In Exercises 33–68, add or subtract as indicated. (4x−10)/(x−2) − (x−4)/(x−2)

Verified step by step guidance

1. Identify the common denominator in the expression: both fractions have the same denominator $x-2$.
2. Since the denominators are the same, you can combine the numerators directly: $(4x-10)-(x-4)$.
3. Distribute the negative sign across the second numerator: $4x-10-x+4$.
4. Combine like terms in the numerator: $(4x-x)+(-10+4)$.
5. Simplify the expression: $(3x-6)/(x-2)$.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions is crucial for performing operations like addition and subtraction. In this case, both expressions share a common denominator, which simplifies the process of combining them.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When adding or subtracting rational expressions, it is essential to have a common denominator to combine the numerators correctly. In the given problem, both fractions have the same denominator, making it straightforward to perform the operation.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting coefficients of the same variable. After obtaining a common denominator, the next step is to combine the numerators by performing the indicated operation and then simplifying the resulting expression. This concept is key to arriving at the final simplified form of the rational expression.