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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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Addition Property of Equality

To solve linear equations in one variable we must use the Basic Properties of Equations:

For any real numbers A, B, C, and D:

1. REFLEXIVE PROPERTY: A = B ⇔ B = A

a) If x = 2 then 2 = x.

b) If x + 3 = 5 then 5 = x + 3

2. ADDITION PROPERTY: If A = B is true then A + C = B + C is true.

a) If x = 2 then x + 5 = 2 + 5

b) If x + 3 = 5 then (x + 3) + (- 3) = 5 + (- 3).

3. TRANSITIVE PROPERTY: If A = B and B = C is true then A = C is true.

[The transitive property is sometimes called the "substitution" property.]

DEFINITION: The solution set for an equation is the set of all numbers that when used is place of the variable make the equation a true statement.

DEFINITION: Two or more equations with the same solution set are said to be equivalent equations.

Example:

a) Is x = 3 a solution to 5x – 7 = 18 ?

5(3) – 7 = 18

25 – 7 = 18

18 = 18

b) Is x = 5 a solution to 9x – 8 = 6x + 7 ?

9(5) – 8 = 6(5) + 7

45 – 8 = 30 + 7

37 = 37

We will also apply the Basic Number Properties. In particular,

Use the "Inverse" Property of Addition

a + (-a) = 0

This will be referred to by a "code": Add Opps

Definition of Subtraction:

To subtract: Change to ADD the opposite
  a – b a + (-b) Say a & (-b)

We can apply these properties "horizontally" one row at a time, as the book does, or

we can apply the properties vertically using a pattern (or balance beam ), where we work on each side of the equation at the same time. Working vertically helps us to see that we are indeed "adding the same thing to both sides".

See the following applications working vertically - using the balance beam

2.2a. Constant added on either side: (b, d are letters to represent integers.).

x + b = d or d = x + b

2.2b Find the variable x. To solve the equation we will keep the variable x on that side of the equation and move the constants to the other side.

To do this we will add opposites on the balance beam below the equation. First look at the pattern then follow the same steps through the example below.

Pattern:

Add opps:

Complete the step:

Then

 
Let B = (d – b)

 the coefficient of x is 1.

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