Problem 23 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{15}{x-2} $$

Step-by-Step Solution

Answer

The expression is undefined at \( x = 2 \).

Step 1: Understanding the Problem

To find where the expression \( \frac{15}{x-2} \) is undefined, we must determine when the denominator equal is zero. A rational expression is undefined where its denominator is zero, as division by zero is not possible.

Step 2: Setting the Denominator to Zero

Take the denominator of the expression, which is \( x-2 \), and set it equal to zero to find the values of \( x \) that result in division by zero. This can be solved by the equation: \( x-2 = 0 \).

Step 3: Solving the Equation

Solve the equation \( x-2 = 0 \) to find the value of \( x \). Add 2 to both sides of the equation to isolate \( x \) resulting in \( x = 2 \).

Step 4: Conclusion

The expression is undefined at \( x = 2 \) because it causes division by zero in the denominator.

Key Concepts

Undefined Expressions

When dealing with rational expressions, understanding when they become undefined is crucial. A rational expression is undefined at points where the denominator equals zero. In simpler terms, dividing by zero in mathematics is not defined. This is because dividing any number by zero does not yield a finite, sensible result.
Understanding this concept helps you avoid errors when working with these kinds of expressions. When you have an expression, like \( \frac{15}{x-2} \), focus on the denominator \( x-2 \). This expression is only undefined when \( x-2 = 0 \), because setting this equal to zero identifies the point where the expression cannot be solved for real numbers. So whenever you're dealing with rational expressions, ask yourself: "When does the denominator equal zero?" Once you know that, you’ve found where the expression is undefined. This key concept is vital in algebra and calculus.

Division by Zero

Division by zero is one of those math topics that often confounds students. It's important to grasp that you simply cannot divide by zero in any mathematical context.
Here's why: when you divide a number by zero, you're asking how many times zero fits into that number. The answer is not defined—it could theoretically fit infinite times, which isn't practical in mathematical terms.
Think of division as distributing items among recipients. If you have \(15\) cookies, and the idea is to divide them among \(0\) people, there's no sensible way to distribute them since there are no recipients. Hence, this operation remains undefined.So always remember, in rational expressions like \( \frac{15}{x-2} \), if the denominator equates to zero, you will have an undefined fraction because it implies that division by zero would occur.

Solving Equations

Solving equations is like unlocking a mystery. It's about finding the values that satisfy the equation. Take, for instance, the denominator of a rational expression \( x-2 \).
To find when the original equation \( \frac{15}{x-2} \) is undefined, you solve \( x-2 = 0 \). This is done by performing operations that isolate \( x \).

• First, set \( x-2 \) equal to zero.
• Next, add \(2\) to both sides to get \( x = 2 \).

This solution illustrates that \( x = 2 \) results in the denominator being zero, thus rendering the expression undefined.Knowing how to solve these equations ensures you're capable of managing all areas where rational expressions become troublesome. This skill not only helps in mathematics but also in logical problem solving across various disciplines.