Intermediate Algebra: Rational Expressions and Equations (Chapter 7 Study Guide)

Rational Expressions

Definition and Domain

A rational expression is an expression of the form $\frac{p}{q}$, where p and q are polynomials and $q \neq 0$. Rational expressions are defined for all values of the variable except those that make the denominator zero.

• Domain: The set of real numbers for which the rational expression is defined (i.e., values that do not make the denominator zero).

• Example: $\frac{1}{x}$ is undefined for $x = 0$.

• Example: $\frac{x+3}{x^2-4}$ is undefined for $x = 2$ and $x = -2$.

Simplifying Rational Expressions

To simplify a rational expression:

1. Factor the numerator and denominator completely.

2. Divide out any common factors.

• Note: Simplify by removing common factors, not common terms.

• Example: $\frac{14x^3y^2}{21x^2y^5}$ simplifies to $\frac{2x}{3y^3}$.

• Negative Fractions: $-\frac{a}{b} = \frac{a}{-b} = -\frac{a}{b}$ for $b \neq 0$.

Multiplication and Division of Rational Expressions

Multiplying Rational Expressions

To multiply two rational expressions:

1. Factor all numerators and denominators completely.

2. Divide out common factors.

3. Multiply numerators together and denominators together.

• Formula: $\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$, $b \neq 0$, $d \neq 0$.

• Example: $\frac{8x^3}{3y^2} \cdot \frac{9y^3}{16x^4} = \frac{3y}{2x}$.

Dividing Rational Expressions

To divide two rational expressions:

1. Multiply the first fraction by the reciprocal of the second fraction.

2. Factor and simplify as in multiplication.

• Formula: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c}$, $b \neq 0$, $d \neq 0$, $c \neq 0$.

• Example: $\frac{15m^3}{6n^2} \div \frac{9m^4}{12n^4} = \frac{10n^2}{9m}$.

Addition and Subtraction of Rational Expressions

With Common Denominator

To add or subtract rational expressions with a common denominator:

1. Add or subtract the numerators.

2. Place the result over the common denominator.

3. Simplify if possible.

• Formula (Addition): $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$, $c \neq 0$.

• Formula (Subtraction): $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$, $c \neq 0$.

• Example: $\frac{5}{x-4} + \frac{x-1}{x-4} = \frac{5 + x - 1}{x-4} = \frac{x+4}{x-4}$.

Least Common Denominator (LCD)

To add or subtract rational expressions with unlike denominators, first determine the LCD:

1. Factor each denominator completely.

2. List all different factors, using the highest power for repeated factors.

3. The LCD is the product of all listed factors.

• Example: For $\frac{1}{9} + \frac{3}{n}$, LCD is $9n$.

• Example: For $\frac{7}{20n^5} + \frac{5}{16n^4}$, LCD is $80n^5$.

With Unlike Denominators

To add or subtract rational expressions with unlike denominators:

1. Find the LCD.

2. Rewrite each fraction with the LCD as the denominator.

3. Add or subtract the numerators.

4. Simplify if possible.

• Example: $\frac{3}{m} + \frac{5}{n}$, LCD is $mn$.

• Opposite Denominators: If denominators are opposites (e.g., $n-8$ and $8-n$), multiply numerator and denominator by $-1$ to match denominators.

Complex Fractions

Definition

A complex fraction is a rational expression with a rational expression in its numerator, denominator, or both.

Method 1: Simplifying by Combining Terms

1. Add or subtract rational expressions in the numerator and denominator to obtain single expressions.

2. Rewrite as a division problem (multiply numerator by reciprocal of denominator).

3. Simplify further if possible.

• Example: $\frac{\frac{9x^2}{25y^2z}}{\frac{3x^2z}{5y^4}}$ simplifies to $\frac{15y^4}{25y^2z} \cdot \frac{9x^2}{3x^2z}$.

Method 2: Multiplying to Clear Fractions

1. Find the LCD of all denominators in the complex fraction.

2. Multiply both numerator and denominator by the LCD.

3. Simplify.

• Example: $\frac{a^3}{1} \div \frac{a-1}{b}$, multiply by $b$ to clear denominators.

Solving Rational Equations

Definition

A rational equation contains one or more rational expressions.

Solving Steps

1. Find the LCD of all fractions in the equation.

2. Multiply both sides by the LCD to eliminate denominators.

3. Solve the resulting equation.

4. Check solutions in the original equation to avoid extraneous solutions (values that make any denominator zero).

• Example: $\frac{x}{9} - \frac{x}{8} = 1$; LCD is 72.

• Extraneous Solutions: Solutions that make any denominator zero must be excluded.

Applications of Rational Equations

Geometry Problems

• Involve areas of geometric shapes.

• Example: The area of a rectangle is 120 sq ft. Width is 2 ft less than two-thirds of the length. Find length and width.

Number Problems

• Relate one number to another.

• Example: One number is 6 times another; sum of reciprocals is $\frac{1}{2}$. Find the numbers.

Motion Problems

• Use the formula $d = rt$ (distance = rate × time).

• Example: Tom and Beth canoe 12 miles downstream and 10 miles upstream in the same time. Find their speed in still water.

Work Problems

• Involve two or more people/machines working together.

• Formula: $\text{rate of work} \times \text{time worked} = \text{part of task completed}$

• Example: A painter can paint a room in 6 hours; apprentice in 9 hours. Together, how long?

Work Problem Table

Person/Machine	Time Alone	Rate
Painter	6 hours	$\frac{1}{6}$ per hour
Apprentice	9 hours	$\frac{1}{9}$ per hour
Together	?	$\frac{1}{6} + \frac{1}{9}$ per hour

Example Solution: $\frac{1}{6} + \frac{1}{9} = \frac{5}{18}$ per hour; time together = $\frac{1}{\frac{5}{18}} = 3.6$ hours.

Additional info: The notes cover all major aspects of rational expressions and equations, including definitions, operations, simplification, solving, and applications. Practice problems are provided for each topic, and methods for handling complex fractions and work problems are included.