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

Verified step by step guidance

1. Identify the least common denominator (LCD) for the fractions \( \frac{2x}{5} \) and \( \frac{x}{4} \). The LCD of 5 and 4 is 20.
2. Rewrite each fraction with the common denominator of 20. Multiply the numerator and denominator of \( \frac{2x}{5} \) by 4 to get \( \frac{8x}{20} \). Multiply the numerator and denominator of \( \frac{x}{4} \) by 5 to get \( \frac{5x}{20} \).
3. Add the two fractions: \( \frac{8x}{20} + \frac{5x}{20} = \frac{(8x + 5x)}{20} \).
4. Combine the numerators: \( 8x + 5x = 13x \), so the expression becomes \( \frac{13x}{20} \).
5. Ensure the fraction is in its lowest terms. Since 13 and 20 have no common factors other than 1, \( \frac{13x}{20} \) is already in its simplest form.

Key Concepts

Fraction Addition

To add fractions, they must have a common denominator. The least common denominator (LCD) is the smallest multiple that both denominators share. Once the fractions are expressed with the LCD, their numerators can be added together while keeping the denominator the same. The result should then be simplified to its lowest terms.

Simplifying Fractions

Simplifying a fraction involves reducing it to its lowest terms, where the numerator and denominator have no common factors other than 1. This is done by dividing both the numerator and denominator by their greatest common divisor (GCD). A simplified fraction is easier to interpret and is often required in mathematical expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In the context of the given question, '2x' represents a variable term that can be manipulated algebraically. Understanding how to combine like terms and perform operations on these expressions is crucial for solving problems involving fractions and algebra.