Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{2(x-1)^{-1}+3}{3(x-1)^{-1}+2} $$

Short answer

The simplified form is \( \frac{3x - 1}{2x + 1} \).

Step-by-step solution

Understand the problem

Identify the expression given: \( \frac{2(x-1)^{-1}+3}{3(x-1)^{-1}+2} \). Notice that \( (x-1)^{-1} = \frac{1}{x-1} \). Replace \( (x-1)^{-1} \) with \( \frac{1}{x-1} \) in the expression.

Replace the negative exponents

Rewrite the expression: \( \frac{2 \frac{1}{x-1} + 3}{3 \frac{1}{x-1} + 2} \).

Combine terms with common denominators

Factor out \( \frac{1}{x-1} \) from the numerator and denominator: \( \frac{\frac{2 + 3(x-1)}{x-1}}{\frac{3 + 2(x-1)}{x-1}} \).

Simplify the simplified fractions

Simplify the numerators and denominators separately: \( \frac{2 + 3(x-1)}{3 + 2(x-1)} \).

Distribute and simplify

Distribute the constants: \( \frac{2 + 3x - 3}{3 + 2x - 2} = \frac{3x - 1}{2x + 1} \).

Factor, if possible

Check if the expression can be factored further. In this case, \( \frac{3x - 1}{2x + 1} \) is already in its simplest form.

Key concepts

Factoring

Factoring is an important concept in algebra that involves breaking down an expression into simpler components, called factors, that can be multiplied together to obtain the original expression. By factoring, complex expressions become easier to manipulate and solve. In this exercise, however, the final expression \(\frac{3x - 1}{2x + 1}\) could not be factored further. Let's understand the process:

• Factor polynomials wherever possible
• Explore common factors in the numerator and denominator
• Ensure the expression is in its simplest form

Remember that factoring can simplify expressions significantly, making it a crucial technique to master in algebra.

Fractions

Dealing with fractions involves understanding and working with the numerator (top part) and the denominator (bottom part) of a fraction. In algebra, fractions often have variables in them too. Here's how we managed the fractions in this exercise:

• Recognize the role of the numerator and denominator in algebraic fractions
• Combine fractions with common denominators
• Simplify complex fractions by manipulating their terms

By following these steps, we broke down the given problem into more manageable parts and combined fractions to simplify the expression.

Negative Exponents

Negative exponents can seem tricky at first, but they're quite straightforward once you get the hang of them. A term with a negative exponent, such as \(a^{-n}\), can be rewritten as \frac{1}{a^n}\. In the given exercise, we replaced \((x-1)^{-1}\) with \frac{1}{x-1}\:

• Understand the notation of negative exponents
• Convert negative exponents into fractions
• Simplify expressions by combining similar terms

By addressing the negative exponents first, we could simplify the given expression step by step.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Simplifying these requires various algebraic techniques including factoring, finding common denominators, and canceling common factors. Here’s the approach we used for rational expressions in this exercise:

• Identify the polynomials in the numerator and denominator
• Factor out common terms
• Combine like terms and simplify the expression

Working with rational expressions can help in greatly simplifying complex algebraic problems.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used for rewriting algebraic expressions in different forms. This involves a variety of methods such as combining like terms, factoring, expanding, and simplifying fractions. Here's what was done in this exercise:

• Replace negative exponents with fractions
• Factor and combine terms with common denominators
• Simplify the final expression by distributing and combining terms

Mastering algebraic manipulation can help you efficiently solve complicated expressions and equations.