Adding and Subtracting Rational Expressions with Like and Unlike Denominators

Chapter 26: Adding and Subtracting Rational Expressions with Like Denominators

Concept Explanation:

Adding and subtracting rational expressions with like denominators follows a similar process to adding and subtracting fractions. When the denominators are the same, you can directly add or subtract the numerators.

For example, given two rational expressions:

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

Common Mistakes:

Ignoring the Denominator: Students may forget to keep the denominator the same.
Simplification Oversight: Not simplifying the result can lead to an improper answer.

Helpful Tips:

Common Denominator: If working with expressions that look complicated, break them down to see the common denominator more clearly.
Factor When Necessary: Before adding or subtracting, consider factoring the numerators for simplification.

Hard Questions:

Q1: Add $\frac{2x}{5} + \frac{3x}{5}$ .

Step-by-Step Solution:

Since the denominators are the same, add the numerators:

$\frac{2x + 3x}{5} = \frac{5x}{5} = x$

Answer:

The result is $x$ .

Q2: Subtract $\frac{4y}{9} - \frac{2y}{9}$ .

Step-by-Step Solution:

Since the denominators are the same, subtract the numerators:

$\frac{4y - 2y}{9} = \frac{2y}{9}$

Answer:

The result is $\frac{2y}{9}$ .

Q3: Add $\frac{x^2}{3} + \frac{2x^2}{3}$ .

Step-by-Step Solution:

Since the denominators are the same, add the numerators:

$\frac{x^2 + 2x^2}{3} = \frac{3x^2}{3} = x^2$

Answer:

The result is $x^2$ .

Chapter 27: Adding and Subtracting Rational Expressions with Unlike Denominators

Concept Explanation:

When dealing with rational expressions that have unlike denominators, you must first find a common denominator. This process involves determining the least common multiple (LCM) of the denominators.

Find the LCM: The least common multiple of the denominators gives you the common denominator.
Rewrite Each Expression: Convert each rational expression to have the common denominator.
Add or Subtract the Numerators: Once the denominators match, add or subtract as necessary.

Common Mistakes:

Not Finding the LCM: Failing to identify the least common denominator can lead to errors.
Improper Conversion: Ensure each expression is properly rewritten with the common denominator.

Helpful Tips:

Cross-Multiplication: When converting expressions, cross-multiplication can help visualize how to rewrite fractions correctly.
Simplify: After finding a common denominator and adding, check if you can simplify the resulting expression.

Hard Questions:

Q1: Add $\frac{1}{4} + \frac{1}{6}$ .

Step-by-Step Solution:

The LCM of 4 and 6 is 12.
Rewrite the fractions:

$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$

Add:

$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Answer:

The result is $\frac{5}{12}$ .

Q2: Subtract $\frac{3x}{5} - \frac{x}{10}$ .

Step-by-Step Solution:

The LCM of 5 and 10 is 10.
Rewrite the first fraction:

$\frac{3x}{5} = \frac{6x}{10}$

Now subtract:

$\frac{6x}{10} - \frac{x}{10} = \frac{6x - x}{10} = \frac{5x}{10} = \frac{x}{2}$

Answer:

The result is $\frac{x}{2}$ .

Q3: Add $\frac{2y}{3} + \frac{3}{4}$ .

Step-by-Step Solution:

The LCM of 3 and 4 is 12.
Rewrite the fractions:

$\frac{2y}{3} = \frac{8y}{12}, \quad \frac{3}{4} = \frac{9}{12}$

Add:

$\frac{8y}{12} + \frac{9}{12} = \frac{8y + 9}{12}$

Answer:

The result is $\frac{8y + 9}{12}$ .

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