# Rational Expressions and Their Simplification

**Mini Lecture 8.1**

Rational Expressions and Their Simplification

Rational Expressions and Their Simplification

**Learning Objectives:**

1. Find numbers for which a rational expression is undefined.

2. Simplify rational expressions.

3. Solve applied problems involving rational expressions.

**Examples:**

Find all the numbers for which the rational expression is undefined. If the
rational expression is

defined for all real numbers, so state.

Simplify.

3. The rational expressiondescribes
the time, in hours, to travel 240 miles at a rate of

(r +20) miles per hour.

a. Determine the value of r that would cause the expression to be undefined.

b. Find the time in hours if r = 40.

**Teaching Notes:**

• Rational expressions are quotients of two polynomials. They indicate division
and

division by zero is undefined. We must always exclude any value(s) of the
variable that

make a denominator zero.

• When simplifying rational expressions, first, factor the numerator and
denominator

completely, then divide both the numerator and the denominator by any common
factors.

A rational expression is simplified if its numerator and denominator have no
common

factors other than 1 and –1.

• When reducing rational expressions, only factors, not common terms, that are
common to

the entire numerator and the entire denominator can be divided out.

**Answers: **1. a. x = –5 b. x = –3 and x = –2 c.
defined for all real numbers

## Mini Lecture 8.2

Multiplying and Dividing Rational Expressions

**Learning Objectives:**

1. Multiply rational expressions.

2. Divide rational expressions.

**Examples:
**Multiply as indicated.

Divide as indicated.

**Teaching Notes:**

• “When in doubt, factor it out.” Factor first – before multiplying.

• Remind students that in order to divide fractions, (in this section, rational
expressions) you

must multiply by the reciprocal of the divisor.

## Mini Lecture 8.3

Adding and Subtracting Rational Expressions with the Same Denominator

**Learning Objectives:**

1. Add rational expressions with the same denominator.

2. Subtract rational expressions with the same denominator.

3. Add and subtract rational expressions with opposite denominators.

**Examples:**

Add. Simplify if possible.

Subtract, simplify if possible.

Add, making sure to find a common denominator first and simplify if possible.

**Teaching Notes:**

• To add rational expressions with the same denominator, add numerators and
place and

sum over the common denominator. Simplify the answer if possible.

• To subtract rational expressions with the same denominator, subtract
numerators and

place the difference over the common denominator. Simplify the answer if
possible.

• When subtracting numerators with a common denominator, make sure to subtract
every

term in that expression.

• When one denominator is the additive inverse of the other, first multiply
either rational

expressions by–1/–1 to obtain a common denominator.

## Mini Lecture 8.4

Adding and Subtracting Rational Expressions with Different Denominators

**Learning Objectives:**

1. Find the least common denominator (LCD).

2. Add and subtract rational expressions with different denominators.

**Examples:**

Find the least common denominator for the rational numbers or rational
expressions. Factor the

denominators first, then build the least common denominator from those factors.

Rewrite each of the following as an equivalent expression with the given denominator.

Add or subtract.

**Teaching Notes:**

• Students may need to be reminded of factoring steps. It is very important to
be able to

factor quickly and completely.

• Students will find this concept easy if they can relate adding and subtracting
rational

expressions to adding and subtracting fractions.

• Watch the signs when subtracting!

• Students need to understand that the LCD is build with the factors of the
denominators.

## Mini Lecture 8.5

Complex Rational Expressions

**Learning Objectives:**

1. Simplify complex rational expressions by dividing.

2. Simplify complex rational expressions by multiplying by the LCD.

**Examples:**

Simplify by dividing; simplify, if possible.

Simplify by the LCD method; simplify, if possible.

**Teaching Notes:**

• Complex rational expressions are called complex fractions. They have
numerators or

denominators containing one or more rational expressions.

• One method for simplifying a complex rational expression is to combine its
numerator

into a single expression and combine its denominator into a single expression,
then divide

by multiplying by the reciprocal of the term in the denominator.

• A second method for simplifying a complex rational expression is to multiply
each term

in the numerator and denominator by the least common denominator (LCD). This
will

produce an equivalent expression that does not contain fractions in the
numerator or

denominator.

• Both methods for simplifying complex rational expressions produce the same
answer.

See which method you prefer.

## Mini Lecture 8.6

Solving Rational Equations

**Learning Objectives:**

1. Solve rational equations.

2. Solve problems involving formulas with rational expressions.

**Examples:**

What is the LCD in each problem?

Solve. Clear each equation of fractions first.

**Teaching Notes:**

• Students may need extra practice finding the LCD and may need to be reminded
to Factor

First.

• When all denominators are in factored form – list the restricted values.
Restricted values

is any numbers that would make any denominator zero.

• Some rational equations can be solved using cross products, but students need
to be

aware that the method can only be used when there is only one rational
expression on

each side of the equation.

• Students need to constantly be reminded to multiply each term or expression on
both

sides of the equation by the LCD to get ride of the fractions.

• Students must check solutions for restricted values.

## Mini Lecture 8.7

Applications Using Rational Equations and Variation

**Learning Objectives:**

1. Solve problems involving motion.

2. Solve problems involving work.

3. Solve problems involving similar triangles.

4. Solve problems involving variation.

**Examples:**

1. A boat travels 5 km upstream in the same amount of time that the boat covers
15 km

downstream. The current in the stream moves at a speed of 2 km/h. What is the
speed of the boat

in still water?

2. John working alone can paint a room in 4 hours. His helper, Luke, would need
6 hours to

do the job by himself. If they work together, how long will the paint job take
to complete?

3. A tree casts a shadow of 7.5 ft. At the same time, Elizabeth measures the
length of her

shadow which is 3 ft. If Elizabeth is 5.5 feet tall, how tall is the tree?

4. If y varies directly as x and y = 200 when x is 300, find y when x = 45.

5. If y varies indirectly as x and y = 42 when x = 50, find y when x = 700.

**Teaching Notes:**

• Time in motion equation:

• Work problem equation:

fractional part of job done + | fractional part of job | = 1 job completed |

by one person | done by the second person |

• Similar Figures:

Corresponding angles have the same measure and the ratios of the lengths of the

corresponding sides are equal. In corresponding figures, the lengths of the
corresponding

sides are proportional. When triangles ABC and DEF are all similar then:

• Direct Variation Equation: y = kx

Where k is the constant of variation, we say that y varies directly as x.

• Inverse Variation Equation:

Where k is the constant of variation, we say that y varies indirectly as x.

• With variation questions, write the equation from the English statements.
Solve for k and

then substitute the value of k back into the original equation to answer the
original

question.