/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Perform the addition or subtract... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 55

Perform the addition or subtraction and simplify. $$\frac{2}{x}+\frac{3}{x-1}-\frac{4}{x^{2}-x}$$

Short Answer

Expert verified
The simplified expression is \(\frac{5x - 6}{x(x-1)}\).

Step by step solution

01

Identify Common Denominator

The first step is to find a common denominator for all three fractions. The denominators are \(x\), \(x-1\), and \(x^2-x\). Rewrite \(x^2-x\) as \(x(x-1)\). Thus, the common denominator is \(x(x-1)\).
02

Rewrite Each Fraction with Common Denominator

Convert each fraction to have the common denominator \(x(x-1)\): - For \(\frac{2}{x}\), multiply both numerator and denominator by \((x-1)\) to get \(\frac{2(x-1)}{x(x-1)}\).- For \(\frac{3}{x-1}\), multiply both numerator and denominator by \(x\) to get \(\frac{3x}{x(x-1)}\).- The fraction \(\frac{4}{x^2-x}\) is already with the denominator \(x(x-1)\).
03

Combine the Fractions

Now that all fractions have the same denominator, combine them into a single fraction:\[ \frac{2(x-1) + 3x - 4}{x(x-1)} \]
04

Simplify the Numerator

Expand and simplify the numerator:- Distribute the \(2\) in \(2(x-1)\) to get \(2x - 2\).- Add like terms in the numerator: \[ 2x - 2 + 3x - 4 = 5x - 6 \]
05

Write the Final Simplified Expression

Now the fraction is:\[ \frac{5x - 6}{x(x-1)} \]This is the simplified form since no common factors appear in the numerator and the denominator.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When dealing with fraction addition and subtraction, finding a common denominator is crucial. This is similar to finding a common language that all fractions can understand. Without a common denominator, it's like trying to add apples and oranges - it just doesn't work easily.
  • Each original fraction might have a different denominator, making direct addition or subtraction impossible.
  • A common denominator is found by identifying the least common multiple (LCM) of the original denominators.
  • In this exercise, the denominators were \(x\), \(x-1\), and \(x^2-x\), where \(x^2-x\) was rewritten as \(x(x-1)\).
By rewriting the fractions with a common denominator, it becomes possible to add or subtract them directly. This step lays a solid foundation for combining fractions.
Simplifying Expressions
After working the fractions into a single expression with a common denominator, the next challenge is simplifying the expression. Simplification requires careful handling of algebraic expressions to make them more digestible.
  • Begin by expanding the expressions, meaning you'll open up terms like \(2(x-1)\) into \(2x - 2\).
  • Combine like terms, which are terms that have the same variables raised to the same power. Here, you combine terms like \(2x\), \(3x\), and \(-4\) to get \(5x - 6\).
  • Write this expression over the common denominator.
Simplifying expressions makes them less cluttered and can often reveal more insights into the nature of the problem, preparing them for further algebraic manipulation if necessary.
Algebraic Fractions
Algebraic fractions behave like numerical fractions but involve variables, introducing a layer of complexity. Understanding them is crucial for a firm grasp of many algebraic techniques and methods.
  • Algebraic fractions can include any polynomial expressions in their numerators and denominators.
  • Operations like addition and subtraction require a special focus on finding a common denominator, unlike straightforward numerical fractions.
  • During calculations, always check that any simplification steps maintain the conditions for non-zero denominators to avoid undefined expressions.
Working with algebraic fractions builds a deeper understanding of both individual and combined algebraic expressions, enhancing one's ability to tackle complex math problems with confidence.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(y-3)(y+3)$$

Perform the indicated operations and simplify. $$(x+2)\left(x^{2}+2 x+3\right)$$

Factor the trinomial. $$3 x^{2}-16 x+5$$

Factor the expression by grouping terms. $$-9 x^{3}-3 x^{2}+3 x+1$$

Factor the expression completely. $$2 x^{3}+4 x^{2}+x+2$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.