Rational Functions and Expressions - Ninth Grade Math
Introduction to Rational Functions and Expressions
Rational Expression: A fraction where the numerator and denominator are polynomials
Rational Function: A function that can be written as the ratio of two polynomials
General Form: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$
Similar to: Working with numerical fractions, but with variables
Rational Function: A function that can be written as the ratio of two polynomials
General Form: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$
Similar to: Working with numerical fractions, but with variables
Definition:
$$f(x) = \frac{P(x)}{Q(x)}$$
where:
• $P(x)$ = polynomial in numerator
• $Q(x)$ = polynomial in denominator
• $Q(x) \neq 0$ (denominator cannot be zero)
Example: $f(x) = \frac{x + 2}{x - 3}$, $g(x) = \frac{x^2 - 4}{x^2 + 1}$
$$f(x) = \frac{P(x)}{Q(x)}$$
where:
• $P(x)$ = polynomial in numerator
• $Q(x)$ = polynomial in denominator
• $Q(x) \neq 0$ (denominator cannot be zero)
Example: $f(x) = \frac{x + 2}{x - 3}$, $g(x) = \frac{x^2 - 4}{x^2 + 1}$
1. Rational Functions: Asymptotes and Excluded Values
Excluded Value: A value of x that makes the denominator zero (undefined)
Vertical Asymptote: A vertical line that the graph approaches but never crosses
Horizontal Asymptote: A horizontal line that the graph approaches as $x \to \pm\infty$
Domain: All real numbers except excluded values
Vertical Asymptote: A vertical line that the graph approaches but never crosses
Horizontal Asymptote: A horizontal line that the graph approaches as $x \to \pm\infty$
Domain: All real numbers except excluded values
Finding Excluded Values
To Find Excluded Values:
Step 1: Set denominator equal to zero
Step 2: Solve for x
Step 3: These x-values are excluded from domain
$$Q(x) = 0 \implies \text{Excluded values}$$
Step 1: Set denominator equal to zero
Step 2: Solve for x
Step 3: These x-values are excluded from domain
$$Q(x) = 0 \implies \text{Excluded values}$$
Example 1: Find excluded values for $f(x) = \frac{3}{x - 5}$
Set denominator = 0:
$x - 5 = 0$
$x = 5$
Excluded value: $x = 5$
Domain: All real numbers except $x = 5$
Set denominator = 0:
$x - 5 = 0$
$x = 5$
Excluded value: $x = 5$
Domain: All real numbers except $x = 5$
Example 2: Find excluded values for $g(x) = \frac{x + 1}{x^2 - 9}$
$x^2 - 9 = 0$
$(x - 3)(x + 3) = 0$
$x = 3$ or $x = -3$
Excluded values: $x = 3$ and $x = -3$
Domain: All real numbers except $x = \pm 3$
$x^2 - 9 = 0$
$(x - 3)(x + 3) = 0$
$x = 3$ or $x = -3$
Excluded values: $x = 3$ and $x = -3$
Domain: All real numbers except $x = \pm 3$
Example 3: Find excluded values for $h(x) = \frac{2x}{x^2 + 4}$
$x^2 + 4 = 0$
$x^2 = -4$
No real solutions (can't have negative under square root)
No excluded values; Domain: All real numbers
$x^2 + 4 = 0$
$x^2 = -4$
No real solutions (can't have negative under square root)
No excluded values; Domain: All real numbers
Vertical Asymptotes
Vertical Asymptote Rule:
A vertical asymptote occurs at $x = a$ if:
• The denominator equals zero at $x = a$
• The numerator does NOT equal zero at $x = a$
• (Factor cannot be cancelled out)
Notation: $x = a$ is the equation of the vertical asymptote
A vertical asymptote occurs at $x = a$ if:
• The denominator equals zero at $x = a$
• The numerator does NOT equal zero at $x = a$
• (Factor cannot be cancelled out)
Notation: $x = a$ is the equation of the vertical asymptote
Example 4: Find vertical asymptotes of $f(x) = \frac{1}{x - 2}$
Denominator = 0: $x = 2$
Numerator at $x = 2$: $1 \neq 0$
Vertical asymptote: $x = 2$
Denominator = 0: $x = 2$
Numerator at $x = 2$: $1 \neq 0$
Vertical asymptote: $x = 2$
Example 5: Find vertical asymptotes of $g(x) = \frac{x + 1}{(x - 1)(x + 3)}$
Denominator = 0: $x = 1$ or $x = -3$
Check numerator: Neither makes numerator zero
Vertical asymptotes: $x = 1$ and $x = -3$
Denominator = 0: $x = 1$ or $x = -3$
Check numerator: Neither makes numerator zero
Vertical asymptotes: $x = 1$ and $x = -3$
Horizontal Asymptotes
Horizontal Asymptote Rules:
Compare degrees of numerator ($n$) and denominator ($m$):
Case 1: If $n < m$ (degree of numerator < degree of denominator)
→ Horizontal asymptote: $y = 0$ (x-axis)
Case 2: If $n = m$ (degrees are equal)
→ Horizontal asymptote: $y = \frac{a}{b}$ (ratio of leading coefficients)
Case 3: If $n > m$ (degree of numerator > degree of denominator)
→ NO horizontal asymptote (may have slant asymptote)
Compare degrees of numerator ($n$) and denominator ($m$):
Case 1: If $n < m$ (degree of numerator < degree of denominator)
→ Horizontal asymptote: $y = 0$ (x-axis)
Case 2: If $n = m$ (degrees are equal)
→ Horizontal asymptote: $y = \frac{a}{b}$ (ratio of leading coefficients)
Case 3: If $n > m$ (degree of numerator > degree of denominator)
→ NO horizontal asymptote (may have slant asymptote)
Example 6: Find horizontal asymptote of $f(x) = \frac{3x}{x^2 + 1}$
Degree of numerator: 1
Degree of denominator: 2
Since $1 < 2$: $y = 0$
Horizontal asymptote: $y = 0$
Degree of numerator: 1
Degree of denominator: 2
Since $1 < 2$: $y = 0$
Horizontal asymptote: $y = 0$
Example 7: Find horizontal asymptote of $g(x) = \frac{2x^2 + 3}{x^2 - 5}$
Both degrees = 2
Leading coefficients: $\frac{2}{1} = 2$
Horizontal asymptote: $y = 2$
Both degrees = 2
Leading coefficients: $\frac{2}{1} = 2$
Horizontal asymptote: $y = 2$
Example 8: Find horizontal asymptote of $h(x) = \frac{x^3}{x + 1}$
Degree of numerator: 3
Degree of denominator: 1
Since $3 > 1$: No horizontal asymptote
No horizontal asymptote
Degree of numerator: 3
Degree of denominator: 1
Since $3 > 1$: No horizontal asymptote
No horizontal asymptote
2. Simplify Rational Expressions
Simplifying: Reducing to lowest terms by canceling common factors
Key Rule: Factor completely, then cancel common factors in numerator and denominator
Key Rule: Factor completely, then cancel common factors in numerator and denominator
Simplification Process:
$$\frac{P(x)}{Q(x)} = \frac{\text{factored numerator}}{\text{factored denominator}}$$
Important Rules:
• Only factors can be cancelled, NOT terms
• $\frac{a \cdot b}{a \cdot c} = \frac{b}{c}$ (can cancel $a$)
• $\frac{a + b}{a + c} \neq \frac{b}{c}$ (CANNOT cancel $a$)
Always state restrictions (excluded values from original denominator)
$$\frac{P(x)}{Q(x)} = \frac{\text{factored numerator}}{\text{factored denominator}}$$
Important Rules:
• Only factors can be cancelled, NOT terms
• $\frac{a \cdot b}{a \cdot c} = \frac{b}{c}$ (can cancel $a$)
• $\frac{a + b}{a + c} \neq \frac{b}{c}$ (CANNOT cancel $a$)
Always state restrictions (excluded values from original denominator)
Steps to Simplify:
Step 1: Factor numerator completely
Step 2: Factor denominator completely
Step 3: Cancel common factors
Step 4: State restrictions (values that make original denominator = 0)
Step 1: Factor numerator completely
Step 2: Factor denominator completely
Step 3: Cancel common factors
Step 4: State restrictions (values that make original denominator = 0)
Example 1: Simplify $\frac{15x^3}{25x^2}$
Factor: $\frac{5 \cdot 3 \cdot x^2 \cdot x}{5 \cdot 5 \cdot x^2}$
Cancel common factors: $\frac{3x}{5}$
Restriction: $x \neq 0$
Answer: $\frac{3x}{5}$, $x \neq 0$
Factor: $\frac{5 \cdot 3 \cdot x^2 \cdot x}{5 \cdot 5 \cdot x^2}$
Cancel common factors: $\frac{3x}{5}$
Restriction: $x \neq 0$
Answer: $\frac{3x}{5}$, $x \neq 0$
Example 2: Simplify $\frac{x^2 - 4}{x^2 - 5x + 6}$
Factor numerator: $(x - 2)(x + 2)$
Factor denominator: $(x - 2)(x - 3)$
$$\frac{(x - 2)(x + 2)}{(x - 2)(x - 3)}$$
Cancel $(x - 2)$: $\frac{x + 2}{x - 3}$
Restrictions: $x \neq 2, 3$
Answer: $\frac{x + 2}{x - 3}$, $x \neq 2, 3$
Factor numerator: $(x - 2)(x + 2)$
Factor denominator: $(x - 2)(x - 3)$
$$\frac{(x - 2)(x + 2)}{(x - 2)(x - 3)}$$
Cancel $(x - 2)$: $\frac{x + 2}{x - 3}$
Restrictions: $x \neq 2, 3$
Answer: $\frac{x + 2}{x - 3}$, $x \neq 2, 3$
Example 3: Simplify $\frac{x^2 + 3x}{x^2 - 9}$
Factor: $\frac{x(x + 3)}{(x - 3)(x + 3)}$
Cancel $(x + 3)$: $\frac{x}{x - 3}$
Answer: $\frac{x}{x - 3}$, $x \neq -3, 3$
Factor: $\frac{x(x + 3)}{(x - 3)(x + 3)}$
Cancel $(x + 3)$: $\frac{x}{x - 3}$
Answer: $\frac{x}{x - 3}$, $x \neq -3, 3$
Example 4: Simplify $\frac{2x - 6}{6 - 2x}$
Factor: $\frac{2(x - 3)}{-2(x - 3)} = \frac{2(x - 3)}{-2(x - 3)}$
Cancel: $\frac{2}{-2} = -1$
Answer: $-1$, $x \neq 3$
Factor: $\frac{2(x - 3)}{-2(x - 3)} = \frac{2(x - 3)}{-2(x - 3)}$
Cancel: $\frac{2}{-2} = -1$
Answer: $-1$, $x \neq 3$
Common Mistake: Cannot cancel terms, only factors!
• WRONG: $\frac{x + 3}{x} = 1 + \frac{3}{x}$ (NOT $1 + 3$)
• WRONG: $\frac{x^2 + 5}{x} \neq x + 5$
• WRONG: $\frac{x + 3}{x} = 1 + \frac{3}{x}$ (NOT $1 + 3$)
• WRONG: $\frac{x^2 + 5}{x} \neq x + 5$
3. Multiply and Divide Rational Expressions
Multiplying Rational Expressions
Multiplication Rule:
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$
Process:
1. Multiply numerators
2. Multiply denominators
3. Simplify (factor and cancel common factors)
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$
Process:
1. Multiply numerators
2. Multiply denominators
3. Simplify (factor and cancel common factors)
Steps to Multiply:
Step 1: Factor all numerators and denominators
Step 2: Cancel common factors
Step 3: Multiply remaining factors
Step 4: State restrictions
Step 1: Factor all numerators and denominators
Step 2: Cancel common factors
Step 3: Multiply remaining factors
Step 4: State restrictions
Example 1: Multiply $\frac{x}{3} \cdot \frac{6}{x^2}$
$$\frac{x \cdot 6}{3 \cdot x^2} = \frac{6x}{3x^2}$$
Simplify: $\frac{2}{x}$
Answer: $\frac{2}{x}$, $x \neq 0$
$$\frac{x \cdot 6}{3 \cdot x^2} = \frac{6x}{3x^2}$$
Simplify: $\frac{2}{x}$
Answer: $\frac{2}{x}$, $x \neq 0$
Example 2: Multiply $\frac{x - 2}{x + 3} \cdot \frac{x + 3}{x - 5}$
$$\frac{(x - 2)(x + 3)}{(x + 3)(x - 5)}$$
Cancel $(x + 3)$: $\frac{x - 2}{x - 5}$
Answer: $\frac{x - 2}{x - 5}$, $x \neq -3, 5$
$$\frac{(x - 2)(x + 3)}{(x + 3)(x - 5)}$$
Cancel $(x + 3)$: $\frac{x - 2}{x - 5}$
Answer: $\frac{x - 2}{x - 5}$, $x \neq -3, 5$
Example 3: Multiply $\frac{x^2 - 9}{x - 2} \cdot \frac{x + 2}{x + 3}$
Factor: $\frac{(x - 3)(x + 3)}{x - 2} \cdot \frac{x + 2}{x + 3}$
Cancel $(x + 3)$: $\frac{(x - 3)(x + 2)}{x - 2}$
Answer: $\frac{(x - 3)(x + 2)}{x - 2}$, $x \neq 2, -3$
Factor: $\frac{(x - 3)(x + 3)}{x - 2} \cdot \frac{x + 2}{x + 3}$
Cancel $(x + 3)$: $\frac{(x - 3)(x + 2)}{x - 2}$
Answer: $\frac{(x - 3)(x + 2)}{x - 2}$, $x \neq 2, -3$
Dividing Rational Expressions
Division Rule:
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$
Key Rule: Multiply by the reciprocal (flip the second fraction)
"Keep, Change, Flip" method
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$
Key Rule: Multiply by the reciprocal (flip the second fraction)
"Keep, Change, Flip" method
Example 4: Divide $\frac{x}{2} \div \frac{x^2}{4}$
Multiply by reciprocal:
$$\frac{x}{2} \cdot \frac{4}{x^2} = \frac{4x}{2x^2} = \frac{2}{x}$$
Answer: $\frac{2}{x}$, $x \neq 0$
Multiply by reciprocal:
$$\frac{x}{2} \cdot \frac{4}{x^2} = \frac{4x}{2x^2} = \frac{2}{x}$$
Answer: $\frac{2}{x}$, $x \neq 0$
Example 5: Divide $\frac{x^2 - 4}{x + 1} \div \frac{x - 2}{x + 1}$
Flip second fraction:
$$\frac{x^2 - 4}{x + 1} \cdot \frac{x + 1}{x - 2}$$
Factor: $\frac{(x - 2)(x + 2)}{x + 1} \cdot \frac{x + 1}{x - 2}$
Cancel: $x + 2$
Answer: $x + 2$, $x \neq -1, 2$
Flip second fraction:
$$\frac{x^2 - 4}{x + 1} \cdot \frac{x + 1}{x - 2}$$
Factor: $\frac{(x - 2)(x + 2)}{x + 1} \cdot \frac{x + 1}{x - 2}$
Cancel: $x + 2$
Answer: $x + 2$, $x \neq -1, 2$
4. Add and Subtract Rational Expressions
Like Denominators: Can add/subtract numerators directly
Unlike Denominators: Must find LCD (Least Common Denominator) first
Unlike Denominators: Must find LCD (Least Common Denominator) first
Same Denominators
Same Denominator Rule:
$$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$
$$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$
Keep denominator, add/subtract numerators
$$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$
$$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$
Keep denominator, add/subtract numerators
Example 1: Add $\frac{3}{x + 2} + \frac{5}{x + 2}$
$$\frac{3 + 5}{x + 2} = \frac{8}{x + 2}$$
Answer: $\frac{8}{x + 2}$, $x \neq -2$
$$\frac{3 + 5}{x + 2} = \frac{8}{x + 2}$$
Answer: $\frac{8}{x + 2}$, $x \neq -2$
Example 2: Subtract $\frac{2x}{x - 1} - \frac{x + 3}{x - 1}$
$$\frac{2x - (x + 3)}{x - 1} = \frac{2x - x - 3}{x - 1} = \frac{x - 3}{x - 1}$$
Answer: $\frac{x - 3}{x - 1}$, $x \neq 1$
$$\frac{2x - (x + 3)}{x - 1} = \frac{2x - x - 3}{x - 1} = \frac{x - 3}{x - 1}$$
Answer: $\frac{x - 3}{x - 1}$, $x \neq 1$
Different Denominators
LCD Method:
Step 1: Find LCD (Least Common Denominator)
• Factor each denominator
• LCD contains all factors, each to highest power
Step 2: Rewrite each fraction with LCD
Step 3: Add/subtract numerators
Step 4: Simplify if possible
Step 1: Find LCD (Least Common Denominator)
• Factor each denominator
• LCD contains all factors, each to highest power
Step 2: Rewrite each fraction with LCD
Step 3: Add/subtract numerators
Step 4: Simplify if possible
Steps:
Step 1: Factor all denominators
Step 2: Find LCD
Step 3: Multiply each fraction by appropriate form of 1 to get LCD
Step 4: Add/subtract numerators
Step 5: Simplify result
Step 1: Factor all denominators
Step 2: Find LCD
Step 3: Multiply each fraction by appropriate form of 1 to get LCD
Step 4: Add/subtract numerators
Step 5: Simplify result
Example 3: Add $\frac{1}{x} + \frac{2}{3x}$
LCD: $3x$
Convert:
$$\frac{1}{x} \cdot \frac{3}{3} + \frac{2}{3x} = \frac{3}{3x} + \frac{2}{3x}$$
$$= \frac{3 + 2}{3x} = \frac{5}{3x}$$
Answer: $\frac{5}{3x}$, $x \neq 0$
LCD: $3x$
Convert:
$$\frac{1}{x} \cdot \frac{3}{3} + \frac{2}{3x} = \frac{3}{3x} + \frac{2}{3x}$$
$$= \frac{3 + 2}{3x} = \frac{5}{3x}$$
Answer: $\frac{5}{3x}$, $x \neq 0$
Example 4: Add $\frac{2}{x - 1} + \frac{3}{x + 2}$
LCD: $(x - 1)(x + 2)$
$$\frac{2(x + 2)}{(x - 1)(x + 2)} + \frac{3(x - 1)}{(x - 1)(x + 2)}$$
$$= \frac{2x + 4 + 3x - 3}{(x - 1)(x + 2)} = \frac{5x + 1}{(x - 1)(x + 2)}$$
Answer: $\frac{5x + 1}{(x - 1)(x + 2)}$, $x \neq 1, -2$
LCD: $(x - 1)(x + 2)$
$$\frac{2(x + 2)}{(x - 1)(x + 2)} + \frac{3(x - 1)}{(x - 1)(x + 2)}$$
$$= \frac{2x + 4 + 3x - 3}{(x - 1)(x + 2)} = \frac{5x + 1}{(x - 1)(x + 2)}$$
Answer: $\frac{5x + 1}{(x - 1)(x + 2)}$, $x \neq 1, -2$
Example 5: Subtract $\frac{x}{x^2 - 4} - \frac{2}{x - 2}$
Factor: $\frac{x}{(x - 2)(x + 2)} - \frac{2}{x - 2}$
LCD: $(x - 2)(x + 2)$
$$\frac{x}{(x - 2)(x + 2)} - \frac{2(x + 2)}{(x - 2)(x + 2)}$$
$$= \frac{x - 2x - 4}{(x - 2)(x + 2)} = \frac{-x - 4}{(x - 2)(x + 2)}$$
Answer: $\frac{-x - 4}{(x - 2)(x + 2)}$, $x \neq 2, -2$
Factor: $\frac{x}{(x - 2)(x + 2)} - \frac{2}{x - 2}$
LCD: $(x - 2)(x + 2)$
$$\frac{x}{(x - 2)(x + 2)} - \frac{2(x + 2)}{(x - 2)(x + 2)}$$
$$= \frac{x - 2x - 4}{(x - 2)(x + 2)} = \frac{-x - 4}{(x - 2)(x + 2)}$$
Answer: $\frac{-x - 4}{(x - 2)(x + 2)}$, $x \neq 2, -2$
5. Simplify Mixed Rational Expressions
Complex Fraction: A fraction that has fractions in numerator and/or denominator
Also called: Compound fraction or mixed rational expression
Example: $\frac{\frac{1}{x}}{\frac{2}{y}}$
Also called: Compound fraction or mixed rational expression
Example: $\frac{\frac{1}{x}}{\frac{2}{y}}$
Two Methods to Simplify:
Method 1 - Multiply by LCD:
1. Find LCD of all fractions (in numerator and denominator)
2. Multiply entire expression by $\frac{\text{LCD}}{\text{LCD}}$
3. Simplify
Method 2 - Divide:
1. Simplify numerator and denominator separately
2. Divide: multiply numerator by reciprocal of denominator
3. Simplify
Method 1 - Multiply by LCD:
1. Find LCD of all fractions (in numerator and denominator)
2. Multiply entire expression by $\frac{\text{LCD}}{\text{LCD}}$
3. Simplify
Method 2 - Divide:
1. Simplify numerator and denominator separately
2. Divide: multiply numerator by reciprocal of denominator
3. Simplify
Example 1: Simplify $\frac{\frac{1}{x}}{\frac{2}{3}}$
Method 2 - Divide:
$$\frac{1}{x} \div \frac{2}{3} = \frac{1}{x} \cdot \frac{3}{2} = \frac{3}{2x}$$
Answer: $\frac{3}{2x}$, $x \neq 0$
Method 2 - Divide:
$$\frac{1}{x} \div \frac{2}{3} = \frac{1}{x} \cdot \frac{3}{2} = \frac{3}{2x}$$
Answer: $\frac{3}{2x}$, $x \neq 0$
Example 2: Simplify $\frac{\frac{x}{2} + \frac{1}{3}}{\frac{x}{4}}$
Method 1 - Multiply by LCD = 12:
$$\frac{12\left(\frac{x}{2} + \frac{1}{3}\right)}{12 \cdot \frac{x}{4}}$$
$$= \frac{6x + 4}{3x} = \frac{2(3x + 2)}{3x}$$
Answer: $\frac{2(3x + 2)}{3x}$, $x \neq 0$
Method 1 - Multiply by LCD = 12:
$$\frac{12\left(\frac{x}{2} + \frac{1}{3}\right)}{12 \cdot \frac{x}{4}}$$
$$= \frac{6x + 4}{3x} = \frac{2(3x + 2)}{3x}$$
Answer: $\frac{2(3x + 2)}{3x}$, $x \neq 0$
Example 3: Simplify $\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$
LCD = $x$, multiply by $\frac{x}{x}$:
$$\frac{x\left(1 + \frac{1}{x}\right)}{x\left(1 - \frac{1}{x}\right)} = \frac{x + 1}{x - 1}$$
Answer: $\frac{x + 1}{x - 1}$, $x \neq 0, 1$
LCD = $x$, multiply by $\frac{x}{x}$:
$$\frac{x\left(1 + \frac{1}{x}\right)}{x\left(1 - \frac{1}{x}\right)} = \frac{x + 1}{x - 1}$$
Answer: $\frac{x + 1}{x - 1}$, $x \neq 0, 1$
6. Solve Rational Equations
Rational Equation: An equation containing rational expressions
Strategy: Eliminate denominators by multiplying by LCD
Important: Always check for extraneous solutions!
Strategy: Eliminate denominators by multiplying by LCD
Important: Always check for extraneous solutions!
Key Method - Clear Denominators:
Step 1: Find LCD of all fractions
Step 2: Multiply EVERY term by LCD
Step 3: Solve resulting equation
Step 4: CHECK solutions in original equation
Step 5: Discard any that make denominator = 0
Step 1: Find LCD of all fractions
Step 2: Multiply EVERY term by LCD
Step 3: Solve resulting equation
Step 4: CHECK solutions in original equation
Step 5: Discard any that make denominator = 0
Complete Steps:
Step 1: Identify restricted values (excluded values)
Step 2: Find LCD of all denominators
Step 3: Multiply every term by LCD
Step 4: Simplify (denominators should cancel)
Step 5: Solve resulting equation
Step 6: Check solutions - discard any that are restricted values
Step 1: Identify restricted values (excluded values)
Step 2: Find LCD of all denominators
Step 3: Multiply every term by LCD
Step 4: Simplify (denominators should cancel)
Step 5: Solve resulting equation
Step 6: Check solutions - discard any that are restricted values
Example 1: Solve $\frac{2}{x} = \frac{6}{x + 3}$
Restricted values: $x \neq 0, -3$
LCD: $x(x + 3)$
Multiply both sides by LCD:
$$x(x + 3) \cdot \frac{2}{x} = x(x + 3) \cdot \frac{6}{x + 3}$$
$$2(x + 3) = 6x$$
$$2x + 6 = 6x$$
$$6 = 4x$$
$$x = \frac{3}{2}$$
Check: Not a restricted value ✓
Answer: $x = \frac{3}{2}$
Restricted values: $x \neq 0, -3$
LCD: $x(x + 3)$
Multiply both sides by LCD:
$$x(x + 3) \cdot \frac{2}{x} = x(x + 3) \cdot \frac{6}{x + 3}$$
$$2(x + 3) = 6x$$
$$2x + 6 = 6x$$
$$6 = 4x$$
$$x = \frac{3}{2}$$
Check: Not a restricted value ✓
Answer: $x = \frac{3}{2}$
Example 2: Solve $\frac{1}{x - 2} + \frac{1}{x + 2} = \frac{4}{x^2 - 4}$
Factor: $x^2 - 4 = (x - 2)(x + 2)$
LCD: $(x - 2)(x + 2)$
Restricted: $x \neq 2, -2$
Multiply by LCD:
$$(x + 2) + (x - 2) = 4$$
$$2x = 4$$
$$x = 2$$
BUT $x = 2$ is restricted!
Answer: No solution (extraneous)
Factor: $x^2 - 4 = (x - 2)(x + 2)$
LCD: $(x - 2)(x + 2)$
Restricted: $x \neq 2, -2$
Multiply by LCD:
$$(x + 2) + (x - 2) = 4$$
$$2x = 4$$
$$x = 2$$
BUT $x = 2$ is restricted!
Answer: No solution (extraneous)
Example 3: Solve $\frac{3}{x} - \frac{2}{x + 1} = 1$
LCD: $x(x + 1)$
Restricted: $x \neq 0, -1$
$$3(x + 1) - 2x = x(x + 1)$$
$$3x + 3 - 2x = x^2 + x$$
$$x + 3 = x^2 + x$$
$$0 = x^2 - 3$$
$$x^2 = 3$$
$$x = \pm\sqrt{3}$$
Check: Neither is restricted
Answer: $x = \sqrt{3}$ or $x = -\sqrt{3}$
LCD: $x(x + 1)$
Restricted: $x \neq 0, -1$
$$3(x + 1) - 2x = x(x + 1)$$
$$3x + 3 - 2x = x^2 + x$$
$$x + 3 = x^2 + x$$
$$0 = x^2 - 3$$
$$x^2 = 3$$
$$x = \pm\sqrt{3}$$
Check: Neither is restricted
Answer: $x = \sqrt{3}$ or $x = -\sqrt{3}$
Example 4: Solve $\frac{x}{x - 3} = \frac{3}{x - 3} + 2$
LCD: $x - 3$
Restricted: $x \neq 3$
Multiply by $(x - 3)$:
$$x = 3 + 2(x - 3)$$
$$x = 3 + 2x - 6$$
$$x = 2x - 3$$
$$-x = -3$$
$$x = 3$$
BUT $x = 3$ is restricted!
Answer: No solution
LCD: $x - 3$
Restricted: $x \neq 3$
Multiply by $(x - 3)$:
$$x = 3 + 2(x - 3)$$
$$x = 3 + 2x - 6$$
$$x = 2x - 3$$
$$-x = -3$$
$$x = 3$$
BUT $x = 3$ is restricted!
Answer: No solution
Warning - Extraneous Solutions:
When you multiply by LCD, you may introduce solutions that make the original denominators zero. ALWAYS check your answers!
When you multiply by LCD, you may introduce solutions that make the original denominators zero. ALWAYS check your answers!
Operations on Rational Expressions Summary
| Operation | Rule | Example |
|---|---|---|
| Simplify | Factor, then cancel common factors | $\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x+2$ |
| Multiply | $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ | $\frac{x}{2} \cdot \frac{3}{x} = \frac{3x}{2x} = \frac{3}{2}$ |
| Divide | Multiply by reciprocal | $\frac{x}{2} \div \frac{x}{3} = \frac{x}{2} \cdot \frac{3}{x} = \frac{3}{2}$ |
| Add/Subtract (same denominator) | $\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$ | $\frac{3}{x} + \frac{2}{x} = \frac{5}{x}$ |
| Add/Subtract (different denominators) | Find LCD, convert, then add/subtract | $\frac{1}{x} + \frac{1}{2} = \frac{2+x}{2x}$ |
Asymptote Quick Reference
| Type | How to Find | Meaning |
|---|---|---|
| Vertical Asymptote | Set denominator = 0, solve for x | Values where function is undefined; graph has vertical line |
| Horizontal Asymptote | Compare degrees: • If $n < m$: $y = 0$ • If $n = m$: $y = \frac{a}{b}$ • If $n > m$: none | Behavior as $x \to \pm\infty$ |
| Excluded Values | Values that make denominator = 0 | Not in domain; may be asymptotes or holes |
Solving Rational Equations Checklist
| Step | Action | Why Important |
|---|---|---|
| 1 | Find restricted values | Know which solutions to reject |
| 2 | Find LCD | Clear all denominators |
| 3 | Multiply every term by LCD | Eliminate fractions |
| 4 | Solve resulting equation | Get potential solutions |
| 5 | CHECK in original equation | Eliminate extraneous solutions |
Success Tips for Rational Functions and Expressions:
✓ Factor completely before simplifying - look for GCF, difference of squares, trinomials
✓ Only factors can be cancelled, NOT terms (can't cancel from addition/subtraction)
✓ Always state restrictions (excluded values) in your answer
✓ For multiplication: factor, cancel, then multiply
✓ For division: multiply by reciprocal (flip second fraction)
✓ For addition/subtraction: need common denominator (LCD)
✓ Vertical asymptotes occur where denominator = 0
✓ Horizontal asymptotes depend on degrees of numerator and denominator
✓ When solving equations: multiply by LCD to clear fractions
✓ ALWAYS check solutions - discard any that make denominator = 0!
✓ Factor completely before simplifying - look for GCF, difference of squares, trinomials
✓ Only factors can be cancelled, NOT terms (can't cancel from addition/subtraction)
✓ Always state restrictions (excluded values) in your answer
✓ For multiplication: factor, cancel, then multiply
✓ For division: multiply by reciprocal (flip second fraction)
✓ For addition/subtraction: need common denominator (LCD)
✓ Vertical asymptotes occur where denominator = 0
✓ Horizontal asymptotes depend on degrees of numerator and denominator
✓ When solving equations: multiply by LCD to clear fractions
✓ ALWAYS check solutions - discard any that make denominator = 0!