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Chapter 7: Problem 13

List all numbers for which each rational expression is undefined. $$ \frac{25}{-7 x} $$

Short Answer

Expert verified
The expression is undefined for x = 0.

Step by step solution

01

Understand the Rational Expression

The expression given is a rational expression: \[\frac{25}{-7x}\]. A rational expression is undefined wherever its denominator is zero.
02

Set the Denominator Equal to Zero

To find when the rational expression is undefined, set the denominator \-7x\ equal to zero: \[-7x = 0\].
03

Solve for x

Solve the equation \[-7x = 0\] for x: \[x = 0\].
04

State the Values

The expression \[\frac{25}{-7x}\] is undefined when x is 0.

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Key Concepts

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

denominator
In mathematics, the denominator is the bottom part of a fraction or rational expression. For the fraction \(\frac{a}{b}\), \b\ is the denominator. It tells you how many equal parts the whole is divided into. More importantly, when the denominator is zero, it makes the fraction undefined. This is because dividing by zero is impossible in standard arithmetic.
To find where a rational expression becomes undefined, always check the denominator. Set the denominator equal to zero and solve for the variable. This value will tell you when the rational expression is not valid.
rational expression
A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, in \(\frac{25}{-7x}\), 25 is the numerator and \-7x\ is the denominator.
Rational expressions are similar to numerical fractions but contain variables. It's crucial to identify where these expressions are undefined, which happens if you divide by zero. Always analyze the denominator: if it's zero at any point, the entire rational expression becomes undefined. Remember that a rational expression can only take on real values when the denominator does not equal zero.
solving equations
Solving equations involves finding the value(s) of the variable(s) that make the equation true. To determine when a rational expression is undefined, you solve for when its denominator equals zero.
Using the example \(\frac{25}{-7x}\), set -7x = 0. Then, solve for x:
\(-7x = 0 \Rightarrow x = 0\).
Therefore, the rational expression \(\frac{25}{-7x}\) is undefined when x = 0. Always follow these steps to find the undefined points for any rational expression: set the denominator to zero and solve the resulting equation.

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