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DrDelMath

Intermediate Algebra 5th Edition
by Elayn Martin-Gay
SUMMARY

Chapter 2: Equations, Inequalities, and Problem Solving

Section 2.1: Linear Equations in One Variable and
Section 2.4 Linear Inequalities in One Variable

Textbook Objectives
  1. Solve linear equations using properties of equality.
  2. Solve linear equations that can be simplified by combining like terms.
  3. Solve linear equations containing fractions or decimals.
  4. Recognize identities and equations with no solutions
Additional Goals
  1. Learn the definition of equation.
  2. Be able to recognize identites, conditional equations, and contradictions.
  3. Learn the definition of equivalent equations.
  4. Learn, understand, and apply the two fundamental properties of equations.
  5. Know the definitions of solution, solution set, and solving.
  6. Use the two fundamental properties of equations to solve equations.
  7. Learn the definition of linear equation in one variable.
  8. Be able to solve any linear equation in one variable.
  9. Know the definiton of graph and be able to graph any linear equation in one variable.
Textbook Objectives
  1. Use interval notation.
    NOTE: All inequalities in this section are inequalities in one variable. There are others later.
  2. Solve linear inequalities in one variable using the addition property of inequality.
  3. Solve linear inequalities in one variable using the multiplication and addition properties of inequality.
  4. Solve problems that can be modeled by linear inequalities in one variable.
Additional Goals
  1. Know and be able to correctly use the symbols for inequality.
  2. Learn the definition of inequality.
  3. Learn the definition of linear inequality in one variable.
  4. Learn definitions of solution, solution set, and graph of an inequality.
  5. Learn the definitions of equivalent inequalities, simplest inequality, and solving inequalities.
  6. Learn three Properties of Inequalities.
  7. Learn and use a procedure to solve linear inequalities in one variable.
  8. Be able to graph a linear inequality in one variable.
  9. Be able to use all four of the forms (graph, interval notation, set builder notation, and description) to discuss intervals of the Real Number line.
  10. Be able to use each of the four forms (graph, interval notation, set builder notation, and description) to describe the solution set for a linear inequality in one variable.

About Equations and Classifying Equations



In Section 1 of this chapter the topic of linear equations in one variable is discussed. These are equations of the form ax + b = 0. In Section 4 of this chapter the topic of linear inequalities in one variable are discussed. These are inequalities of the form ax + b < 0 and ax + b > 0. Note that in each of these three statements, an algebraic expression ax + b is compared to the number 0. Clearly these three statements resemble the Law of Trichotomy. In fact The Law of Trichotomy tells us that for each real number x:

As we examine linear equations in one variable and linear inequalities in one variable side by side we will uncover even stronger ties between the three expressions ax + b < 0, ax + b = 0 and ax + b > 0

When solving ax + b = 0, the other two siblings ax + b < 0 and ax + b > 0 cry out for attention.
When solving ax + b < 0, the other two siblings ax + b = 0 and ax + b > 0 cry out for attention.
When solving ax + b > 0, the other two siblings ax + b = 0 and ax + b < 0 cry out for attention.

You should recognize that when you solve any one of the three, solutions to the other two are close at hand and may be determined without another "solution process" or another bunch of calculations.

The relationship between solution sets for linear equations in one variable and linear inequalities in one variable is an example of relationships between equations and inequalites of any degree involving any number of variables.
It is an important principle to observe and learn!

Fundamentals of Equations:
Fundamentals of Inequalities:

About Solutions of Equations and Inequalities. Graphs of Equations and Inequalities. Equivalent Equations and Inequalities

The following properties of equations and inequalities are extremely important. These properties are the first things that should come to mind when you encounter an equation or inequality in any context.

These properties are the basic tools for:

Properties of Equations:
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Properties of Inequalities:

About Basic Properties of Equations and Inequalities

Linear Equations in One Variable:
Linear Inequalities in One Variable:

About Linear Equations and Inequalities in One Variable

Putting It Together

Law of Trichotomy and Boundary Equations.

It is appropriate that you read the first 3 pages of this essay.

About Putting It All Together

Minimal List of Exercises Page 53.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Some of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 12
Section 2.1 Exercise Set: 4, 12, 15, 17, 21, 23, 27, 28, 31, 32, 33, 34, 35, 37, 41, 43, 45, 51, 55, 71, 73, 74, 75, 76, 78, 81, 82, 83, 84, 87, 88, 89, 90.

Minimal List of Exercises Page 83.

Vocabulary and Readiness Check: 1 - 12
For each inequality that you solve in the following list you should write the solution set in both interval notation and set builder notation. You should also graph the inequality. Finally you should write a description of the solutions set.

Section 2.4 Exercise Set: 1, 5, 9, 13, 17, 18, 25, 31, 33, 41, 45, 53, 54, 57, 65, 67, 69, 71, 89, 90, 91, 92, 103, 104, 105.

How do exercises 103, 104, and 105 relate to the Law of Trichotomy?

 

Section 2.2: An Introduction to Problem Solving

Textbook Objectives
  1. Write algebraic expressions that can be simplified.
  2. Apply the steps for problem solving.
Additional Goals
  1. Learn steps/guidelines for problem solving.
  2. Be able to translate a problem into an equation which models the problem.
  3. Solve various simple "word" problems.li

Additional worked out exercises.

Additional worked out examples are available. Some of them are interactive.

Click HERE to study some exercises.
Click HERE to see some worked exercises.
For some worked Mixture Problems click HERE.

Minimal List of Exercises Page 62.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Some of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 12
Section 2.2 Exercise Set: 5, 9, 13, 15, 17, 18, 20, 21, 22, 23, 25, 26, 29, 31, 42, 43, 49, 51, 59,81, 82, 83.

Section 2.3: Formulas and Problem Solving

Textbook Objectives
  1. 1. Solve a formula for a specified variable.
  2. 2. Use formulas to solve problems.
Additional Goals
  1. Learn or review formulas for area, perimeter, and volume of certain geometric shapes.
  2. Learn the Pythagorean Theorem.
  3. Be able to use temperature conversion formulas.
  4. Learn and be able to use the formula for percent.
  5. Learn the formula which relates to distance, rate, and time.
  6. Learn and be able to use the formula for simple interest.

The following are links to other web pages which are related to the topics in this section.
The alternate or supplemental discussion presented on these other pages should be helpful.

Certain basic knowledge is required to do any mathematical modeling. For example to model a problem in mechanical engineering, you must know enough about mechanical engineering to understand the problem and convert it to mathematical statements. The very nature of the world in which we live requires that you must know and be able to use the basic formulas presented in this section.

Math Open Ref interactive page about area of squares Math Open Ref interactive page about area of parallelogram Math Open Ref interactive page about area of rectangles Math Open Ref interactive page about area of triangles Math Open Ref interactive page about area of trapezoids Math Open Ref interactive page about area of circles Math Open Ref interactive page about circumference of a circle

 

About Variables, Algebraic Expressions, and Basic Formulas



It is not required to memorize the following three temperature conversion formulas, but you should be able to work with them.

Percent

Each percent problem is solved by referring to a basic formula that relates percentage, percent, and base.

Use this verbal statement and the basic percent formula to help translate percent problems into an equation. Finally remember the meaning of percent is "per 100" and convert all percents to decimals.

Additional worked out exercises related to percent.

Additional worked out examples are available. Some of them are presented in a diagramatic manner which may help you understand percent problems a bit better. Click HERE.

 

About Percent

Minimal List of Exercises Page 72.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Some of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.
Each of the individually listed exercises should be completed. In each of the lists (like 1 - 24) work as many exercises as needed to insure that you understand the concept being illustrated or utilized.

Section 2.3 Exercise Set: 1 - 24, 29, 30, 33, 35, 37, 39, 48, 51, 53.

 

Section 2.5: Compound Inequalities

Textbook Objectives
  1. Find the intersection of two sets.
  2. Solve compound inequalities containing and.
  3. Find the union of two sets.
  4. Solve compound inequalities containing or.
Additional Goals
  1. Know the definition of compound inequality.
  2. Know the definition of union of sets.
  3. Know the definition of intersection of sets.
  4. Know and be able to correctly use the symbols for intersection and union.
  5. Use and interpret Venn diagrams to illustrate union and intersection of sets.
  6. Be able to correctly and interchangably use intersection and the conjunction and.
  7. Be able to correctly and interchangably use union and the conjunction or.

A Power Point Introduction to Sets.

Before beginning the study of these last three sections of the text you should study the follwing power point introduction to sets. HERE

 

About Introduction to Sets

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About Compound Inequalities

Minimal List of Exercises Page 92.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Some of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.
Each of the individually listed exercises should be completed. In each of the lists (like 1 - 24) work as many exercises as needed to insure that you understand the concept being illustrated or utilized.

Vocabulary and Readiness Check: 1 - 8

For each inequality that you solve in the following list you should write the solution set in both interval notation and set builder notation. You should also graph the inequality. Finally you should write a description of the solutions set.

Section 2.5 Exercise Set: 1, 5, 7, 9, 15, 21, 27, 41, 51, 55, 63, 87, 89.

 

Section 2.6: Absolute Value Equations and
Section 2.7: Absolute Value Inequalities

Textbook Objectives
  1. Solve absolute value equations.
Additional Goals
  1. Know the definition of absolute value.
  2. Recognize and use the fact that if k > 0 then |X| = k is equivalent to the pair of equations X = k or X = -k
  3. Recognize and use the fact that if k < 0 then an equation of the type |X| = k has no solution.
  4. Recognize and use the fact that an equation of the type |X| = 0 is equivalent to X = 0.
Textbook Objectives
  1. Solve absolute value inequalities of the form |X| < a.
  2. Solve absolute value inequalities of the form |X| > a.
Additional Goals
  1. Recognize and use the fact that |X| < 0 has no solution.
  2. Recognize and use the fact that the solution set for |X| > 0 is all real numbers for which X is not 0.
  3. Recognize and use the fact that if k < 0 then |X| < k has no solution.
  4. Recognize and use the fact that if k < 0 then the solution set for |X| > k is R, the set of Real numbers.
  5. Recognize and use the fact that if k > 0 then |X| < k is equivalent to the compound inequality -k < X < k
  6. Recognize and use the fact that if k > 0 then |X| > k is equivalent to the compound inequality X < -k OR X > k

Solving equations or inequalities involving absolute values is based on the definition of absolute value.

Notice the precise definition of absolute value has two cases:
    Case 1: The expression inside the absolute value symbol is positive or zero.
    Case 2: The expression inside the absolute value symbol is negative.

Procedure: To solve an equation or an inequality involving absolute values of variable expressions, it is necessary to consider the two equations which naturally result from the definition of absolute value.

Every equation or inequality involving absolute value is solved by considering the two cases as in this general example:
Generic Example: To solve an equation or inequality involving |something|, two cases must be considered. The two cases arise from the definition of absolute value. Therefore to solve any equation or inequality involving |something|, we consider:
    Case 1: The equation that results from relacing |something| with (something)
    Case 2: The equation that results from relacing |something| with (the opposite of something).

The following two properties of equations and three properties of inequalities are also important tools when solving equations involving absolute value.

generating equivalent inequalitiesshow button generating equivalent equationsshow button

These five properties might be used to:
    Simplify the equation or inequality before considering the two cases.
    Solve the equations or inequalities in Case 1 and Case 2.

In this section of the textbook the expression inside the absolute value symbol is always a linear expresion in one variable. Each of these equations or inequalities can be solved by directly using the two cases that arise from the definition of absolute value together with the two properties for generating equivalent equations and the three properties for generating equivalent inequalities. HOWEVER, there is an easier method which will be proven in the next chapter. For the time being we will simply explain and demonstrate the easier method.

Solving Equations and Inequalities of the form |ax + b| = k, |ax + b| < k, and |ax + b| > k
Begin to understand this topic by recalling the Law of Trichotomy.

The Law of Trichotomy informs us that the constant k on the right side is negative, zero, or positive.
If k is negative, we easily observe:

If k is 0, then we observe that the absolute value of an expression is 0 if and only if the expression is 0. Therefore if k is 0, the equations and inequalities are equivalent to ax + b < 0, ax + b = 0, and ax + b > 0. These are simple linear equations and linear inequalities which were previously mastered.

The only remaining case is when k is positive. Therefore in the remaining discussion we will assume that the constant k on the right is positive.

Fact:When k is positive:

The word equivalent as used in the above fact means | X | < k has the same solution set as -k < X < k.

Fact: Neither the corresponding "greater than" inequality nor the corresponding equality can be written in such a compact form.

The equation is called the boundary equation because its graph forms a boundary between the graphs of the two inequalities.

interval all three

The fact that the "less than" inequality is equivalent to a compact compound inequality makes it the easiest to solve because the compact form incorporates both cases into one computational structure.

The Law of Trichotomy informs us that when we consider any one of |ax + b| = k, |ax + b| < k, or |ax + b| > k we should, in fact, consider all three of them.

The Law of Trichotomy informs us that each real number is a solution to one of |ax + b| = k, |ax + b| < k, or |ax + b| > k.
Therefore the union of the three solution sets is R.
Therefore when all three are graphed on the same number line, the entire number line is used.

interval all three

The Law of Trichotomy informs us that each real number is a solution to only one of the three.
Therefore the intersection of any two of the solution sets is the empty set.
Therefore when all three are graphed on the same number line, none of the graphs overlap.

interval all three

The Process

  1. Whether solving |ax + b| = k, |ax + b| < k, or |ax + b| > k always solve |ax + b| < k computationally and then solve the other two by using deductive reasoning without any additional computation.

  2. Convert |ax + b| < k to its compact compound inequality form - k < ax + b < k.

  3. Solve the compact compound inequality by using the three methods to generate equivalent inequalities until simplest inequalities are obtained. The solution set will be an interval, call it S.
  4. Use Deductive Reasoning to conclude the solution set for the equality is the set containing the endpoints of S.

  5. Use Deductive Reasoning to conclude the solution set for the "greater than" inequality is everything else. So the solution set for the "greater than" inequality will be the two rays to the left of and to the right of S.

 

About Absolute Value Equations and Inequalities

Minimal List of Exercises Page 98.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Some of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.
Each of the individually listed exercises should be completed. In each of the lists (like 1 - 24) work as many exercises as needed to insure that you understand the concept being illustrated or utilized. Vocabulary and Readiness Check Page 98: 1 - 5
Section 2.6 Exercise Set: 5, 11, 17, 21, 39, 47, 51, 57, 59, 71, 82, 83, 84, 85.

Vocabulary and Readiness Check Page 104: 1 - 5
Section 2.7 Exercise Set Page 104: 5, 13, 23, 43, 47, 49, 53, 69, 72, 77, 93, 94, 95, 96, 99.
Answer Questions 65, 66, and 67. Sketch graphs. How are these three exercises related to the Law of Trichotomy?