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
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.
Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer sections 2.1-2.3.)
 | ch 2.1-2.2 each introduce a single property for solving simple linear equations. Ch 2.3 puts those properties together to solve more complicated linear equations in a multi-step process. Ch 2.3 pulls together collecting like terms, the distributive law, simplifying expressions, and solving equations :) |
| ch 2.3 pg 129-130 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. However, MyMathLab of course adopts the text's terminology. I will share examples of each "special case" equation and how to determine if it is no or all solutions 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 CS 3rd ed by Blitzer sections 2.4.)
| ch 2.4 pg 135-138 covers solving a formula for a specified variable. Be sure to try a variety. |
| ch 2.4 pg 136-137 includes the geometry formulas for the perimeter and area of a rectangle, but the text does not use these formulas in an application problem until the next section. |
| ch 2.4 pg 138-143 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 "180o 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 CS 3rd ed by Blitzer sections 2.5-2.6.)
| ch 2.5 pg 148 has a chart for translating words to algebraic expressions. |
| ch 2.5 includes examples of the following application problems: consecutive integers, business apps, geometry, percent/money. Any of these may appear on the quiz! |
| ch 2.6 pg 159-160 has the formulas for perimeter and area of four basic 2D shapes. These geometry formulas and their applications are important and may appear on the quiz! |
| ch 2.6 pg 162-164 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. |
| ch 2.6 pg 164-166 covers the angles of a triangle theorem, complementary angles, and supplementary angles. You should cover know these terms and how to solve application problems involving angles and angles in triangles. |
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 graphing 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 CS 3rd ed by Blitzer sections 2.7)
| ch 2.7 starts by introducing the three ways of expressing the solution to an inequality: graphing on a number line, interval notation, and set-builder notation (which is really an expanded form of inequality notation). Note that rather than writing a solution in inequality notation e.g. x ≤ 1, the text often writes it using set-builder notation { 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". The most common notation used especially in later courses is interval notation e.g. (-∞,1]. The chart on page 174 compares the different notations. |
| ch 2.7 starting on page 175 the text covers solving linear inequalities via three properties summarized in a chart on page 176. I recommend that you put the "additional property of inequality" in your own words instead of making this so complicated. For instance, "It is OK to add (or subtract) a number to both sides of an inequality." Similarly, the "multiplication properties of inequality" could be combined as, "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." Remembering to reverse the direction of the inequality sign whenever you multiply or divide by a negative number is very important! |
| ch 2.7 pg 181 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. |
| ch 9.1 is related to ch 2.7. See below for notes on 9.1. There is plenty to skip! |
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 CS 3rd ed by Blitzer sections 9.1-9.4.)
| Note that we jumped to chapter 9! It seems logical to do section 9.2 now while we are studying inequalities. |
| ch 9.1 is somewhat optional. It basically reviews solving linear inequalities covered in 2.5-2.7. It you do choose to study 9.1 SKIP the business applications, revenue function, cost function, profit function, and all the graphing stuff! The only problems from 9.1 that might be included on a test are those that are just like those in 2.5-2.7. |
| ch 9.2 is important. It covers intersections and unions. It focuses on solving "compound inequalities" including those connected by and, or, and double inequalities. |
| ch 9.2 pg 621 skip the "using technology" box and any exercises involving graphs. |
| ch 9.3-9.4 SKIP |
Originally written: 2006-006-15
Last revision:
2009-01-14 12:41 PM