Real Numbers & Sets, Properties, Expressions,
Equations, Inequalities, Absolute Value
1.1 Sets of Numbers
Real Numbers (R)
Natural Numbers (N) –
Whole Numbers (W) –
Integers (Z) –
Rational Numbers (Q) –
Irrational Numbers (I) –
Tree Diagram
Venn Diagram
Example 1: To which number set(s) does each belong?
a) -7
b) ¾
c)
d) 3.16̅
6
̅ , π, √−5, −
Example 2: Consider the numbers 2. 3
11
2
e) 0
, and 2.7652.
a) Order the numbers from least to greatest.
b) Classify each number by the subsets of the real numbers to which it
belongs.
2
Number Sets
A set is a collection of items called elements. The empty set (Ø) contains no
elements.
Sets can be finite, with a countable number of elements:
Eg. The set of all possible rolls of a single di.
Or sets can be infinite, with unlimited elements:
Ex. The set of all real numbers between 1 and 6.
Sets can be described in 4 ways:
1. Words – The set of all real numbers less than three.
The set of all even numbers between 2 and 20.
2. Roster Notation – {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
-can be used for finite and infinite sets
3. Interval Notation – set of all numbers btwn 2 endpoints
-always represents an infinite set
- curved ( symbol means the endpt is not in the set
-square [ symbol means the endpt is in set.
Example 3: Describe each set in interval notation.
a.
-1
4
b.
-1
________
c.
4
________
d.
-1
-1
4
________
4
________
3
4. Set Builder Notation – uses inequalities (usually) to define the set
-may contain the element symbol (є)
{x | 8 < x ≤ 15 and x  N}
Read the above as “the set of all numbers x such that x is greater than 8
and less than or equal to 15 and x is a natural number.”
The symbol  means “is an element of.” So x  N is read “x is an element
of the set of natural numbers,” or “x is a natural number.”
The set of all numbers less than 3
The set of all positive even numbers
{x| x < 3}
{x| x = 2n and n єN}
Example 4: Describe the sets pictured in Ex. 3 in set-builder notation.
a.
b.
c.
d.
Example 5: Write each set in the indicated notations.
Representing Intervals
Number Line
Interval
Notation
Inequality
Set-Builder
Notation
x2
x2
x2
1  x  1
OR
3x4
Brackets [ ] include the endpoints.
Parentheses ( ) do not include endpoints.
4
Words
Could the roster notation be used to represent any of the sets in this table?
Why or why not?
1.2 Properties
Properties of Real Numbers
Property
Closure
Commutative
Associative
Identity
Inverse
Distributive
For Addition
For Multiplication
Example 6: Which property is illustrated?
a) 9a + (6a + 4) = (9a + 6a) + 4
b) 8(2y – 6) = 16y – 48
4𝑚
∙
3
=1
c) (7x + 3) + 0 = 7x + 3
d)
e) 5a + (4 + 7b) = 5a + (7b + 4)
f) (9s + t2)(1) = 9s + t2
3
4𝑚
Properties of Equality
Property
Reflexive
Symmetric
Transitive
Example
5
Example 7: Which property is illustrated?
a) 5(a + 7b) = 5(a + 7b)
b) If 2m = n and n = 6, then 2m = 6
c) If 9x – 8 = 10, then 10 = 9x – 8
d) If y = 3x + 2 and y = 4x – 7 then 3x + 2 = 4x – 7.
6
1.4 Simplifying Algebraic Expressions
Variable – a symbol or letter that represents a number
Term – a number, a variable, or the product of the two
Ex: 2, 2x,
x, 4xy2
Coefficient – the numerical factor in a term (2x2)
Algebraic expression – an expression that contains one or more terms.
Ex: x, 2x2, 5xy2z3 + 4x
To evaluate an algebraic expression – substitute numbers for variables and
follow the order of operations.
Example 8:
Evaluate (k – 18)2 – 4k for k = 6.
Like Terms – terms that have the same variables with the same exponents
(2x and 4x; 3y2 and -9y2, -5xyz3 and 2xyz3)
To simplify an algebraic expression - combine all like terms.
Example 9: Simplify 2h – 3k + 7(2h – 3k)
Perimeter – the distance around a figure. Add up all sides.
Example 10: Find the perimeter of this figure. Simplify the answer.
c
c
2
d–c
d
c
2
d
c
7
1.4 Practice Problems
1. Evaluate each expression for c = -3 and d = 5.
a. c2 – d2
b. c(3 – d) – c2
c. – d2 – 4(d – 2c)
2. The expression – 0.08y2 + 3y models the percent increase of voters in a
town from 1990 to 2000. In the expression, y represents the number of
years since 1990 (so 1992 would be y=2).
a. Find the approximate percent of increase of voters by 1998.
b. Approximately what percent of the eligible voters will vote in 2012?
c. Approximately what percent of the eligible voters will vote in 2020?
3. Simplify by combining like terms.
a. 2x2 + 5x – 4x2 + x – x2
b. – 2(r + s) – (2r + 2s)
c. y(1 + y) – 3y2 – (y + 1)
4. Find the perimeter of each figure. Simplify your answer.
a.
3x
3c
b.
2x – y
2c
2x
d
d
y
3x – y
8
6c - 2d
3
2c
d
3c - d
3
1.6/1.7 Functions
A relation is a pairing of input values with output values. It can be shown as:
 a set of ordered pairs (x,y), where x is an input and y is an output.
 A table of values where the x column lists inputs and the y column lists
output values.
 A mapping diagram where corresponding input (x) and output (y) values
are connected with arrows.
 A graph where (x,y) pairs are plotted as points.
The set of input values (x) for a relation is called the ________, and the set
of output values (y) is called the _________.
 The domain and range are represented in roster notation whenever the
relation is a finite set of ordered pairs.
 When a relation is an infinite line or curve on a graph, the domain and
range can be represented as infinite sets using either interval or setbuilder notation.
Example 1: Give the domain and range of each relation.
a) {(2,3), (3,1), (-1,2), (4,-2)}
b)
y
x
y
c)
-5
-5
x
-5
-5
5
9
5
A relation in which the first coordinate is never repeated is called a function.
In a function, there is only one output for each input, so each element of the
domain is mapped to exactly one element in the range.
Two ways to determine whether a relation is a function:
1. List the ordered pairs and determine whether any domain (x) value is
repeated. (If any x value occurs more than once, the relation is not a
function.)
Ex: Is the following relation a function? {(3,1), (5,-2), (4,1), (-2,0)}
2. Use the vertical line test on the graph of the relation. (If any
vertical line passes through more than one point on the graph of a
relation, then the relation is not a function.)
Ex: Are the following
y relations functions?
y
x
x
-5
-5
-5
-5
5
5
Example 2: Find the domain and range in roster notation. (If needed write the
ordered pairs for the graphs.) Then tell if the relation is a function.
y
y
a)
b)
x
x
c) {(4, 2), (1, 3), (-5, 0), (4, 3)}
d) {(0, 4), (1, 5), (0, 2), (5, 1)}
10
Example 3: Use the vertical-line test to determine if each graph is a function.
Then find the domain & range in interval and/or set-builder notation.
a)
y
y
y
b)
c)
-5
x
x
x
-5
-5
-5
-5
5
( Interval & Set-builder)
-5
5
(Interval)
(Set-Builder)
5
When the ordered pairs of a function can be defined by an equation, the equation can be
written in function notation.
f(x) = 2x + 1 is read “the function of x equals 2x +1” or “f of x equals 2x + 1”
Output value
Input value
To find the value of a function for a given input, or x value, simply substitute the value into
the function rule, or equation.
f(3) = 2(3) + 1
Example 4: Find f(3) and f(-5) for each function.
a) f(x) = 3x – 5
c) f(y) = y2 + 2y
b) f(a) = ¾ a – 1
y
y
c)
d)
x
x
11
-5
-5
-5
5
-5
5
2.1 Solving Equations & Inequalities
Steps for solving equations:
1. Simplify each side of the equation separately.
2. Move all variable terms to one side of the equation.
3. Isolate the variable using inverse operations in the reverse order of
operations.
Examples: Solve.
1. 9x – 6 = 12
2. 2x + 3 = 5x – 1
3. 2(3y – 1) = 4y + 7
4. -2 (x - 3) = -4
5. 12 - 3(w + 7) = 15
6. 4(8 - p) - (7 - p) = 22
7. 18 - 4y = -2(6 + 2y)
3t
8. 7t + 6 - 2  5   = 5t - 11

9. 32 + 4 (c -1) = - (4c + 5)
12
2
Identities and Contradictions
 When solving an equation results in a statement that is true for all values,
the equation is an identity and the solution set is all real numbers ().
 When solving an equation results in a statement that is false for all values,
the equation is a contradiction and the solution set is the empty set ( ).
Examples: Solve.
1. 3x + 4x + 5 = 7x + 5
2. 8(y + 7) = 6y – 8 + 2y
3. 5(x – 6) = 3x – 18 + 2x
4. 3(2 – 3x) = -7x – 2(x – 3)
Solving Inequalities
Follow the same steps as solving equations, EXCEPT: If you multiply or divide
both sides by a negative number, you must reverse the inequality symbol.
Examples: Solve and graph the solution on a number line.
1. 8a –2 ≥ 13a + 8
2. x + 8 ≥ 4x + 17.
3. 12 + 3q > 9q – 18
4. -18d + 5 (8 + 3d) ≤ 7 (3d - 8)
13
2.8 Solving Absolute Value Equations & Inequalities
Compound Inequalities
 A disjunction is a compound statement that uses the word or.
x ≤ –3 OR x > 2
Set builder notation: {x|x ≤ –3 OR x > 2}
Interval notation: (-∞,-2] OR (2, ∞)
 A conjunction is a compound statement that uses the word and.
x ≥ –3 AND x < 2 (or -3≤ x< 2)
Set builder notation: {x|x ≥ –3 AND x < 2} or {x| -3≤ x< 2 }
Interval notation: [-3,2)
To solve compound inequalities, solve each inequality and graph the solution.
Examples: Solve and describe the solution in interval notation.
1. 9x < 54 and – 4x < 12
2. 6(x + 2) ≥ 24 or 5x + 10 ≤ 15
3. 3x – 5 ≥ – 8 and 3x – 5 ≤ 1
4. x – 5 < –2 or –2x ≤ –10
14
Absolute Value
Recall that the absolute value of a number x, written |x|, is the distance from
x to zero on the number line. Absolute-value equations and inequalities can
be represented by compound statements.
Consider the equation |x| = 3:
The solutions of |x| = 3 are the
two points that are 3 units from
zero. The solution is a disjunction:
x = –3 or x = 3.
The solutions of |x| < 3 are the
points that are less than 3 units
from zero. The solution is a
conjunction: –3 < x < 3.
The solutions of |x| > 3 are the
points that are more than 3 units
from zero. The solution is a
disjunction: x < –3 or x > 3.
Solving Absolute Value Equations & Inequalities:
1. Isolate the absolute value expression if necessary.
2. Rewrite as a compound equation/inequality.
 Equation – disjunction with 2nd equation set equal to the opposite.
 Less Than(d) inequalities ( < or ≤ ) – conjunction with 2nd inequality
reversed and set to the opposite.
 Greator inequalities ( > or ≥ ) – disjunction with 2nd inequality
reversed and set to the opposite.
3. Solve both equations/inequalities in the compound statement.
4. Graph the solution and describe the solution set.
For Example:
a) |2x + 1| = 3
b) |2x + 1| < 3
15
c) |2x + 1| ≥ 3
Examples: Solve and describe the solution set.
1.
k +6 =9
2.
2x +3  5
3. -2 4m+3 < -14
4. m- 4 - 5 = 0
5. 5z +3 - 7 < 34
6. -2 x +1 +5  -3
7. m- 4 - 5 = 0
8. 3 |6y – 9| + 12 > 24
16