chapter 3: equations and inequations this chapter begins on page 126

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Chapter 3: Equations and Inequations This chapter begins on page 126

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Page 1: Chapter 3: Equations and Inequations This chapter begins on page 126

Chapter 3: Equations and Inequations

This chapter begins on page 126

Page 2: Chapter 3: Equations and Inequations This chapter begins on page 126

Chapter 3 Get Ready These concepts need to be reviewed

before beginning Chapter 3:1. Inequality statements2. The zero principle3. Use systematic trial and the Cover-up

method to solve equations4. Use Algebra tiles and algebraic

symbols to solve equations5. Expand Algebraic Expressions

Page 3: Chapter 3: Equations and Inequations This chapter begins on page 126

3.1: Solve Single Variable Equations #1

An equation is a statement formed by two expressions related by an equal sign.

For example, 3x + 3 = 2x – 1 is an equation.

Page 4: Chapter 3: Equations and Inequations This chapter begins on page 126

3.1 #2 To solve an equation means to find the

number that can be substituted for the variable to make the equation true.

The number that makes the equation true is called the solution for the equation.

For example, for the equation, x + 2 = 6, the solution is x = 4 because 4 + 2 = 6

Page 5: Chapter 3: Equations and Inequations This chapter begins on page 126

3.1 #3 An equation can be thought of as a

balanced scale. In order to maintain the balance,

whatever is done to one side must also be done to the other side in the same step. This creates simpler equivalent equations.

Equivalent equations will have the same solution.

Page 6: Chapter 3: Equations and Inequations This chapter begins on page 126

3.1 #4

An inverse operation is a mathematical operation that undoes a related operation.

For example, addition and subtraction are inverse operations; multiplication and division are also inverse operations.

Page 7: Chapter 3: Equations and Inequations This chapter begins on page 126

3.1 #5

Within this section, there are three types of problems that you will be required to solve.

1. Solving a multi-step equation2. Solving an equation with fractions3. Solving a word problem by using

an equation

Page 8: Chapter 3: Equations and Inequations This chapter begins on page 126

3.1 #6

Suggested strategy: 1. When solving equations, I suggest

to always check your answer by substituting the exact value of the variable directly into the equation.

2. If both sides yield the same value, then your solution is correct.

Page 9: Chapter 3: Equations and Inequations This chapter begins on page 126

3.2: Represent Sets Graphically and Symbolically#1

An inequality is a mathematical statement relating expressions by using one or more inequality symbols <, >, ≤, or ≥

The symbol ≥ means “is greater than or equal to” and the symbol ≤ means “is less than or equal to”.

Page 10: Chapter 3: Equations and Inequations This chapter begins on page 126

3.2 #2

The following are examples of inequalities:

4 < 5 x ≥ 3 -2 ≤ a ≤ 6

Page 11: Chapter 3: Equations and Inequations This chapter begins on page 126

3.2 #3

Set notation is a mathematical statement that shows an inequality or equation and the set of numbers to which the variable belongs.

Page 12: Chapter 3: Equations and Inequations This chapter begins on page 126

3.2 #4

Here is an example of set notation:

{x| -2 ≤ x < 5, x ε R}

The ε sign (epsilon in Greek) means “belongs to” or “is an element of”.

Page 13: Chapter 3: Equations and Inequations This chapter begins on page 126

3.2 #5

Set notation can be expressed in two different ways:

1. Symbolically as an inequation (for example, {x| -2 ≤ x < 5, x ε R}) and

2. Graphically as a number line (see page 147)

Page 14: Chapter 3: Equations and Inequations This chapter begins on page 126

3.2 #6

When representing a set graphically (i.e. with a number line), an open circle shows that the number is not included in the set and a closed circle shows that the number is included in the set.

Page 15: Chapter 3: Equations and Inequations This chapter begins on page 126

3.2 #7 Important note: in Chapter 1, we learned

about subsets of the real numbers: natural numbers (N) whole numbers (W) integers (I) rational numbers (Q) irrational numbers (Q with a bar above it)

Page 16: Chapter 3: Equations and Inequations This chapter begins on page 126

3.3: Solve Single Variable Inequations #1

Solving an inequality is similar to solving an equation:

You still isolate the variable on one side of the inequality by performing inverse operations to both sides of the inequality.

Page 17: Chapter 3: Equations and Inequations This chapter begins on page 126

3.3 #2

However, when multiplying or dividing both sides of an inequation by a negative number, you must reverse the inequality sign.

This is very important and one must pay close attention to this fact if one wants to solve these inequalities properly.

Page 18: Chapter 3: Equations and Inequations This chapter begins on page 126

3.3 #3

The solution to an inequality is a set of values of a variable that make the inequality true.

For this reason it may be referred to as the solution set for the inequality.

Page 19: Chapter 3: Equations and Inequations This chapter begins on page 126

3.4: Problem Solving with Linear Equations and Inequalities #1

Being able to solve problems is an important skill in your daily life.

One goal of mathematics education is to help you develop a variety of problem-solving strategies.

Page 20: Chapter 3: Equations and Inequations This chapter begins on page 126

3.4 #2

There are several strategies available to solve problems. Here are four of them you may have already learned:

1. Make a table of values2. Use systematic trial and error3. Looking for a pattern4. Set up and solve an algebraic

equation

Page 21: Chapter 3: Equations and Inequations This chapter begins on page 126

3.4 #3

In this section, you should know how to solve 2 types of word problems:

1. Solving a word problem with the help of an equation.

2. Solving a word problem with the help of an inequation.

Page 22: Chapter 3: Equations and Inequations This chapter begins on page 126

3.4 #4 To solve a problem using an

equation, I suggest following these five simple steps:

1. Read the problem completely a minimum of three times.

2. Choose a variable (typically a letter of the alphabet) to represent the unknown.

Page 23: Chapter 3: Equations and Inequations This chapter begins on page 126

3.4 #5

3. Write an equation. (this is the difficult part)

4. Solve the equation algebraically.5. Write a conclusion. This means

verifying your solution by direct substitution into your equation and stating this solution in a complete sentence.

Page 24: Chapter 3: Equations and Inequations This chapter begins on page 126

Summary of Chapter 3

What subjects did we learn about in Chapter 3?