# Math 150 Lecture Notes for Chapter 2 Equations and Inequalities

**Math 150 Lecture Notes for Section 2A Solving Equations
**Introduction and Review of Linear Equations

Solve the equation for x. Check your answer.

Solving Quadratic Equations

A

**quadratic equation**is an equation that can be written in the form .

I. Solve by Factoring

II. Solve by Completing the Square

III. Solve by Using the Quadratic Formula

The solution to any quadratic equation of the form
is

Solve:

Equations in Quadratic Form

Solve:

Rational Equations

A **rational equation **is an equation which involves rational expressions, or
fractions.

Solve:

Radical Equations

Strategy to solve radical equations:

1. Isolate one radical.

2. Raise both sides of the equation to the appropriate power to remove the
radical.

3. Repeat the process until all radicals have been removed.

4. Check for extraneous solutions!

Solve:

Absolute Value Equation

Solve:

Equations in Several Variables

Solve:

a. Solvefor y where

b. Solve for b. Note
that S is the surface area of a square right

pyramid where b is the length of a side of the square base and l is the slant
height.

**Math 150 Lecture Notes for Section 2B Solving
Inequalities**

Introduction

Find the values of x which satisfies the inequality
.

Answer as a statement:

Answer as number line graph:

Answer using interval notation:

Linear Inequalities

Note: When you multiply or divide an inequality by a negative number, you must
reverse

the inequality sign.

Multiply both sides of
by
-1.

Solve:

Absolute Value Inequalities

can be thought of as “what numbers are less than 5 units from 0 on the
number

line?”

can be thought of as “what numbers are more than 5 units from 0 on the
number

line?”

is equivalent to

is equivalent to

Solve:

Nonlinear Inequalities

Strategy to solve nonlinear inequalities:

1. Move every term to one side (make one side zero).

2. If possible, factor the expression on the nonzero side.

3. Find the critical values or values for which the expression is zero or
undefined.

4. Draw a number line and let the critical numbers divide the number line into

intervals.

5. Determine the sign for each factor in each interval.

6. Determine if the sign for all factors in each interval is positive or
negative and

compare to our inequality to see if it is more than or less than zero.

Solve:

(Why don’t we just multiple both sides by
to clear all

denominators?)

(Why don’t we just divide both sides by x?)