Problem 38 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{44}{57-x} $$

Step-by-Step Solution

Answer

The expression is undefined for \( x = 57 \).

Step 1: Identify the Expression to be Analyzed

The expression we are given is \( \frac{44}{57-x} \). Our task is to determine when this expression becomes undefined.

Step 2: Determine When the Denominator is Zero

A rational expression is undefined whenever its denominator is zero. We set the denominator equal to zero and solve for \( x \):\[57 - x = 0\]

Step 3: Solve the Equation for x

To isolate \( x \), add \( x \) to both sides of the equation, then subtract 57 from both sides:\[x = 57\]

Step 4: Conclusion

The expression \( \frac{44}{57-x} \) is undefined when \( x = 57 \). This is because substituting 57 for \( x \) makes the denominator zero.

Key Concepts

Understanding Undefined Expressions

Rational expressions, like any mathematical expression, have certain conditions that must be met for them to be valid. Specifically, a rational expression becomes undefined when its denominator is equal to zero. This is because division by zero is not allowed in mathematics—it produces an undefined value. Therefore, identifying when a denominator equals zero is a critical step in understanding when a rational expression is undefined.
Imagine you're solving an equation, and you encounter a denominator of zero. This is a red flag, signaling that the expression at that value is not meaningful within the set of real numbers.
To find the undefined expressions in a problem:

• Focus on the denominator.
• Set the denominator equal to zero.
• Solve for the variable to identify the undefined value.

By pinpointing these undefined points, you ensure accurate and reliable solutions to mathematical problems.

Why Does the Denominator Matter?

In a rational expression, the denominator holds crucial significance. It determines where the expression is valid or invalid because it should never be zero. Let's take a peek at the irrational nature of division by zero:
When the denominator becomes zero, something curious happens. The expression essentially "blows up" into something undefined, breaking the rules of arithmetic that aid us in solving rational expressions logically. For instance, the basic form of \[\frac{a}{b}\]where \( b eq 0 \) defines a valid expression. Should \( b \) become zero, the expression fails. Therefore, always checking for zero in the denominator is a must when solving any rational expression.
Remember these steps to address potential undefined scenarios:

• Set the denominator equal to zero.
• Isolate the variable.
• Determine what values make the denominator zero, identifying where it's undefined.

Solving Equations in Rational Expressions

When dealing with rational expressions, solving equations effectively requires careful attention to the denominator. Take the equation \[57 - x = 0\]. By isolating \( x \), we are discovering the value that will make the denominator zero, and consequently where the expression becomes undefined. This process uncovers critical insights about the behavior of the expression that we are analyzing.
Here's how you solve such equations step-by-step:

• First, identify the denominator and set it equal to zero, which helps in finding potentially undefined values.
• Next, manipulate the equation to isolate the variable—perform operations like addition or subtraction as needed.
• Finally, identify the solution, which in our example turns out to be \( x = 57 \).

Evidently, solving equations in rational expressions is like being a detective, tracing the point(s) of undefined values to keep your calculations on track and flawless. Simplifying complex expressions begins with this fundamental understanding, making future mathematical challenges more approachable.