Add or subtract as indicated. 2/5x − (x+1)/4x

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{2}{5x} - \frac{x+1}{4x}\).
2. Find the least common denominator (LCD) of the two fractions. Since the denominators are \$5x\( and \)4x\(, the LCD is \)20x$.
3. Rewrite each fraction with the common denominator \$20x$: multiply numerator and denominator of \(\frac{2}{5x}\) by 4, and multiply numerator and denominator of \(\frac{x+1}{4x}\) by 5.
4. Express the subtraction with the common denominator: \(\frac{2 \times 4}{20x} - \frac{(x+1) \times 5}{20x}\), which simplifies to \(\frac{8}{20x} - \frac{5(x+1)}{20x}\).
5. Combine the numerators over the common denominator: \(\frac{8 - 5(x+1)}{20x}\), then simplify the numerator by distributing and combining like terms.

Key Concepts

Like Terms and Variable Expressions

Understanding how to identify and combine like terms is essential when working with algebraic expressions. Terms with the same variable and exponent can be added or subtracted directly, while unlike terms must be handled separately.

Operations with Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, denominator, or both. Adding or subtracting them requires finding a common denominator to combine the expressions properly.

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, enabling addition or subtraction of the numerators.