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

Verified step by step guidance

1. Identify the given expression: \(\frac{2x}{5} + \frac{x}{4}\).
2. Find the least common denominator (LCD) of the fractions. Since the denominators are 5 and 4, the LCD is \$20$.
3. Rewrite each fraction with the denominator \$20$ by multiplying numerator and denominator appropriately: \(\frac{2x}{5} = \frac{2x \times 4}{5 \times 4} = \frac{8x}{20}\) and \(\frac{x}{4} = \frac{x \times 5}{4 \times 5} = \frac{5x}{20}\).
4. Add the numerators over the common denominator: \(\frac{8x}{20} + \frac{5x}{20} = \frac{8x + 5x}{20} = \frac{13x}{20}\).
5. Check if the fraction \(\frac{13x}{20}\) can be simplified further by factoring numerator and denominator; since 13 is prime and does not divide 20, the expression is already in lowest terms.

Key Concepts

Adding Rational Expressions

Adding rational expressions involves combining fractions with variable expressions in the numerator and denominator. To add them, you must find a common denominator, rewrite each fraction with this denominator, and then add the numerators while keeping the denominator the same.

Finding the Least Common Denominator (LCD)

The least common denominator is the smallest expression that both denominators divide into evenly. Identifying the LCD allows you to rewrite each fraction with a common denominator, which is essential for performing addition or subtraction of rational expressions.

Simplifying Rational Expressions

After performing operations on rational expressions, simplifying involves factoring numerators and denominators and canceling common factors. This process reduces the expression to its lowest terms, making the answer clearer and more concise.