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CHAPTER 6

CHAPTER 6. Algebra: Equations and Inequalities. 6.1. Algebraic Expressions and Formulas. Objectives Evaluate algebraic expressions. Use mathematical models. Understand the vocabulary of algebraic expressions. Simplify algebraic expressions. Algebraic Expressions.

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CHAPTER 6

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  1. CHAPTER 6 Algebra: Equations and Inequalities

  2. 6.1 • Algebraic Expressions and Formulas

  3. Objectives Evaluate algebraic expressions. Use mathematical models. Understand the vocabulary of algebraic expressions. Simplify algebraic expressions.

  4. Algebraic Expressions • Algebra uses letters, calledvariables, such as x and y, to represent numbers. • An algebraic expression is a combination of variables and numbers using the operations of addition, subtraction, multiplication, or division as well as powers or roots. • Examples of algebraic expressions:

  5. Order of Operations Agreement • Perform operations within the innermost parentheses and work outward. If the algebraic expression involves a fraction, treat the numerator and the denominator as if they were each enclosed in parentheses. • Evaluate all exponential expressions. • Perform multiplications and divisions as they occur, working from left to right. • Perform additions and subtractions as they occur, working from left to right.

  6. Example: Evaluating an Algebraic Expression • Evaluate 7 + 5 (x – 4)3 for x = 6 • Solution: • 7 + 5 (x – 4)3 = 7 + 5(6 – 4)3 • = 7 + 5(2)3 • = 7 + 5(8) • = 7 + 40 • = 47 Replace x with 6. First work inside the parentheses. Evaluate the exponential expression. Multiply 5(8) = 40. Add.

  7. Formulas and Mathematical Models • An equation is formed when an equal sign is placed between two algebraic expressions. • A formula is an equation that uses letters to express a relationship between two or more variables. • Mathematical modeling is the process of finding formulas to describe real-world phenomena.

  8. Example: Modeling Caloric Needs • The bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderately active lifestyles. • The mathematical model W = 66x2 + 526x + 1030 describes the number of calories needed per day by women in age group x with moderately active lifestyles. • According to the • model, how many calories per day are • needed by women • between the ages • of 19 and 30, • inclusive, with this • lifestyle?

  9. Example continued • Solution: Because 19-30 is designated as group 4, we substitute 4 for x in the given model. • The formula indicates that 2078 calories are needed per day by women in the 19-30 age range with moderately active lifestyle.

  10. Vocabulary of Algebraic Expressions • Term: Those parts of an algebraic expression separated by addition. • Example: in the expression 7x – 9y – 3 • Coefficient: The numerical part of a term. • 7, –9 • Constant: A term that consists of just a number, also called a constant term. –3 • Like terms: Terms that have the exact same variable factors. 7xand 3x • Factors: Parts of each term that are multiplied.

  11. Properties of Real Numbers

  12. Example: Simplifying an Algebraic Expression • Simplify: 5(3x – 7) – 6x • Solution: • 5(3x – 7) – 6x • = 5∙3x– 5∙7 – 6xdistributive property • = 15x – 35 – 6xmultiply • = (15x – 6x) – 35group like terms • = 9x – 35 combine like terms

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