/*! 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 28 Simplify. If possible, use a sec... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 28

Simplify. If possible, use a second method, evaluation, or a graphing calculator as a check. $$ \frac{\frac{x}{x^{2}+5 x-6}+\frac{6}{x^{2}+5 x-6}}{\frac{x}{x^{2}-5 x+4}-\frac{2}{x^{2}-5 x+4}} $$

Short Answer

Expert verified
\(\frac{x-4}{x-2}\)

Step by step solution

01

Factor the Denominators

Identify the quadratic expressions in the denominators and factor them. Note that the common factorizations are:\[ x^2 + 5x - 6 = (x+6)(x-1) \]\[ x^2 - 5x + 4 = (x-4)(x-1) \]
02

Rewrite the Fractions

Rewrite each fraction with the factored denominators:\[ \frac{x}{(x+6)(x-1)} + \frac{6}{(x+6)(x-1)} \quad \text{and} \quad \frac{x}{(x-4)(x-1)} - \frac{2}{(x-4)(x-1)} \]
03

Combine the Numerators

Combine the numerators since the denominators are the same:\[ \frac{x + 6}{(x+6)(x-1)} \quad \text{and} \quad \frac{x - 2}{(x-4)(x-1)} \]
04

Form the Complex Fraction

Form the complex fraction by placing the two results from Step 3 into the original fraction:\[ \frac{\frac{x+6}{(x+6)(x-1)}}{\frac{x-2}{(x-4)(x-1)}} \]
05

Simplify the Complex Fraction

To simplify the complex fraction, multiply by the reciprocal of the denominator fraction:\[ \frac{x+6}{(x+6)(x-1)} \cdot \frac{(x-4)(x-1)}{x-2} \]
06

Cancel Common Factors

Cancel the common factors of \((x-1)\) from the numerators and the denominators:\[ \frac{(x+6)(x-4)}{(x+6)(x-2)} \]
07

Simplify the Expression

After canceling the common \((x+6)\) term, simplify the expression to:\[ \frac{x-4}{x-2} \]
08

Check the Simplification

Verify the result using either a second method, evaluation, or a graphing calculator to confirm the simplification is correct.

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.

Factoring Quadratics
Factoring quadratics is a crucial step in simplifying complex fractions. When a quadratic expression can be rewritten as a product of two binomials, it's easier to work with. For example, the quadratic expressions \[ x^2 + 5x - 6 \] and \[ x^2 - 5x + 4 \] can be factored as: \[ x^2 + 5x - 6 = (x + 6)(x - 1) \]and \[ x^2 - 5x + 4 = (x - 4)(x - 1) \]. This step transforms the denominators into manageable parts for further simplification.
Combining Fractions
When fractions have the same denominators, you can combine them by adding or subtracting the numerators. In our problem, after factoring, the fractions become easier to manage: \[ \frac{x}{(x+6)(x-1)} + \frac{6}{(x+6)(x-1)} \] and \[ \frac{x}{(x-4)(x-1)} - \frac{2}{(x-4)(x-1)} \]. Combining these fractions means adding or subtracting the numerators: \[ \frac{x + 6}{(x+6)(x-1)} \]and \[ \frac{x - 2}{(x-4)(x-1)} \]. Doing this keeps the problem structured and clear.
Canceling Common Factors
Canceling common factors simplifies both the numerators and the denominators. This helps to reduce complexity and visualize the problem better. For instance, when forming the complex fraction, \[ \frac{\frac{x + 6}{(x + 6)(x - 1)}}{\frac{x - 2}{(x - 4)(x - 1)}} \], you multiply by the reciprocal of the second fraction to get: \[ \frac{x+6}{(x+6)(x-1)} \times \frac{(x-4)(x-1)}{x-2} \]. Notice \( (x - 1) \) appears in both numerator and denominator, allowing you to cancel it out.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing them to their simplest form. After canceling common factors, you're left with a simplified fraction. For example, \[ \frac{(x+6)(x-4)}{(x+6)(x-2)} \] can be reduced by canceling the common \( (x+6) \) term: \[ \frac{x-4}{x-2} \]. This reduces the complexity of calculations and helps verify the result using another method, such as evaluation or a graphing calculator.

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

Is the following statement true or false: "For any real numbers \(a, b,\) and \(c,\) if \(a c=b c,\) then \(a=b^" ?\) Explain why you answered as you did.

Escalators. Together, a \(100-\mathrm{cm}\) wide escalator and a \(60-\mathrm{cm}\) wide escalator can empty a 1575 -person auditorium in 14 min. The wider escalator moves twice as many people as the narrower one. How many people per hour does the \(60-\mathrm{cm}\) wide escalator move?

One factor influencing urban planning is VMT, or vehicle miles traveled. The table below lists the annual VMT per household for various densities for a typical urban area. $$\begin{array}{c|c}{\text { Population Density }} \\ {\text { (in number of households }} & {\text { Annual VMT }} \\ { \text { per residential acre } )} & {\text { per Household }} \\ {25} & {12,000} \\ {50} & {6,000} \\ {100} & {3,000} \\ {200} & {1,500}\end{array}$$ a) Determine whether the data indicate direct variation or inverse variation. b) Find an equation of variation that describes the data. c) Use the equation to estimate the annual VMT per household for areas with 10 households per residential acre.

Perform the indicated operations. Simplify, if possible. $$\frac{2 x+11}{x-3} \cdot \frac{3}{x+4}+\frac{-1}{4+x} \cdot \frac{6 x+3}{x-3}$$

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

Calculus

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.