Hartfield – Intermediate Algebra (Version 2014-2D) Unit 3 ...faculty.ung.edu/thartfield/courses/2014-1-spring/0099_notes_unit3.pdf · Hartfield – Intermediate Algebra (Version

Hartfield – Intermediate Algebra (Version 2014-2D) Unit 3 | Page 1

Topic 3‐1 Review of Rational Expressions, part 1

Definition: A rational expression is a quotient of polynomials.

What are some examples of rational expressions?

What is a rational function? Recall the concept of the domain of a function and how it is related to rational functions.


Simplify rational expressions. To simplify a rational expression, factor the numerator and denominator and cross out any common factors. Simplify.

2 64 10xx

22 64

x xx

Remember: Factoring for the sake of factoring is not simplification. Simplification only occurs if factors are crossed out.

Simplify.

2

27

8 7x x

x x

2

212

12x

x


Simplify.

2

3 23 6 12

2 4x xx x x

2

23 2

2x xx x

Multiplying and Dividing Rational Expressions. Recall the rules for multiplying and dividing fractions.

A C A CB D B D

A C A D A DB D B C B C

Since simplification is expected any time you work with rational expressions, multiplication and division problems almost always become simplification problems which focus on factoring!


Multiply.

2 2 2

2 2 23 6

3 3 5 6x y x xx xy x xy y

Multiply.

22

14 162 65 4

xxxx x


Divide.

2

2 22 2

2 18 4 16 12xy y x y xy yx x x

Divide.

2 2

2 23 9 3 6 3

9 2 2y y y yy y y


Topic 3‐2 Review of Rational Expressions, part 2

Adding and Subtracting Rational Expressions

Recall the rules for adding and subtracting fractions.

A C A CB B B

A C A CB B B

If the expressions have like denominators, adding and subtracting will be easy but simplification concerns may loom on the horizon.

Add.

5 1 31 1

x xx x


Subtract.

2 23 5 1

4 4x xx x

If the expressions have unlike denominators, you will have to find a least common denominator which again focuses on factoring.

Add.

1 24 3

xx


Subtract.

5 32 1

x xx x


Add.

2 22 1

4 2x x

x x x


Subtract.

22 12 6 4x xx


Topic 3‐3 Rational Equations

Process for solving rational equations:

1. Find the LCD of the denominators. 2. Multiply the LCD through the equation to clear all fractions.

3. Solve the resulting equation. 4. Check to make sure the proposed solution doesn’t make a fraction in the original equation undefined.

Solve.

3 1 52 2x x


Solve.

2 75

2 2x x

x x

Solve.

2 55 5 2 2 4 4x xx x x


Solve.

2 82

2 2x x

x x


Solve.

23 10 19 53 47 12

xx xx x

Solve.

24 1 1

2 24 x xx


Solve for R.

1M

DR

Solve for M.

1 1P

M N


Topic 3‐4 Proportions

Proportions are a special kind of rational equations. Definition: A proportion is an equality of two ratios. Definition: A ratio is a comparison of two quantities,

often expressed using fractions.

To solve a proportion A CB D ,

apply cross‐multiplication: AD BC

Solve.

23 2 5

xx

1 1

5 3x x


A machine can process 300 parts in 20 minutes. Find how many parts can be processed in 45 minutes.

In Quack County with a voting population of 50,000, a pre‐election poll of 250 eligible voters was taken. Donald Duck® had the support of 39 voters in the poll. Estimate the number of voters in Quack County who would vote for Donald.


Topic 3‐5 Applications of Rational Equations

Part a: Shared Work Problems

t1 & t2 are the times necessary for individuals to complete a job T is the time needed working together to complete the job

An experienced bricklayer can construct a long wall in 6 days. The apprentice can complete the job in 12 days. Find how long it will take if they work together.

1 2

1 1 1t t T


Mr. Dodson can paint a bedroom by himself in 4 hours. With his son helping him, the bedroom can be painted in 3 hours. How long would Mr. Dodson’s son need to paint the bedroom by himself?


One pipe fills a storage pool in 20 minutes. A second pipe fills the same pool in 15 minutes. When a third pipe is added, and all three are used to fill the pool, it only takes 5 minutes. How long does it take the third pipe to do the job alone?


Part b: distance = rate X time

The current on a portion of the Mississippi River is 3 miles per hour. A barge can go 6 miles upstream in the same amount of time as it takes to go 10 miles downstream. Find the speed of the barge in still water. The use of a chart will make it easier to set up the problem.


In the time it takes Bob to ride his bicycle 150 miles over the flat countryside he can climb 90 miles of hills and small mountains. If Bob cycles 10 mph slower on the hills than he does on level terrain, how fast can Bob cycle on the flat countryside?


A marketing manager travels 1080 miles in a corporate jet and then an additional 240 miles by car. If the car ride takes one hour longer than the jet ride takes, and if the rate of the jet is 6 times the rate of the car, find the time the manager travels by jet and find the time the manager travels by car.


Topic 3‐6 Complex Fractions

A complex fraction is a fraction with at least one fraction inside its numerator or denominator. When we simplify a complex fraction, we want to rewrite the expression so that neither the numerator nor denominator has a fraction in it. There are two families of complex fractions which each have a preferred strategy for simplification:

Group 1:

312

4 515

x

x

or

2

2

2

7yx xyyx

Group 2: 2

2

2 1

4 1

xx

x

o r

3 24 42 24

x x

x x

Simplifying by treating like a division problem (Group 1) If a complex fraction is composed of one fraction over one fraction, then treat the complex fraction like a division problem.

Process: Rewrite ND as .N D Then take the reciprocal

of D and multiply it to N. Simplify the product completely.

Simplify.

312

4 515

x

x


Simplify.

2

2

2

7yx xyyx

While this process works very well for expressions in group 1, the problem is that most complex fractions have multiple expressions in their numerator or denominator. While it is possible to combine the numerator and denominator expressions so as to create a group 1 example, a better strategy emerges for the group 2 examples.

Simplify by using a LCD (Group 2) For most complex fractions, finding the LCD of the fractions in the numerator and denominator—then multiplying the LCD to each—is the best method for simplification.

Process: Take all fractions in the numerator and denominator and find their LCD. Then multiply the complex fraction by a unit fraction of the LCD/LCD and simplify completely.


Simplify.

2

2

2 1

4 1

xx

x

Simplify:

2

2 45 33 138 15

x xx

x x


Simplify.

3 24 42 24

x x

x x


Simplify.

23 2

2

42

x x xx

x


If a fraction has variables/expressions with negative exponents, recognize that the fraction is complex by rewriting the expression with positive exponents.

Simplify.

2

13x

x x

Simplify.

1 2

11

2

2 2

x x

x x

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Hartfield – Intermediate Algebra (Version 2014-2D) Unit 3 ...faculty.ung.edu/thartfield/courses/2014-1-spring/0099_notes_unit3.pdf · Hartfield – Intermediate Algebra (Version