Chapter 3: Equations and Inequations

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
• Inequality statements
• The zero principle
• Use systematic trial and the Cover-up method to
  solve equations
• Use Algebra tiles and algebraic symbols to solve
  equations
• 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.
• Solving a multi-step equation
• Solving an equation with fractions
• Solving a word problem by using an equation

8
3.1 6

• Suggested strategy
• When solving equations, I suggest to always check
  your answer by substituting the exact value of
  the variable directly into the equation.
• If both sides yield the same value, then your
  solution is correct.

9
3.2 Represent Sets Graphically and Symbolically1

• An inequality is a mathematical statement
  relating expressions by using one or more
  inequality symbols lt, gt, , 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 lt 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 lt 5, x e R
• The e 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
• Symbolically as an inequation (for example, x
  -2 x lt 5, x e R) and
• 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
• Make a table of values
• Use systematic trial and error
• Looking for a pattern
• 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
• Solving a word problem with the help of an
  equation.
• 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
• Read the problem completely a minimum of three
  times.
• Choose a variable (typically a letter of the
  alphabet) to represent the unknown.

23
3.4 5

• Write an equation. (this is the difficult part)
• Solve the equation algebraically.
• 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?