Rational Expressions and Equations

RATIONAL EXPRESSIONS AND EQUATIONS

Unit Overview
In this unit you will learn how to multiply, divide, add, and subtract rational expressions. You will also simplify rational expressions. The unit concludes with solving rational equations.

Simplifying Rational Expression

A rational expression is in simplest form when its numerator and denominator are polynomials that have no common divisors.

To Simplify:

*factor the numerator and denominator

*cancel any common factors

Example #1: Simplify

Stop! Go to Questions #1-3 about this section, then return to continue on to the next section.

Multiplying and Dividing Rational Expressions

Multiplying and dividing rational expressions is similar to multiplying and dividing rational numbers.

To Multiply:

*factor all numerators and denominators

*cancel any common factors

*multiply straight across

Example #2: Multiply

To divide rational expressions you multiply by the reciprocal of the divisor, just as you do when you divide rational numbers.

To Divide:

*multiply by the reciprocal, change the problem to a multiplication problem and use the reciprocal (flip) of the rational expression you are dividing by

*factor all numerators and denominators

*cancel any common factors

*multiply straight across

Example #3: Divide

Multiplying and Dividing Rational Functions (04:22)

Stop! Go to Questions #4-10 about this section, then return to continue on to the next section.

Adding and Subtracting Rational Expressions

To add and subtract rational expressions, you must have a common denominator. If you do, just add the numerators together and keep the common denominator.

Example #1: Find the sum:

Example #2: Find the difference:

.

After adding or subtracting the numerators, if possible, factor and cancel the common factors.

*If the denominators are not the same:

a.) factor all denominators

b.) find all common factors

c.) multiply the denominators by what is missing in the common denominator

Example #3: Find the difference:

Let’s try another example.

Example #3: Find the difference:

The examples illustrate that rational expressions form a system similar to the rational numbers, in that rational expressions are closed under addition, subtraction, multiplication, and division by a nonzero rational expression.

Adding and Subtracting Rational Functions (08:54)

Stop! Go to Questions #11-16 about this section, then return to continue on to the next section.

Solving Rational Equations

To solve rational equations:

a.) find all excluded values by factoring the denominator and setting all factors equal to zero and solving

b.) find the LCD and multiply all terms by the LCD to eliminate fractions

c.) solve the equation and check your answer

Example #1: Solve

x + 3 = 0	x – 1 = 0	Both of these values will produce a zero in the denominator, so they must be excluded from the solution.

x ≠ –3	x ≠ 1

The LCD will be (x + 3)(x – 1).

Multiply and solve.

x² – x = 6x + 18

x² – 7x – 18 = 0

(x – 9)(x + 2) = 0

x –9 = 0 x + 2 = 0

x = 9 and x = –2

Check your answers by replacing x with 9 first then replacing it with –2.

Example #2: Solve

Step #1: Factor all denominators to find the LCD and excluded values. In this case the LCD is (x – 2)(x + 2) and the excluded values are 2 and –2 because if x is equal to either of these, the denominator will become zero.

Step #2: Multiply each term by the LCD to eliminate all fractions.

Step #3: Combine all like terms, factor and solve.

x² – 4x – 12 = 0

(x – 6)(x + 12) = 0

x – 6 = 0 x + 2 = 0

x = 6 and x = –2

Since –2 is an excluded value, the only solution is 6.

Using Rational Equations

Uniform Motion is motion at a constant speed. In uniform motion problems, we use the formula d = rt (distance = rate × time). In problems where there rate or time is unknown, we use the formula t =

or r =

. Both of these formulas use equations involving rational expressions.

Example #1: Allie drove 120 miles to visit her grandmother. On the return trip home, it was raining so she averaged 10 miles per hour less. The drive home took her 24 minutes longer. Find the average speed for both trips.

Use the formula t =

Let r = the rate Allie travels to her grandmother’s house

= time Allie travels to her grandmother’s house

r – 10 = rate on the return trip

= time for the return trip

Since rate is usually measured in miles per hour change 24 minutes to hours, 24 minutes out of 60 minute is

.

Because the difference between the two time is

, the set up for the equation will be: longer time – shorter time =

–

Multiply each term by the LCD 5(r)(r – 10)
120(5r) – 120(5)(r – 10) = 2r(r – 10)

Simplify each term.
600r – 600r – 6000 = 2r²– 20r

Group all terms on one side of the equation.
2r²– 20r + 6000 = 0

Divide each side by 2.
r²– 10r + 3000 = 0

Factor and solve.
(r – 60)(r +50) = 0
r – 60 = 0 or r +50 = 0
r = 60 or r = –50

The answer r = –50 is a solution to the equation, but it cannot be the average speed of a car. Her average speed going to her grandmother’s was 60 mph. Her average speed on the return trip was 50 mph.

Check:

t = = = 2 hours	Time it takes Allie to go to grandmother's

t = = = 2 hours	Time it takes Allie to return home

2 hours – 2 hours = hours more

Work Problem

Work problems typically involve two people or machine working together to complete a task. The basic set up for a word problem is

Portion of job completed by
1st person

Portion of job completed by
2nd person

1 (one completed job)

Tyler can shovel a rake the leaves in the yard in 30 minutes. If Josh helps, the job takes only 20 minutes. How long would it take Josh to do the job by himself?

Let x = number of minutes it takes Josh to rake the leaves working alone