CONCEPT PREVIEW Perform the indicated operation, and write each answer in lowest terms 2x/5 + x/4

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Identify the terms to be added: \( \frac{2x}{5} + \frac{x}{4} \). Since these are fractions with different denominators, we need to find a common denominator before adding.

Find the least common denominator (LCD) of 5 and 4. The LCD is the smallest number that both denominators divide into evenly. Calculate the LCD as \( \text{LCD} = 20 \).

Rewrite each fraction with the denominator 20 by multiplying numerator and denominator appropriately: \( \frac{2x}{5} = \frac{2x \times 4}{5 \times 4} = \frac{8x}{20} \) and \( \frac{x}{4} = \frac{x \times 5}{4 \times 5} = \frac{5x}{20} \).

Add the two fractions now that they have the same denominator: \( \frac{8x}{20} + \frac{5x}{20} = \frac{8x + 5x}{20} = \frac{13x}{20} \).

Check if the resulting fraction \( \frac{13x}{20} \) can be simplified further by finding the greatest common divisor (GCD) of 13 and 20. Since 13 is a prime number and does not divide 20, the fraction is already in lowest terms.

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주요 개념

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Adding Rational Expressions

Adding rational expressions involves combining fractions with variable terms. To add them, you must have a common denominator, just like with numerical fractions, so the expressions can be combined into a single fraction.

Finding the Least Common Denominator (LCD)

The least common denominator is the smallest expression that both denominators divide into evenly. Finding the LCD allows you to rewrite each fraction with the same denominator, enabling straightforward addition of the numerators.

Simplifying Algebraic Fractions

After adding fractions, simplify the resulting expression by factoring and reducing common factors in the numerator and denominator. This ensures the answer is in lowest terms, making it easier to interpret and use.