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Avon High School Section: P.1 ACE COLLEGE ALGEBRA II - NOTES Algebraic Expressions, Mathematical Models, and Real Numbers Mr. Record: Room ALC-129 Day 1 Algebraic Expressions & Evaluating Algebraic Expressions Algebra uses combinations of variables and numbers with operations (addition, subtraction, multiplication, division, powers, roots, etc.) to form algebraic expressions. A couple of important ideas from Algebra 1 are found in the boxes below. Example 1 a. Evaluate 3x y for x 4 and y 2. x 1 b. Simplify 15 9 3(22 3) 12 8 4 2 Formulas & Math Models An ________________ is formed when an equals sign is placed between two algebraic expressions. A _________________ is an equation that uses variables to express a relationship between two or more quantities. Two examples of a formula are given to the right? What is your value of H in the two situations? Example 2 Modeling the Cost of Attending a Public College The bar graph in Figure P.1 shows the average cost of tuition and fees for public four-year colleges, adjusted for inflation. The formula T 17 x 2 261x 3257 models the average cost of tuition and fees, T, for public U.S. colleges for the school year ending x years after 2000. a. Use the formula to find the average cost of tuition and fees, T, for public U.S. colleges for the school year ending in 2007? b. By how much does the formula underestimate or overestimate the actual cost shown in Figure P.1? Sets A ___________ is a collection of objects whose contents can be clearly determined. The objects in a set are called _________________. Here are two different ways to write the set consisting of the elements 1, 2, 3, 4, 5 Roster Method Set-Builder Notation 1, 2,3, 4,5 x | x is a counting number less than 6 Note: It is important that you use these symbols { } rather than ( ) or [ ]. The set of all x such that x is a counting number less than 6. A set that contains no elements can be denoted by the symbol _________ . Example 3 a. Finding the Intersection and Union of Two Sets Find the intersection: 2, 4,6,8,10 3, 4,5,6,7 b. Find the union: 2, 4,6,8,10 3, 4,5,6,7 The Set of Real Numbers The subsets that make up the real number system are summarized in the following table. The natural numbers are sometimes called _____________ numbers. The whole numbers are the same as the natural numbers except for the inclusion of ______ . A rational number can always be written as a _______________ . An irrational number, when written as a decimal will never ________________ nor __________________ . Example 4 Finding Subsets of Two Numbers Consider the following set of numbers: 3, 0, 1 , .95, , 8, 16 . List the numbers in the set that are: 2 a. natural numbers b. whole numbers c. integers d. rational numbers e. irrational numbers f. real numbers Absolute Values Up to now, you probably only think of absolute value in the follow manner: 3 3 5 5 Or perhaps, you perceive absolute value to be the distance the number within the symbols is from 0. The above facts are true, but as you progress through higher levels of mathematics, it becomes important to think of absolute value in a more formal way. Definition of Absolute Value x if x 0 x x if x 0 If the contents of this box look a little unusual, don’t worry. We will discuss these types of functions (called “piecewise functions”) in more detail later on. Other important properties of absolute value are described in the following tables. Properties of Negatives Pay very close attention to these properties when working with algebraic expressions containing negative signs. Example 5 Simplifying Algebraic Expressions Simplify 18 x 2 4 6 x 2 2 5 Using Algebra to Solve Problems in Your Everyday Life