Properties of Equations:
DrDelMathIntermediate Algebra 5th Edition
|
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!
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:
It is appropriate that you read the first 3 pages of this essay.
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.
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?
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.
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.
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.
It is not required to memorize the following three temperature conversion formulas, but you should be able to work with them.
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 examples are available. Some of them are presented in a diagramatic manner which may help you understand percent problems a bit better. Click HERE.
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.
Before beginning the study of these last three sections of the text you should study the follwing power point introduction to sets. HERE
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.
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 inequalities
generating equivalent equations
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.
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:
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.
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.
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.
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?