Problem 13 - Algebra A Combined Function

Question

Find any numbers for which each rational expression is undefined. $$ \frac{x-4}{2 x-5} $$

Step-by-Step Solution

Answer

The expression is undefined at \( x = \frac{5}{2} \).

Step 1: Identify when the denominator is zero

A rational expression is undefined when its denominator is zero. For the expression \( \frac{x-4}{2x-5} \), we need to find the value of \( x \) such that the denominator \( 2x-5 = 0 \).

Step 2: Solve the equation

Now solve the equation from Step 1. Set \( 2x-5 = 0 \) and solve for \( x \). First, add 5 to both sides, giving you \( 2x = 5 \). Then, divide both sides of the equation by 2 to isolate \( x \), resulting in \( x = \frac{5}{2} \).

Step 3: Conclusion

The solution to the equation \( 2x-5 = 0 \) tells us the expression is undefined at \( x = \frac{5}{2} \). Therefore, \( x = \frac{5}{2} \) is the value that makes the rational expression undefined.

Key Concepts

Understanding Undefined Expressions

When dealing with rational expressions, understanding when they are undefined is crucial. A rational expression becomes undefined when its denominator is equal to zero. This is because division by zero is not possible in mathematics—it doesn't give a meaningful result. Hence, whenever you encounter a rational expression, your first step should be to identify values of the variable that make the denominator zero.

In our specific example, we looked at the rational expression \( \frac{x-4}{2x-5} \). To determine when this expression is undefined, we turn our focus to the denominator. Setting the denominator equal to zero allows us to find these critical values. Here, \( 2x-5 = 0 \) provides the moment that the rational expression loses its definition. Thus, solving this equation is essential to uncover these values.

Techniques for Solving Equations

Solving equations forms a foundational skill in math, particularly when working with rational expressions. In this context, once the equation has been derived from setting the denominator equal to zero—such as \( 2x-5 = 0 \)—we treat it as a simple linear equation.

Below are clear steps to solve such equations:

• First, isolate the term with the variable. Add or subtract values from both sides of the equation to do this. In our example, we added 5 to both sides to get \( 2x = 5 \).
• Next, perform operations to solve for the variable completely. This might involve dividing or multiplying. For our equation, we divided both sides by 2, arriving at \( x = \frac{5}{2} \).

These steps are integral for correctly identifying values where the rational expression is undefined.

The Critical Role of a Zero Denominator

The denominator of a fraction being zero is a red flag in mathematics, leading to undefined expressions. This situation is crucial when analyzing rational expressions as it determines the expression's viability or legality.

Why does the denominator matter so much? Essentially, division by zero doesn't produce a meaningful result, and mathematically, it is considered an operation that cannot be performed. Any rational expression relying on a denominator must have it non-zero to remain valid.

For \( \frac{x-4}{2x-5} \), we discovered the expression becomes undefined when \( x = \frac{5}{2} \). Such solutions signal points where the expression ceases to behave normally and should be treated with caution. Understanding and identifying these points ensure a proper handling of rational expressions, preserving mathematical integrity.