/*! 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 For what values of the variable ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 26

For what values of the variable is the rational expression undefined? $$\frac{4}{x^{2}-1}$$

Short Answer

Expert verified
The values of x for which the rational expression \( \frac{4}{x^{2}-1} \) is undefined are \( x = 1 \) and \( x = -1 \).

Step by step solution

01

Identify the Denominator

The denominator of the given rational expression: \( \frac{4}{x^{2}-1} \) is \( x^{2}-1 \). This expression will be undefined when its denominator is zero.
02

Set the Denominator Equal to Zero

To find the roots of the denominator, set \( x^{2}-1 = 0 \). This will give the values of \( x \) for which the rational expression is undefined.
03

Solve the Equation

Solving the equation \( x^{2}-1 = 0 \), we get \( x^{2} = 1 \). Thus, \( x = \sqrt{1} \) & \( x = -\sqrt{1} \) Hence, \( x = 1 \) & \( x = -1 \).

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.

Undefined Values
The concept of undefined values is crucial in understanding rational expressions. A rational expression is undefined when the denominator equals zero because division by zero is not possible in mathematics. This results in a value that cannot be determined or interpreted.
For instance, consider the expression \(\frac{4}{x^2-1}\). To find out when this expression is undefined, we look for values that make the denominator \(x^2-1\) equal to zero.
Here's a simple method to determine undefined values:
  • Identify the denominator of the rational expression.
  • Set the denominator equal to zero.
  • Solve the resulting equation to find the values that cause the expression to be undefined.
This allows you to safely determine when the expression cannot be evaluated.
Denominator
Understanding the role of the denominator in rational expressions is fundamental. The denominator is the part of a fraction situated below the division line, and it indicates the divisor in the expression.
In the rational expression \(\frac{4}{x^2-1}\), the denominator is \(x^2-1\). Mathematical operations require that the denominator never equals zero since it represents the divisor. When it becomes zero, the expression loses its meaning.
To manage this, follow these steps:
  • Identify the denominator in the rational expression.
  • Determine when the denominator equals zero, as this indicates undefined status.
  • Investigate and solve the equation that results from setting the denominator to zero.
Comprehending the denominator's role is essential in avoiding undefined results in calculations.
Quadratic Equations
Quadratic equations frequently appear in algebra, including solving for when certain expressions are undefined. A quadratic equation generally takes the form \(ax^2 + bx + c = 0\).
In the context of rational expressions like \(\frac{4}{x^2-1}\), solving the equation \(x^2-1 = 0\) plays a critical role. This is a simple quadratic equation, as it only involves \(x^2\) and a constant.
Here's how to handle solving these:
  • Isolate the quadratic term by setting the equation to \(x^2 = 1\).
  • Solve for \(x\) by taking the square root of both sides, giving \(x = \sqrt{1}\) and \(x = -\sqrt{1}\).
  • Conclude with \(x = 1\) and \(x = -1\), which make the denominator zero, rendering the expression undefined at these points.
Understanding quadratic equations allows you to solve for undefined values in rational expressions effectively.

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

The population \(P\) of Texas (in thousands) for 1995 projected through 2025 can be modeled by \(P=18,870(1.0124)^{t},\) where \(t=0\) represents \(1995 .\) Find the ratio of the population in 2025 to the population in 2000 . (Review 8.3) P Source: U.S. Bureau of the Census.

Simplify the expression. $$\frac{x+6}{x+1}-\frac{4}{2 x+3}$$

Simplify. \(-18 c^{3} \div \frac{-27 c}{-4}\)

Simplify the expression. $$\frac{7 x}{x^{3}}-\frac{6 x}{x^{3}}$$

You can use \(x-3\) as the LCD when finding the sum \(\frac{5}{x-3}+\frac{2}{3-x}\) What number can you multiply the numerator and the denominator of the second fraction by to get an equivalent fraction with \(x-3\) as the new denominator?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.