Equations and Inequalities

Prof. Keely's Math Online Lecture Notes
Clark College, Vancouver WA
Copyright © 2000 Sally J. Keely. All Rights Reserved.

ELEMENTARY & INTERMEDIATE ALGEBRA

These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: This material is extensively elaborated upon in my optional e-book GOLDen Mathematics: Elementary Algebra. This section of material only is downloadable for nominal fee at www.lulu.com/content/431008. See "Tell me more about Keely's GM book".

Solving Linear Equations

GOLDen Mathematics - Elementary Algebra: Section 2.1
Supplemental Sites: MathOL Links - Alg 2.1

Topics of Importance
Definition of "linear"
Solve linear equations isolating the x using the "Undo Property"
Solve for x involved in: addition, subtraction, multiplication, division, fractional coefficients
Solve for x involved in a combination of operations
Solve for x by simplifying first and then isolating
Special cases: "no solution" vs "all solutions"
Converting any repeating decimal

fraction (algebraically)

Comments and Cautions
Solving linear equations is one of the most important topics in the entire course especially since the next course concentrates on solving a variety of more complicated equations naturally extending the linear ones studied here. Practice a variety of problems especially those that contain fractions and decimals. Be sure to study the "no solution" and "all solution" special cases too. I have an optional powerpoint presentation (web-based, use MSIE for best results) that covers the basics available for your viewing pleasure.

Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 2.1-2.3.)

ch 2.3 pg 122-123 The text defines a "contradiction" (an equation that has no solution a.k.a. an inconsistent equation), an "identity" (an equation that is true for all real numbers, i.e. has an infinite number of solutions), and a "conditional equation" (which has a finite number of solutions, in fact just one solution when it is a linear conditional equation). I won't use these formal terms, but I do want you to be able to recognize how many solutions an equation has (none, one, two, ..., an infinite number). I prefer to refer to the two "special cases" as "no solution" equations and "all solution" equations. Let's talk more about these in class.

Formulas - Elementary

GOLDen Mathematics - Elementary Algebra: Section 2.2
Supplemental Sites: MathOL Links - Alg 2.2

Topics of Importance
Evaluate formulas given specific input values
Solve formulas for a specific variable

Comments and Cautions
Examples of "formulas" that you have likely seen are a²+b²=c², D=RT, A=½bh, etc. Today's material concentrates on evaluating such formulas for given values of the variables. Watch your units when you are evaluating! We also learn to rearrange the formulas by solving for a specific variable like rewriting A=½bh as h=2A/b. This will be useful when dealing with real-world applications :)

Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 2.4.)

ch 2.4 pg 130-134 covers percents, the percent formula, and percent increase/decrease. Some of this material should be review from a pre-algebra course. But don't skim over it! There is a lot going on here and several new algebraic processes. Percent problems are some of the most useful real-life applications that we will cover. (More on percent problems below.)

Applications of Linear Equations

GOLDen Mathematics - Elementary Algebra: Section 2.3
Supplemental Sites: MathOL Links - Alg 2.3

Topics of Importance
Solving elementary word problems of the following types:
Number problems (including consecutive)
Money problems
Percent problems
Geometry problems including perimeter, area
Triangle problems including the "180^o triangle theorem", complementary and supplementary angles
Modeling data with a linear equation

Comments and Cautions
Our first set of word problems! Be sure to concentrate on those types listed above. The key is in the translating step when the word problem is translated into an equation. Read the problem carefully, assign variables, know what you are looking to solve, take your time! You can do it if you take one step at a time and not allow yourself to get overwhelmed.

One caution when working with percent problems: remember that percents must be taken of something - so the percent must be multiplied by something. One of the most common errors is to take the percent of the wrong thing. For example, in "A TV is marked up 50% over its wholesale price. It sells for $200, what is the wholesale price?" It is WRONG to take 50% of $200 - that would be 50% of the retail price. Instead start with the unknown wholesale price, x, add the mark-up, 50% of x, to get the retail. I.e., solve x+0.50x=200.

Word problems is often a sticking point for students, but if you keep an optimistic attitude, take one step at a time, and read the problem carefully, you can do it! If you need additional assistance, I recommend a little book: How to Solve Word Problems in Algebra by Mildred Johnson. There are others available that are good too, but this one is easy to follow and full of examples like those we cover in this class.

Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 2.5-2.6.)

ch 2.6 pg 152-153 covers volume formulas. I personally think that you have enough to learn without including the volume formulas. You can SKIP the volume formulas, problems involving volume, and examples 4-5 in the text. I will not test you on volume.

Solving Linear Inequalities

Textbook correspondence: Chapter 2.7+9.1
GOLDen Mathematics - Elementary Algebra: Section 2.4
Supplemental Sites: MathOL Links - Alg 2.4

Topics of Importance
Definition of linear inequality in one variable
Notation: number line vs. inequality vs. interval
What a "solution set" to a linear inequality means
Solving linear inequalities algebraically
Applications of linear inequalities

Comments and Cautions
There are two major concepts covered in this section. First you must be able to represent sets by drawing them on a number line (watch "open" vs. "closed" circles), writing them using an inequality (like x<1), or writing them in interval notation (which is the most commonly used method). Second you must learn to solve a linear inequality.

Caution: When isolating the variable in an inequality, don't forget that if you multiply or divide both sides by a negative number you must reverse the direction of the inequality!

Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 2.7 + 9.1.)

	ch 2.7 pg 162 introduces "set-builder notation" which is really an expanded form of "inequality notation". For instance rather than writing a solution in inequality notation e.g. x ≤ 1, the text writes it using set-builder notation like { x \| x ≤ 1 } which is read "the set of all x such that x is less than or equal to 1". The \| bar means "such that". It is far more common in later courses to use interval notation e.g. (-∞,1] which this text covers in ch 9.1.
	ch 2.7 pg 163 The "additional property of inequality" -- Put this in your own words instead of making this so complicated like, "It is OK to add (or subtract) a number to both sides of an inequality." Similarly, the "multiplication property of inequality" could be stated, "It is OK to multiply (or divide) both sides of an inequality by a positive number, but if you multiply (or divide) both sides by a negative number you must reverse the direction of the inequality."
	ch 2.7 pg 168-169 Example 8 shows a "no solution" case for linear inequalities. Example 9 shows an "all solution" case for linear inequalities. Pay attention to these "special cases".
	Note that we jumped to section 9.1! It seems logical to do this section and 9.2 now while we are studying inequalities.
	ch 9.1 covers "interval notation" which is the most common way of writing the answer to an inequality.
	ch 9.1 SKIP the business applications, revenue function, cost function, profit function (page 584-588, examples 5-6). You will not be tested on any of these applications.

Solving Compound Linear Inequalities

GOLDen Mathematics - Elementary Algebra: Section 2.5
Supplemental Sites: MathOL Links - Alg 2.5

Topics of Importance
Combining solutions sets connected by "and", "or"
Solving compound linear inequalities: "and", "or", double

Comments and Cautions
"Compound inequalities" means combining two or more inequalities together. There are three ways to do this. Two inequalities can be connected by an "and" (intersection ) which means that the final answer will be those numbers that are in both sets (the overlapping region). Two inequalities can be connected by an "or" (union ) which means that the final answer will be those numbers that are in one set or the other or both -- basically the answer will be both sets all lumped together. The third way to compound inequalities is the "double" kind which means that there are multiple inequalities in one statement like 4<2x-1<10. The key to solving double inequalities is simply to get x isolated in the middle. The final answers to all of these compound inequalities will usually be written in interval notation. Treat this section as an introduction to compound inequalities. This topic will be covered more thoroughly in a College Algebra class.

Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 9.2.)

ch 9.2's exercises include some that involve solving inequalities algebraically, graphically, and using tables. You will be tested primarily on the algebraic methods of solving. You may be tested on solving an inequality graphically given the graph. You will not be test on solving an inequality using the table feature of your calculator; this method is completely optional.