1. 1.1 Some Basics of Algebra Algebraic Expressions and Their Use
Translating to Algebraic Expressions
Evaluating Algebraic Expressions
Sets of Numbers

2. Terminology
A letter that can be any one of various numbers is called a variable. If a letter always represents a particular number that never changes, it is called a constant.

3. Algebraic Expressions
An algebraic expression consists of variables, numbers, and operation signs.
Examples:
When an equal sign is placed between two expressions, an equation is formed.

4. Translating to Algebraic Expressions
Key Words
per of
less than
more than
ratio
twice
decreased by
increased by
quotient of
times
minus
plus
divided by
product of
difference of
sum of
divide
multiply
subtract
add
Division
Multiplication
Subtraction
Addition

5. Example Translate to an algebraic expression:
Eight more than twice the product of 5 and a number.
Solution
Eight more than twice the product of 5 and a number.

6. Evaluating Algebraic Expressions
When we replace a variable with a number, we are substituting for the variable. The calculation that follows is called evaluating the expression.

7. Example Evaluate the expression Solution 8xz – y = 8·2·3 – 7 = 48 – 7
Substituting
= 48 – 7
Multiplying
= 41
Subtracting

8. Example
The base of a triangle is 10 feet and the height is 3.1 feet. Find the area of the triangle.
Solution h
10·3.1 b
= 15.5 square feet

9. Exponential Notation n factors
The expression an, in which n is a counting number means
n factors
In an, a is called the base and n is called the exponent, or power. When no exponent appears, it is assumed to be 1. Thus a1 = a.

10. Rules for Order of Operations
1. Simplify within any grouping symbols.
2. Simplify all exponential expressions.
3. Perform all multiplication and division working from left to right.
4. Perform all addition and subtraction working from left to right.

11. Example Evaluate the expression Solution 2(x + 3)2 – 12 x2
= 2(2 + 3)2 –
Substituting
Working within parentheses
Simplifying 52 and 22
Multiplying and Dividing
Subtracting

12. Example Evaluate the expression Solution
4x2 + 2xy – z = 4·32 + 2·3·2 – 8
Substituting
= 4·9 + 2·3·2 – 8
Simplifying 32
= – 8
Multiplying
= 40
Adding and Subtracting

13. Part 2 of 1.1
Sets of Numbers

14. Sets of Numbers Natural Numbers (Counting Numbers) Whole Numbers
Numbers used for counting: {1, 2, 3,…}
Whole Numbers
The set of natural numbers with 0 included: {0, 1, 2, 3,…}
Integers
The set of all whole numbers and their opposites: {…,-3, -2, -1, 0, 1, 2, 3,…}

15. Sets of Numbers Rational Numbers
Numbers that can be expressed as an integer divided by a nonzero integer are called rational numbers:

16. Converting Fractions to Decimals
Divide the numerator by the denominator
4 8
2 0
1 8

17. Any fraction can be converted to a repeating decimal or a terminating decimal..

18. Look at the following conclusion.
All integers can be written as fractions. Insert a denominator of 1.
Look at the following conclusion.

19. Rationals include the following.
All integers (-2, 5, 17, 0)
All fractions (proper, improper, or mixed)
All terminating decimals
All repeating decimals
-2.34,

20. Sets of Numbers Real Numbers
Numbers like are said to be irrational. Decimal notation for irrational numbers neither terminates nor repeats.
Real Numbers
Numbers that are either rational or irrational are called real numbers:

21. Identify as natural, whole, integers, rational, or irrational.-3 10
0.457
6.2

22. Answers Natural: 10 Whole: 10 0 Integers: 10 0 -3 Rational:
Irrational:

23. Set Notation Roster notation: {2, 4, 6, 8}
Set-builder notation: {x | x is an even number between 1 and 9}
“The set of all x
such that
x is an even number between 1 and 9”

24. Write with Roster Notation

25. Write with Set-Builder Notation
1) The set of all integers between -6 and 1 5)

26. Elements and Subsets
If B = { 1, 3, 5, 7}, we can write B to indicate that 3 is an element or member of set B. We can also write B to indicate that 4 is not an element of set B.
When all the members of one set are members of a second set, the first is a subset of the second. If A = {1, 3} and B = { 1, 3, 5, 7}, we write A B to indicate that A is a subset of B.

28. True or False? Use the following sets: N= Naturals, W = Wholes,
Z = integers, Q = Rationals, H = Irrationals,
and R = Reals

29. Answers
False
True