Chapter 1

Overview

This chapter focuses on finding the solution set for various types of Equations and Inequalities. (Linear, Quadratic, Polynomial, Rational, Absolute Value, Radical).

The general method for solving any equation is to start with the given equation and produce a sequence or equivalent equations (equations that have the same solution set) until the solution set is easily seen.

For example:

For many types of equations we use a property that is often taken for granted:

The Zero–Product Property

This property is used, in conjunction with factoring, to solve many types of equations.

Another very important method, which is used for solving Quadratic Equations, is:

The Quadratic Formula

This formula works for ANY Quadratic equation, but there are often other methods to solve quadratic equation that are shorter and simpler.

To solve a Rational equation, the first step in making it a simpler equation, we multiply each term of the equation by the Least Common Multiple of all the denominators in the equation. Then, by canceling the common terms, we have produced an equivalent equation that is no longer rational and we continue our sequence of equivalent equations until the solutions are evident.

To solve equations involving Absolute Value we must first understand the definition :

The key to understanding this definition is to think of –x as “the opposite of x ” and NOT “negative x”. So if x is less than zero (i.e. x is negative), then –x, “the opposite of x”, is greater that zero (i.e. –x is positive)! There for |x|= -x.

For solving Radical equations you will need to square the entirety of both sides of the equation. Squaring each term of the equation is a very common mistake!

For example:

Solving Inequalities involving these types of equations is very similar to solving the equalities. The one main difference is that when you Multiply or Divide each side (part) of the inequality by a negative quantity you must reverse the direction of the inequality symbol.

Here is an example that will help you remember:

These are a few of the Main Points of this chapter. If you work to UNDERSTAND these ideas the material in this chapter will be much easier!