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 DiagramVenn Diagram

Example 1: To which number set(s) does each belong?

a) -7b) ¾ c) d) e) 0

Example 2: Consider the numbers , π, , , 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.

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.

(-1 4) (-1 4)

a.b.

________________

(-1 4) (-1 4) c.d.

________________

4. Set Builder Notation – uses inequalities (usually) to define the set

({x | 8 < x ≤ 15 and x N})-may contain the element symbol (є)

(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.” ) (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 set of all numbers less than 3{x| x < 3}

The set of all positive even numbers{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

Inequality

Interval Notation

Set-Builder Notation

Words

x 2

x 2

x 2

(Brackets [ ] include the endpoints.Parentheses ( ) do not include endpoints.)1 x 1 OR 3 x 4

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

For Addition

For Multiplication

Closure

Commutative

Associative

Identity

Inverse

Distributive

Example 6: Which property is illustrated?

a) 9a + (6a + 4) = (9a + 6a) + 4b) 8(2y – 6) = 16y – 48

c) (7x + 3) + 0 = 7x + 3d)

e) 5a + (4 + 7b) = 5a + (7b + 4)f) (9s + t2)(1) = 9s + t2

Properties of Equality

Property

Example

Reflexive

Symmetric

Transitive

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.

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

d d – c

d

c

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.

( 3c2c d 2cd d )

a. 3xb.

2x – y

2x

y 3x – y

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 set-builder notation.

(yx5-5-5)Example 1: Give the domain and range of each relation.

a) {(2,3), (3,1), (-1,2), (4,-2)} b)

(yx5-5-5)

c)

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.)

(yx5-5-5) (yx5-5-5)Ex: Are the following relations functions?

(yx) (yx)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.

a) b)

c) {(4, 2), (1, 3), (-5, 0), (4, 3)}d) {(0, 4), (1, 5), (0, 2), (5, 1)}

(yx5-5-5) (yx5-5-5) (yx5-5-5)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) b) c)

( Interval & Set-builder) (Interval)(Set-Builder)

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”

(Input value) (Output 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 b) f(a) = ¾ a – 1 c) f(y) = y2 + 2y

(yx5-5-5) (yx5-5-5)

c) d)

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 = 122. 2x + 3 = 5x – 1 3. 2(3y – 1) = 4y + 7

4. -2 (x - 3) = -45. 12 - 3(w + 7) = 15 6. 4(8 - p) - (7 - p) = 22

7. 18 - 4y = -2(6 + 2y)8. 7t + 6 - 2 = 5t - 11

9. 32 + 4 (c -1) = - (4c + 5)

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 + 52. 8(y + 7) = 6y – 8 + 2y

3. 5(x – 6) = 3x – 18 + 2x4. 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 + 82. x + 8 ≥ 4x + 17.

3. 12 + 3q > 9q – 184. -18d + 5 (8 + 3d) ≤ 7 (3d - 8)

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 ≤ 14. x – 5 < –2 or –2x ≤ –10

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| = 3b) |2x + 1| < 3c) |2x + 1| ≥ 3

Examples: Solve and describe the solution set.

1. 2.

3. 4.

5. 6.

7. 8. 3 |6y – 9| + 12 > 24

2

c

2

6c-2d

3

3c-d

3

æö

+

ç÷

èø

3

5

2

t

k+6=9

5

£

2x+3

-24m+3<-14

m-4-5=0

5z+3-7<34

³

-2x+1+5-3

6