand B are bought at $1.25 each.

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Writing Algebraic Expressions (Example)

Ex 6: Each package of paper contains x pieces. If there are 1200 pieces of paper, in terms of x, how many packages are there?

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Writing Algebraic Expressions (Example)

Ex 7: Each package of paper contains 35 pieces. If there are x packages, in terms of x, how many pieces of paper are there?

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Writing Algebraic Expressions (Example)

Ex 8: Two angles are supplementary if their measures sum to 180. Given that y is the measure of one of the two angles, represent the other angle in terms of y.

Combining Like Terms

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Combining Like Terms

• Two terms are said to be like terms if they satisfy BOTH of the following conditions.– The variables must be the same– The variables must be raised to the same

power

• To add like terms, add the coefficients and retain the variable base.– Apply the distributive property first if

applicable

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Combining Like Terms (Example)

Ex 9: Simplify: 3x + 6x – 2y + 5 – x + 7

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Combining Like Terms (Example)

Ex 10: Simplify 2(x + 4) – 3(x – 1)

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Combining Like Terms (Example)

Ex 11: Simplify 2x2 – 7x + 3 – (2x – 2x2)

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Summary

• After studying these slides, you should know how to do the following:– Specifically identify, by operation, instances of the

commutative and associative properties– Write an algebraic expression given a problem

description– Simplify an expression by combining like terms

• Additional Practice– Attempt the suggested problems found on the course

website.