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

2. Chapter 3 Get Ready These concepts need to be reviewed before beginning Chapter 3: 1. Inequality statements 2. The zero principle 3. Use systematic trial and the Cover-up method to solve equations 4. Use Algebra tiles and algebraic symbols to solve equations 5. Expand Algebraic Expressions

3. 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.

4. 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

5. 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.

6. 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.

7. 3.1 #5 Within this section, there are three types of problems that you will be required to solve. 1. Solving a multi-step equation 2. Solving an equation with fractions 3. Solving a word problem by using an equation

8. 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.

9. 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”.

10. 3.2 #2 The following are examples of inequalities: 4 < 5 x ≥ 3 -2 ≤ a ≤ 6

11. 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.

12. 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”.

13. 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)

14. 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.

15. 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)

16. 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.

17. 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.

18. 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.

19. 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.

20. 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 values 2. Use systematic trial and error 3. Looking for a pattern 4. Set up and solve an algebraic equation

21. 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.

22. 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.

23. 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.

24. Summary of Chapter 3 What subjects did we learn about in Chapter 3?