/*! 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 26 Perform the indicated operation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 26

Perform the indicated operation. Simplify, if possible. $$ \frac{3}{x-2}+\frac{3}{x+2} $$

Short Answer

Expert verified
\( \frac{x-3}{x^2-4x+3} = \frac{1}{x-1} \)

Step by step solution

01

Identify the common denominator

Notice that both fractions have the same denominator: \[ x^2 - 4x + 3 \].
02

Combine the numerators

Since the denominators are the same, combine the numerators: \[ \frac{(x-5) + 2}{x^2 - 4x + 3} \].
03

Simplify the numerator

Simplify the expression in the numerator by combining like terms: \[ (x-5) + 2 = x - 3 \]. This gives us: \[ \frac{x-3}{x^2 - 4x + 3} \].
04

Simplify the denominator

Factor the denominator, if possible. The quadratic expression can be factored as: \[ x^2 - 4x + 3 = (x - 3)(x - 1) \]. Thus, the final fraction is: \[ \frac{x-3}{(x-3)(x-1)} \].
05

Simplify the whole fraction

Cancel out the common term in the numerator and denominator: \[ \frac{x-3}{(x-3)(x-1)} = \frac{1}{x-1} \] (assuming \( x eq 3 \) to avoid division by zero).

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 performing operations with fractions, it's essential to have the same denominator for all fractions involved. This makes it easier to combine them.
In this exercise, we notice that the denominators of both fractions are already the same: \ta) \(\frac{x-5}{x^{2}-4 x+3}\)\tb) \(\frac{2}{x^{2}-4 x+3}\)
Having a common denominator allows you to directly combine the numerators. This is a valuable skill that simplifies adding, subtracting, and comparing fractions.
Simplify Fractions
Simplifying fractions involves making the fraction as simple as possible. This could mean combining like terms in the numerator or reducing the fraction.
In our example, once we have a common denominator, we add the numerators:\t\[ \frac{(x-5) + 2}{x^{2}-4 x+3} = \frac{x-3}{x^{2}-4 x+3} \]
Notice how we simplify \( (x - 5) + 2 \) to \( x - 3 \). This is a crucial step. Always look out for opportunities to combine like terms and reduce your fractions to their simplest form.
Factoring Quadratics
Factoring quadratics is an essential step in simplifying algebraic fractions. A quadratic expression like \( x^{2}-4x+3 \) can often be factored into the product of two binomials.
In our example, \( x^{2}-4x+3 \) factors to: \t\[ x^{2}-4x+3 = (x-3)(x-1) \]
This step makes it easier to see if we can cancel out any common terms from the numerator and denominator. Factoring quadratics makes complex fractions more manageable and reveals potential simplifications.
Combining Like Terms
Combining like terms means simplifying expressions by adding or subtracting similar terms.
In our example, the expression \( (x - 5) + 2 \) involves combining terms with and without variables. By doing this, we get: \t\[ (x-5) + 2 = x - 3 \]
This is a critical algebraic skill that helps simplify expressions and solve equations efficiently. Identifying and combining like terms reduces complexity and makes it easier to solve or further manipulate the expression.

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

To prepare for Chapter \(8,\) review solving inequalities \((\text {Section } 2.6)\) Solve. $$ 2(x-1) \geq 4(x+1) $$

Find the LCM. $$2(y-3), 6(y-3)$$

To prepare for Chapter \(8,\) review solving inequalities \((\text {Section } 2.6)\) Solve. $$ \frac{2}{3} x+1 \leq x+4 $$

Find the LCM. $$10,15$$

Young's rule for determining the size of a particular child's medicine dosage \(c\) is $$ c=\frac{a}{a+12} \cdot d $$ where \(a\) is the child's age and \(d\) is the typical adult dosage. If a child's age is doubled, the dosage increases. Find the ratio of the larger dosage to the smaller dosage. By what percent does the dosage increase?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.