Slide 1

A set is a collection of objects.An object in a set is called an element of that set.Different type of integers:

The real-number line is shown as 0.1 Sets of Real Numbers

11Important properties of real numbersThe Transitive Property of Equality

The Closure Properties of Addition and Multiplication

The Commutative Properties of Addition and Multiplication0.2 Some Properties of Real Numbers

The Commutative Properties of Addition and Multiplication

The Identity Properties

The Inverse Properties

The Distributive Properties

Example 1 Applying Properties of Real Numbers

Example 3 Applying Properties of Real Numbers

Solution:Show thatSolution:

Show that

Solution:

Properties:0.3 Exponents and Radicals

exponentbase6Example 1 Exponents

The symbol is called a radical. n is the index, x is the radicand, and is the radical sign.

8Example 3 Rationalizing Denominators

Solution:

Example 5 Exponents

Eliminate negative exponents in and simplify.Solution:

9b. Simplify by using the distributive law.Solution:

Eliminate negative exponents inSolution:

d. Eliminate negative exponents inSolution:

e. Apply the distributive law toSolution:

If symbols are combined by any or all of the operations, the resulting expression is called an algebraic expression.A polynomial in x is an algebraic expression of the form: where n = non-negative integer cn = constants0.4 Operations with Algebraic Expressions

is an algebraic expression in the

variable x.

is an algebraic expression in the

variable y.

is an algebraic expression in the

variables x and y.

A list of products may be obtained from the distributive property:

By Rule 2,

By Rule 3,

Example 5 Special ProductsExample 7 Dividing a Multinomial by a Monomial

16If two or more expressions are multiplied together, the expressions are called the factors of the product.0.5 Factoring

Example 1 Common Factors a. Factor completely. Solution:

b. Factor completely. Solution:

Example 3 Factoring

Simplifying FractionsAllows us to multiply/divide the numerator and denominator by the same nonzero quantity.Multiplication and Division of FractionsThe rule for multiplying and dividing is0.6 Fractions

Rationalizing the DenominatorFor a denominator with square roots, it may be rationalized by multiplying an expression that makes the denominator a difference of two squares.Addition and Subtraction of FractionsIf we add two fractions having the same denominator, we get a fraction whose denominator is the common denominator.Example 1 Simplifying Fractions a. Simplify Solution:

b. Simplify

Solution:

Example 3 Dividing Fractions

Example 5 Adding and Subtracting Fractions

24Example 7 Subtracting Fractions

EquationsAn equation is a statement that two expressions are equal. The two expressions that make up an equation are called its sides. They are separated by the equality sign, =.0.7 Equations, in Particular Linear EquationsExample 1 Examples of Equations

A variable (e.g. x, y) is a symbol that can be replaced by any one of a set of different numbers.Equivalent EquationsTwo equations are said to be equivalent if they have exactly the same solutions.There are three operations that guarantee equivalence:Adding/subtracting the same polynomial to/from both sides of an equation.Multiplying/dividing both sides of an equation by the same nonzero constant.Replacing either side of an equation by an equal expression.Operations That May Not Produce Equivalent Equations

Multiplying both sides of an equation by an expression involving the variable.Dividing both sides of an equation by an expression involving the variable.Raising both sides of an equation to equal powers.Linear EquationsA linear equation in the variable x can be written in the form

where a and b are constants and .A linear equation is also called a first-degree equation or an equation of degree one.

Example 3 Solving a Linear EquationSolveSolution:

Example 5 Solving a Linear EquationsSolve

Solution:

Literal EquationsEquations where constants are not specified, but are represented as a, b, c, d, etc. are called literal equations.The letters are called literal constants.Example 7 Solving a Literal EquationSolve for x.

Solution:

Example 9 Solving a Fractional Equation Solve Solution:Fractional EquationsA fractional equation is an equation in which an unknown is in a denominator

35Radical Equations A radical equation is one in which an unknown occurs in a radicand.Example 13 Solving a Radical Equation Solve Solution:

A quadratic equation in the variable x is an equation that can be written in the form

where a, b, and c are constants andA quadratic equation is also called a second-degree equation or an equation of degree two.0.8 Quadratic Equations

Example 1 Solving a Quadratic Equation by Factoring a. Solve Solution: Factor the left side factor: Whenever the product of two or more quantities is zero, at least one of the quantities must be zero.

Example 3 Solving a Higher-Degree Equation by Factoring

a. Solve Solution:

Quadratic Formula

The roots of the quadratic equation

can be given as

Example 7 A Quadratic Equation with One Real Root Solve by the quadratic formula. Solution: Here a = 9, b = 62, and c = 2. The roots are

Quadratic-Form EquationWhen a non-quadratic equation can be transformed into a quadratic equation by an appropriate substitution, the given equation is said to have quadratic-form.Example 9 Solving a Quadratic-Form Equation Solve

Solution: This equation can be written as Substituting w =1/x3, we have

Thus, the roots are