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: Intermediate Algebra. This section of material only is downloadable for nominal fee at www.lulu.com/content/378809. See "Tell me more about Keely's GM book".
Rational Expression Operations
GOLDen Mathematics - Intermediate Algebra: Section 7.1
Supplemental Sites: MathOL Links - Alg
7.1
Topics of Importance
Rational expressions/functions: definition, domain
restrictions, evaluate
Rational expression ops: reduce, multiply, divide, add, subtract including
working with opposites
Introduction to modeling data with a rational function
Comments and Cautions
Our goal today is to work with algebraic fractions. All
the operations you learned to do with numeric fractions in grade school
(multiply, "flip and multiply" to divide, LCDs to add, etc.) we will
be learning except that the numerator and denominator of our fractions will be
polynomials. Concentrate on the operations: reducing to simplify, multiply and
divide, add and subtract (with same denominators and with different denominators), etc. Be sure
that you recognize opposites (like x-1 and 1-x) and how to handle them in each
of these operations. Don't worry too much about the graphical representation of "excluded values"
a.k.a. "domain restrictions", but you should recognize algebraically that there are x-values that can't be plugged into a
rational expression because they cause the denominator to be zero (eg., in
2/(x+5) x cannot be -5). You can find the domain restrictions by factoring the
denominator of the rational expression and determining what x-values would make
it zero, i.e. set it equal to zero and solve for x.
Caution: when reducing a rational expression like (2x2-50)/(x+5) do not cancel the x's nor reduce the 5's! This is a very common error, tempting, but extremely illegal. Remember that you can only cancel factors not terms. So before canceling you must factor completely! Never reduce a rational expression without factoring top and bottom completely first. Let's talk about this more on the boards.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 7.1-7.4.)
 | ch 7.1 covers simplifying rationals, 7.2 multiplying/dividing, 7.3 adding with same denominators, and 7.4 adding with different denominators. |
| ch 7.1 pg 444 Notice that the domain restriction(s) of a rational function is/are represented graphically by a vertical asymptote line at that x-value. This will be covered more thoroughly in a college algebra class. For now it suffices to find the domain restriction(s) algebraically by finding the x-value(s) that make the denominator zero. |
| ch 7.1 pg 444 The graph of this function is drawn in "dot mode" so that the vertical asymptote is not shown as a vertical line. If you recreate this graph in your calculator you are likely not in "dot mode" and the domain restriction will be represented as a vertical line at x=4. Note that this vertical line represents the domain restriction and is not part of the actual graph of the function. |
| ch 7.1 pg 447 has a very important "caution"! This is a very common error. Remember, you can only cancel factors not terms. |
| ch 7.1 ex 5 I agree with the author that the easiest way to reduce the opposites (like x-3 and 3-x) is to cancel them leaving a -1 factor on the top or bottom. However, if you are used to factoring a -1 out from the top or bottom, or if you are used to multiplying the fraction by -1/-1, then please stick to the method that works best for you. |
| ch 7.2-7.4 Pay special attention to the examples that contain "opposites". Watch your signs! |
| ch 7.3 pg 461 cautions you against a common sign error when subtracting. Don't forget to distribute the minus sign throughout the numerator when combining the fractions into one. |
| ch 7.4 is a very important section. You must learn to find the LCD, build each fraction up to have the LCD by multiplying each fraction by an expression equivalent to 1 e.g. (x(x-1))/(x(x-1)), combine into a single fraction, simplify top and bottom, factor each completely, and reduce if necessary. These problems can get quite long! Plan your schedule accordingly. |
Compound Fractions
GOLDen Mathematics - Intermediate Algebra: Section 7.2
Supplemental Sites: MathOL Links - Alg
7.2
Topics of Importance
Simplify "type 1" compound fractions by the flip 'n
multiply method.
Simplify "type 2" compound fractions by the LCD method.
Evaluate expressions/functions containing compound fractions.
Comments and Cautions
A compound fraction is a fraction within a fraction (yikes!). There are two
types. Type 1 has a single fraction in the numerator and a single fraction in
the denominator. This type is best simplified using the method of flip 'n
multiply where you flip the denominator and multiply it by the numerator. Type 2
has more than one fraction being added in the numerator or in the denominator or
both, basically little fractions all over the place. The fastest way to
eliminate all the little fractions is to multiply through by 1 in the form of
the LCD/LCD. This method clears the little fractions in one fell swoop. Some
people prefer though to convert type 2's into type 1's and then use flip 'n
multiply. Let's compare these two approaches on the boards.
This section is quite involved and can be a bit overwhelming. Try to treat it as an introduction to compound fractions and don’t get too bogged down in the details. You will see them again in a 100-level math course where the processes will become more fluid and you will be motivated by real-world applications. For now, just try to get the main ideas and processes down.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 7.5.)
| ch 7.5 Examples 1-3 treat the numerator and denominator separately at first. For each they add/subtract the terms to combine into a single fraction. Then put these two fractions back into the compound fraction so that it is a (type 1) single fraction over single fraction. This can then be treated as a division problem or better yet the top fraction multiplied by the flip of the bottom fraction. |
| ch 7.5 Examples 4-6 eliminate the denominators of the little fractions all at once by multiplying the numerator and denominator of the big compound fraction by the LCD of all the little fractions. This is the preferred way to work all type 2 compound fractions because there are fewer steps overall. But if this way doesn't make sense to you, feel free to do them all by the example 1-3 way. |
Originally written: 2006-006-15
Last revision:
2008-01-12 04:47 PM