Chapter 1 Algebra, Mathematical Models, and Problem Solving

§ 1.1 Algebraic Expressions and Real Numbers

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 1.1 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, Intermediate Algebra, 5e – Slide #3 Section 1.1

Blitzer, Intermediate Algebra, 5e – Slide #4 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, Intermediate Algebra, 5e – Slide #4 Section 1.1

Algebraic Expressions 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, Intermediate Algebra, 5e – Slide #5 Section 1.1

Mathematical Operation Translating Phrases into Expressions English Phrase Mathematical 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 Blitzer, Intermediate Algebra, 5e – Slide #6 Section 1.1

Translating Phrases into Expressions EXAMPLE Write the English phrase as an algebraic expression. Let x represent the number. Four more than five times a number SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #7 Section 1.1

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 1.1 Evaluating an Algebraic Expression EXAMPLE 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, Intermediate Algebra, 5e – Slide #8 Section 1.1

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 1.1 Evaluating an Algebraic Expression CONTINUED Subtract Multiply Therefore . Blitzer, Intermediate Algebra, 5e – Slide #9 Section 1.1

In evaluating expressions, what comes first? #1 Start with the parentheses. Parentheses say “Me First!” #2 Then evaluate the exponential expressions. #4 Additions and subtractions are also equal to each other in order – and they come last. #3 Multiplications and divisions are equal in the order of operations – Perform them next. Remember by “PEMDAS” - parentheses, exponents, multiplication, division, addition, subtraction Blitzer, Intermediate Algebra, 5e – Slide #10 Section 1.1

Blitzer, Intermediate Algebra, 5e – 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, Intermediate Algebra, 5e – Slide #11 Section 1.1

Order of Operations - PEMDAS EXAMPLE Evaluate for . SOLUTION Replace R with 3 Evaluate inside parentheses first Evaluate – first exponent Blitzer, Intermediate Algebra, 5e – Slide #12 Section 1.1

Order of Operations - PEMDAS CONTINUED Evaluate – second exponent 27-2(81) 27-162 Multiply -135 Subtract Blitzer, Intermediate Algebra, 5e – Slide #13 Section 1.1

Number Sets Sets of Numbers Definition Natural Numbers All numbers in the set {1,2,3,4,…} Whole Numbers All numbers in the set {0,1,2,3,4,…} Integers All numbers in the set {…-3,-2,-1,0,1,2,3,…} Rational Numbers All numbers a/b such that a and b are integers Irrational Numbers All numbers whose decimal representation neither terminate nor repeat Real Numbers All numbers that are rational or irrational NOTE: “…” means continue without end Blitzer, Intermediate Algebra, 5e – Slide #14 Section 1.1

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 1.1 Three Common Number Sets Note that… 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… Blitzer, Intermediate Algebra, 5e – Slide #15 Section 1.1

Set-Builder Notation Express x > 10 using set-builder notation EXAMPLE Express x > 10 using set-builder notation SOLUTION {x | x is a real number and greater than 10} Blitzer, Intermediate Algebra, 5e – Slide #16 Section 1.1

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 1.1 Rational Numbers Definition 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. 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. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 1.1

Blitzer, Intermediate Algebra, 5e – 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, Intermediate Algebra, 5e – Slide #18 Section 1.1

Inequalities < > Inequalities Meanings Examples is less than 10 < 32 -5 < 3 -7 < -2 > is greater than 6 > -4 11 > 8 -6 > -12 is less than or is equal to 3.4 4.5 -2 -2 is greater than or is equal to 5 5 0 -3 Blitzer, Intermediate Algebra, 5e – Slide #19 Section 1.1