Adding Rational Expressions (Unlike Denominators)

Learn how to add rational expressions with different denominators by finding the LCD.

Advanced25 minLesson

Definition

To add rational expressions with unlike denominators, we must first find a common denominator. The process is similar to adding numerical fractions:

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

For rational expressions, we:

1.Factor each denominator completely
2.Find the LCD (Least Common Denominator)
3.Rewrite each fraction with the LCD
4.Add the numerators
5.Simplify if possible

\frac{2}{x+1} + \frac{3}{x-1} = \frac{2(x-1) + 3(x+1)}{(x+1)(x-1)} = \frac{5x+1}{x^2-1}

Try it now

What is the LCD of

\frac{1}{x}

and

\frac{1}{x+2}

Worked Examples

Simplify: $\frac{2}{x+3} + \frac{5}{x-2}$

Identify the denominators

Denominator 1: $(x+3)$ , Denominator 2: $(x-2)$ → Both are linear, no common factors

Find the LCD

LCD = $(x+3)(x-2)$ → Multiply the distinct factors

Rewrite first fraction

$\frac{2}{x+3} \cdot \frac{x-2}{x-2} = \frac{2(x-2)}{(x+3)(x-2)}$ → $\frac{2x-4}{(x+3)(x-2)}$

Rewrite second fraction

$\frac{5}{x-2} \cdot \frac{x+3}{x+3} = \frac{5(x+3)}{(x+3)(x-2)}$ → $\frac{5x+15}{(x+3)(x-2)}$

Add the numerators

$\frac{2x-4+5x+15}{(x+3)(x-2)}$ → $\frac{7x+11}{(x+3)(x-2)}$

Answer: $\frac{7x+11}{(x+3)(x-2)}$ or $\frac{7x+11}{x^2+x-6}$

Common Mistakes

Forgetting to factor denominators first

Why it's wrong: Without factoring, you might miss common factors and create an LCD that is larger than necessary, making the problem harder.

Correct: Always factor each denominator completely before finding the LCD. For example, $x^2-9 = (x+3)(x-3)$ .

Adding denominators instead of finding LCD

Why it's wrong: Unlike numerators, denominators are not added together. The LCD must contain all factors from each denominator.

Correct: $\frac{1}{x+2} + \frac{1}{x+3} \neq \frac{2}{2x+5}$ . The LCD is $(x+2)(x+3)$ .

Forgetting to multiply both numerator and denominator

Why it's wrong: When converting to the LCD, you must multiply by a form of 1 (same expression over itself) to keep the value unchanged.

Correct: $\frac{2}{x+1} = \frac{2(x-1)}{(x+1)(x-1)}$ , not $\frac{2}{(x+1)(x-1)}$ .

Not simplifying the final answer

Why it's wrong: After adding, the resulting numerator might share a common factor with the denominator.

Correct: Always check if the numerator can be factored and if any factors cancel with the denominator.

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Practice Problems

17 problems

Problem 1 of 17

Easy

What is the LCD of $\frac{1}{x}$ and $\frac{1}{x+2}$ ?

Why It Matters

Adding rational expressions with unlike denominators appears in many real-world applications:

Physics: Combining resistances in parallel circuits: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}$
Work problems: If one worker completes a job in $x$ hours and another in $y$ hours, together they complete $\frac{1}{x} + \frac{1}{y}$ of the job per hour
Optics: The thin lens equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
Rate problems: Combining different rates of production or travel

Mastering this skill is essential for solving complex algebraic equations and real-world optimization problems.

Real World Applications

Parallel Resistors in Electronics

When resistors are connected in parallel, the total resistance is found using the formula with unlike denominators.

Example:

For two resistors of $R_1 = x$ ohms and $R_2 = x+2$ ohms in parallel: $\frac{1}{R_{total}} = \frac{1}{x} + \frac{1}{x+2} = \frac{2x+2}{x(x+2)}$

1Try It Yourself

You have two resistors: one with resistance $x$ ohms and another with resistance $(x+3)$ ohms connected in parallel.

Find an expression for the total resistance.

Step 1: Write the mathematical expression

First find $\frac{1}{R_{total}} = \frac{1}{x} + \frac{1}{x+3}$

Combined Work Rate Problems

When two people work together, their combined rate involves adding fractions with different denominators.

Example:

If Worker A completes a job in $x$ hours and Worker B in $(x+2)$ hours, their combined rate is $\frac{1}{x} + \frac{1}{x+2} = \frac{2x+2}{x(x+2)}$ jobs per hour.

2Try It Yourself

Machine A produces a batch in $t$ hours. Machine B produces the same batch in $(t+4)$ hours.

What fraction of the batch do both machines produce together in one hour?

Step 1: Write the mathematical expression

Combined rate = $\frac{1}{t} + \frac{1}{t+4}$

Key Takeaways

1To add rational expressions with unlike denominators, first factor all denominators completely
2Find the LCD by including each factor the maximum number of times it appears in any denominator
3Multiply each fraction by a form of 1 to convert to the LCD
4Add the numerators while keeping the common denominator
5Simplify the result by factoring and canceling common factors

Frequently Asked Questions

Factor each denominator completely. The LCD contains each factor raised to the highest power it appears in any denominator. For example, for

\frac{1}{x^2(x+1)}

and

\frac{1}{x(x+1)^2}

, the LCD is

x^2(x+1)^2

Glossary

Rational expression: A fraction where the numerator and/or denominator contains a polynomial
LCD (Least Common Denominator): The smallest expression that is divisible by all denominators in a problem
Unlike denominators: Denominators that are not identical and require finding a common denominator
Factor: To write an expression as a product of simpler expressions

Formula Card

Adding Two Rational Expressions

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

General formula when denominators have no common factors

LCD Method

\frac{a}{P} + \frac{b}{Q} = \frac{a \cdot \frac{LCD}{P} + b \cdot \frac{LCD}{Q}}{LCD}

Rewrite each fraction using the Least Common Denominator