1. Chapter 1 Algebra, Mathematical Models, and Problem Solving

2. § 1.1
Algebraic Expressions and Real Numbers

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

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

19. 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
is greater than or is equal to
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 1.1