DrDelMathIntermediate Algebra 5th Edition
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Embedded in the following diagram are some obvious facts about the Cartesian Coordinate system,
points in the system, and the Real Number coordinates of those points. Because these facts are
obvious, the beginning student has a tendency to overlook them at times when their application might be appropriate.
You are advised to study these facts and look for applications.
Location of GraphsThe preceding six definitions of solution, solution set, and graph make it clear that the graph of an equation in two variables or an inequality in two variables consists of points represented by ordered pairs of real numbers. Because every ordered pair of real numbers is associated with a point in the Cartesian Coordinate System (and visa versa), we deduce that the graph of any equation in two variables or inequality in two variables must appear in the Cartesian Coordinate System.
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Properties of Equations: |
Properties of Inequalities: |
An understanding of the Cartesian Coordinate System permits a development and an understanding of some basic formulas and concepts in an algebraic framework. We begin with an ancient formula (The Pythagorean Theorem) and use it to develop a few useful algebraic formulas.
The first two formulas for a circle should be memorized. They are used frequently in this course and the next as well as applications outside the classroom.
If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most 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 - 20
Section 3.1 Exercise Set: 1, 5, 6, 11, 13, 15, 17, 19, 21, 25, 27, 33, 39, 45, 49, 57, 59, 61.
Section 3.7 Exercise Set: 23, 29, 33, 35, 37, 43, 45
If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most 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 Concept Check: 1 - 10
Section 3.2 Exercise Set: 3, 5, 6, 9, 12, 17, 20, 22, 23, 25, 29-37, 39, 41, 43, 45, 50, 51, 52, 55, 67-69, 73, 75, 77, 81, 83, 93, 95.
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To rewrite an equation or inequality means to change the form of the equation without changing the solution set. Another way to understand the idea of rewriting an equation or inequality is to view it as generating an equivalent equation or inequality. Therefore the two Properties of Equations and Three Properties of Inequalities which are used to generate equivalent equations and inequalities are again important. When an equation in two variables is not written in the form y = mx + b, these basic properties are used to determine if it is a linear equation in two variables. Similar statements are valid for inequalities in two variables. |
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The symbols = and < are frequently combined to mean the union of the solution set
of the inequality and the solution set of the equality. The result is the
solution set of the inequality and its boundary. Therefore you may see statements of the form y ≤ mx + b or Ax + By ≤ C. Although an equation is involved, these are usually called inequalities. In the same way the symbols = and > are frequently combined to mean the union of the solution set of the inequality and the solution set of the equality. The result is the solution set of the inequality and its boundary. Therefore you may see statements of the form y ≥ mx + b or Ax + By ≥ C. Although an equation is involved, these are usually called inequalities. |
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When two linear expressions are compared, the Law of Trichotomy dictates that we consider three cases. We refer to the equation in the triple as the boundary equation because its graph forms a boundary between the graphs of the two inequalities.
When considering linear equations and inequalities in two variables, the graph of the boundary equation divides the coordinate plane into two half-planes, one of which is the graph of the "less-than" inequality and the other is the graph of the "greater-than" inequality.
Sketching the graph of the boundary equation (accomplished by plotting two points) and testing one point determines the solution sets and graphs for each of the three.
Observation: The above procedure considers, and solves, both inequalities as
well as the corresponding equation as suggested by The Law of Trichotomy.
If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most 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 Concept Check: 1 - 6
Section 3.3 Exercise Set: For the set {1, 7, 9, 11} replace f(x) with y and sketch the resulting linear equation.
For the set {13, 15, 16, 17, 19, 20} sketch the graph only.
35, 39, 43, 51, 57, 75.
If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most 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 Concept Check: 1 - 12
Section 3.4 Exercise Set: 9, 10, 11, 13, 16
For the set { 25, 27, 29} replace f(x) with y and then find the slope, the x-intercept, and the y-intercept.
48, 49, 61, 62, 63, 64, 71, 74, 79, 85, 87, 89, 91, 93, 95.
The little apprentice owl tells you how to convert the equation of a line from one form
to another.
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In Special Topics there are two short papers about linear equations in two variables.
Each of them contains everything about linear equations in two variables.
Individually or collectively they provide a quick summary of all information about linear equations in two variables.
They are Paper 1 and Paper 2.
If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most 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 Concept Check: 1 - 10
Section 3.5 Exercise Set: 3, 5, 9, 13, 15, 17, 19, 21, 25, 27, 29, 35, 37, 39, 41, 43, 65.
Section 3.6: Graphing Piecewise Defined Functions
If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most 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 Concept Check:
Section 3.6 Exercise Set: