Linear Equations
Literal Equations
Simplifying Expressions & Solving Equations
Two Equations containing Two Variables
Solving Linear Equations
Plane Curves Parametric Equation
Linear Equations and Matrices
Trigonometric Identities and Conditional Equations
Solving Quadratic Equation
Solving Systems of Linear Equations by Graphing
Exponential and Logarithmic Equations
Quadratic Equations
Homework problems on homogeneous linear equations
Solving Quadratic Equations
Functions, Equations, and Inequalities
Solving Multiple-Step Equations
Test Description for Quadratic Equations and Functions
Solving Exponential Equations
Linear Equations
Linear Equations and Inequalities
Literal Equations
Quadratic Equations
Linear Equations in Linear Algebra
Investigating Liner Equations Using Graphing Calculator
represent slope in a linear equation
Linear Equations as Models
Solving Quadratic Equations by Factoring
Solving Equations with Rational Expressions
Solving Linear Equations
Solve Quadratic Equations by Completing the Square
Solving a Quadratic Equation

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Functions, Equations, and Inequalities

2.5 More Equation Solving

I. Rational Equations

Equations which contain rational expressions are called rational equations. We will use a four
step process for solving rational equations.

Step 1:
Identify any restrictions on the domain by setting each denominator equal to zero and
solving for x.

Step 2:
Multiply every term on both sides of the equation by the Least Common Denominator
(LCD) to eliminate all of the denominators.

Step 3: Solve the resulting equation.

Step 4: Exclude any solutions which are restrictions on the domain.

Example 1

Restrictions on the Domain: x ≠ –7 and x ≠ –1 LCD: (x + 7)·(x + 1)

Since neither of these x-values is a restriction on the domain, they are both valid

Example 2

Restrictions on the Domain: x ≠ – 4 and x ≠ 4 LCD: (x + 4)·(x – 4)

Since both of these x-values are restrictions on the domain, this equation has
no solution.

II. Radical Equations

A radical equation is an equation in which at least one term has a variable under a radical.
We will use a four step process for solving radical equations.

Step 1: Isolate the radical on one side of the equal sign.

Step 2: Raise both sides of the equation to the appropriate power to clear the radical.

Step 3: If the resulting equation still contains a radical, repeat steps 1 and 2.
When the equation contains no radical, solve the equation for x.

Step 4:
Anytime you raise both sides of an equation to an even power, you must check your
solutions in the original equation.

Example 3

checking 0:

0 is not a valid solution.

checking 5:

5 is a valid solution.

Example 4

checking 2:

checking 42:

is not a valid solution.

III. Solving an Equation for a Specified Variable

When we solve an equation for a specified variable, we must undo the operations which have
been done on the specified variable by doing the opposite operations in the opposite order.
We work from the outside in. First we undo Addition / Subtraction, then Multiplication / Division,
then Exponents, and finally Parentheses.

Example 5