Chapter 1 Algebra, Mathematical Models, and Problem Solving

§ 1.1 Algebraic Expressions and Real Numbers

Variables in Algebra Algebra uses letters such as x and y to represent numbers. If a letter is used to represent various numbers, it is called a variable. For example, the variable x might represent the number of minutes you can lie in the sun without burning when you are not wearing sunscreen. Blitzer, Algebra for College Students, 6e – Slide #3 Section 1.1

Suppose you are wearing number 6 sunscreen. If you can normally lie in the sun x minutes without burning, with the number 6 sunscreen, you can lie in the sun 6 times as long without burning - that is, 6 times x or 6x would represent your exposure time without burning. Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.1

A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression. Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.1 Algebraic Expressions

Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.1 English PhraseMathematical Operation sum plus increased by more than Addition difference minus decreased by less than Subtraction product times of (used with fractions) twice Multiplication quotient divide per ratio Division Translating Phrases into Expressions

Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.1 Translating Phrases into ExpressionsEXAMPLE Write the English phrase as an algebraic expression. Let x represent Four more than five times a number SOLUTION the number.

Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.1 Evaluating an Algebraic ExpressionEXAMPLE The formula expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. Use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale. SOLUTION Replace F with 50

Blitzer, Algebra for College Students, 6e – Slide #9 Section 1.1 Evaluating an Algebraic Expression Multiply Therefore CONTINUED Subtract.

In evaluating expressions, what comes first? #1 Start with the parentheses. Parentheses say “Me First!” #2 Then evaluate the exponential expressions. #3 Multiplications and divisions are equal in the order of operations – Perform them next. #4 Additions and subtractions are also equal to each other in order – and they come last. Remember by “PEMDAS” - parentheses, exponents, multiplication, division, addition, subtraction Blitzer, Algebra for College Students, 6e – Slide #10 Section 1.1

Blitzer, Algebra for College Students, 6e – Slide #11 Section 1.1 Order of Operations - PEMDAS Order of Operations 1) First, perform all operations within grouping symbols 2) Next, Evaluate all exponential expressions. 3) Next, do all multiplications and divisions in the order in which they occur working from left to right. 4) Finally, do all additions and subtractions in the order in which they occur, working from left to right.

Blitzer, Algebra for College Students, 6e – Slide #12 Section 1.1 Order of Operations - PEMDAS Evaluate for. EXAMPLE Replace R with 3 Evaluate inside parentheses first Evaluate – first exponent SOLUTION

Blitzer, Algebra for College Students, 6e – Slide #13 Section 1.1 Order of Operations - PEMDAS -135Subtract Multiply CONTINUED (81) Evaluate – second exponent

Blitzer, Algebra for College Students, 6e – Slide #14 Section 1.1 Number Sets Sets of NumbersDefinition Natural NumbersAll numbers in the set {1,2,3,4,…} Whole NumbersAll numbers in the set {0,1,2,3,4,…} IntegersAll numbers in the set {…-3,-2,-1,0,1,2,3,…} Rational NumbersAll numbers a/b such that a and b are integers Irrational NumbersAll numbers whose decimal representation neither terminate nor repeat Real NumbersAll numbers that are rational or irrational NOTE: “…” means continue without end

Blitzer, Algebra for College Students, 6e – Slide #15 Section 1.1 Three Common Number Sets The natural numbers are the numbers we use for counting. The set of whole numbers includes the natural numbers and 0. Zero is a whole number, but is not a natural number. The set of integers includes all the whole numbers and their negatives. Every whole number is an integer, and every natural number is an integer. These sets are just getting bigger and bigger… Note that…

Blitzer, Algebra for College Students, 6e – Slide #16 Section 1.1 Set-Builder Notation {x | x is a real number and greater than 10} Express x > 10 using set-builder notation EXAMPLE SOLUTION

Blitzer, Algebra for College Students, 6e – Slide #17 Section 1.1 Rational Numbers Rational numbers can be expressed either in fraction or in decimal notation. Every integer is rational because it can be written in terms of division by one. The set of rational numbers is the set of all numbers that can be expressed as the quotient of two integers with the denominator not zero. That is, a rational number is any number that can be written in the form a/b where a and b are integers and b is not zero. Definition

Blitzer, Algebra for College Students, 6e – Slide #18 Section 1.1 Symbols and The symbol is used to indicate that a number or object is in a particular set. Here is an example: 7 {1,2,5,7,9} The symbol is used to indicate that a number or object is not in a particular set. For example: 3 {4,6}

Blitzer, Algebra for College Students, 6e – Slide #19 Section 1.1 Inequalities MeaningsExamples < is less than 10 < < 3 -7 < -2 > is greater than 6 > > 8 -6 > -12 is less than or is equal to is greater than or is equal to