Problem 5 - Algebra 1

Question

For what values of the variable is the rational expression undefined? $$\frac{x-1}{x-5}$$

Step-by-Step Solution

Answer

The rational expression is undefined when \(x = 5\).

Step 1 - Identify the Denominator

In the given rational expression \(\frac{x-1}{x-5}\), the denominator is \(x - 5\).

Step 2 - Set the Denominator Equal to Zero and Solve for x

In order for the expression to be undefined, the denominator should equal zero. So, we set \(x - 5 = 0\). Solving for x, we add 5 to both sides to get \(x = 5\).

Key Concepts

Understanding Undefined Expressions

In mathematics, undefined expressions occur when values are substituted into an expression that make it impossible to determine a valid result. For rational expressions, a common cause for an undefined expression is when the denominator equals zero.
Such occurrences are undefined because division by zero is not possible. When you divide by zero, you are essentially trying to distribute something into no parts, which is mathematically nonsensical and yields no valid number.

• Example: Consider \( \frac{1}{0} \). This expression is undefined because there is no real number that you can multiply by zero to get 1.

Thus, when working with rational expressions, it's critical to find values that cause the denominator to be zero, as these values make the expression undefined.

The Role of the Denominator

The denominator is a key component of any fraction or rational expression which is found below the fraction line. In \( \frac{x-1}{x-5} \), the expression \( x-5 \) functions as the denominator.
For a rational expression to be valid, the denominator must not be zero. This is because zero in the denominator leads to an undefined expression. Thus, understanding the denominator is the first step in ensuring a well-defined expression.

• Identify the denominator in the expression.
• Determine which values of the variable make the denominator zero.

Knowing the values that zero out the denominator helps identify potential points of undefined expressions, ensuring the rational expression is evaluated correctly without errors.

Steps to Solving Equations

Solving equations is a systematic process that allows you to find the set values that satisfy an equation. To ensure rational expressions are defined, it's often necessary to solve equations involving the denominator.
Here’s a simple guide to solving equations like \( x-5 = 0 \):

• Step 1: Set the equation based on the denominator. For our expression, the equation is \( x-5 = 0 \).
• Step 2: Isolate the variable. We add 5 to both sides to solve for \( x \), leading to \( x = 5 \).

At \( x = 5 \), the denominator turns zero, making the rational expression undefined. Thus, \( x = 5 \) is precisely what makes the expression \( \frac{x-1}{x-5} \) undefined, which is a crucial step in handling such mathematical challenges.