Problem 24 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{5 x}{x+5} $$

Step-by-Step Solution

Answer

The expression is undefined for \( x = -5 \).

Step 1: Understand the Question

We are asked:
Find all real numbers for which the rational expression is undefined. See
Example 2.
$$
\frac{5 x}{x+5}
$$

Step 2: Recall the Definition

We recall the relevant mathematical definition or concept.

Step 3: State the Answer

The expression is undefined for \( x = -5 \).

Key Concepts

Undefined Expressions

In mathematics, especially when dealing with rational expressions, understanding when an expression is undefined is crucial. A rational expression is a fraction where both the numerator and the denominator are polynomials. However, the key concern here is the denominator. An expression becomes undefined when it causes division by zero because division by zero is not possible within the real number system. To determine when an expression is undefined, you solve for the values that make the denominator zero.

For instance, consider the expression \( \frac{5x}{x+5} \). The expression is undefined for any real number that makes the denominator, \( x+5 \), equal to zero. Finding when this occurs involves setting the denominator to zero and solving the resulting equation. This clear identification helps you to understand the limits of the rational expression.

Denominator Identification

Identifying the denominator in a rational expression is the first step in determining the values for which the expression might be undefined. The denominator is the part of the fraction found below the division line. In our example, the rational expression is \( \frac{5x}{x+5} \). Here, \( x+5 \) serves as the denominator.

Identifying the denominator helps you set the foundation for evaluating undefined conditions. By setting \( x+5 = 0 \), you see that the expression is undefined when \( x = -5 \). This process of setting the denominator equal to zero and solving for \( x \) allows you to pinpoint the precise values that will make the rational expression undefined due to division by zero.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes both rational numbers (such as fractions and integers) and irrational numbers (such as the square roots of non-perfect squares). Understanding real numbers is essential when working with rational expressions since it concerns the values for which certain operations, like division, are valid.

In the context of the expression \( \frac{5x}{x+5} \), we are interested in identifying which real numbers would make it undefined. For this expression, it becomes clear when solving for the denominator \( x+5 \) that subtraction of 5 from both sides gives \( x = -5 \) as the point where the expression is undefined. Thus, for the real numbers analyzed, \( x = -5 \) is excluded from the domain of this rational expression, reinforcing the understanding that division by zero is not possible.