Problem 30 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{-6 x}{3 x-1} $$

Step-by-Step Solution

Answer

The expression is undefined for \(x = \frac{1}{3}\).

Step 1: Analyzing the Denominator

A rational expression is undefined when its denominator is zero. So, the first step is to set the denominator equal to zero. The denominator of the given rational expression is \(3x - 1\).

Step 2: Setting the Equation

Set the equation for the denominator to zero: \(3x - 1 = 0\).

Step 3: Solving for x

Solve the equation \(3x - 1 = 0\). First, add 1 to both sides to get \(3x = 1\).

Step 4: Isolating the Variable

Divide both sides by 3 to solve for \(x\): \(x = \frac{1}{3}\).

Step 5: Conclusion

The rational expression is undefined for \(x = \frac{1}{3}\), because this value makes the denominator zero and the expression invalid.

Key Concepts

Undefined Expressions

In mathematics, undefined expressions occur when calculations lead to an illogical or indeterminate form. A common scenario where this arises is when we encounter division by zero. Rational expressions, which are fractions involving polynomials, must have a non-zero denominator to be defined. Consequently, if the denominator of a rational expression equals zero for certain values of the variable, those particular values will make the expression undefined. Understanding this concept helps in identifying potential domain restrictions in algebraic functions.

Recognizing when an expression is undefined is crucial. This is not just about spotting division by zero but understanding that it leads to inconsistency and mathematical impossibility. Whenever you are given a rational expression to evaluate or simplify, be sure to check for any undefined values by setting the denominator equal to zero and solving for the variable.

Denominator Zero

The denominator of a rational expression plays a vital role in determining whether the expression is defined or undefined. Simply put, a rational expression becomes undefined when its denominator equals zero. For instance, in the expression \( \frac{-6x}{3x-1} \), the denominator is \( 3x - 1 \).

To find when the expression is undefined, we equate the denominator to zero:

• Set \(3x - 1 = 0\).
• Solve the resulting equation to find the critical value of \(x\).

This process ensures you are aware of the value of \(x\) that would make the expression invalid, as division by zero cannot be performed. Keep in mind that even a single number that zeroes the denominator will render the entire expression undefined at that point.

Solving Equations

Solving equations involving rational expressions requires careful manipulation to avoid undefined conditions. From our previous steps, once the denominator is zero (\(3x - 1 = 0\)), we solve for the variable to identify the point of undefinedness.

Here's how we solve the equation:

• Add 1 to both sides to get \(3x = 1\).
• Divide both sides by 3, which gives us \(x = \frac{1}{3}\).

These steps are simple, yet powerful in determining critical values. They shed light on how minor changes in the variable can switch an expression from defined to undefined. Always remember, such solutions help us identify restrictions or holes in the domain of the function.

Real Numbers

Real numbers encompass all the values needed in practical and theoretical scenarios, including both rational and irrational numbers. When tackling rational expressions, we typically seek values from the set of real numbers that will keep the expression defined.

Rational expressions require careful attention to avoid undefined values, generally arising from zero denominators.

• We only consider values of \(x\) that maintain a non-zero denominator.
• This allows us to confidently compute or simplify the expression.

In our example, \(x = \frac{1}{3}\) is omitted, as it zeroes the denominator, making the expression undefined. Therefore, the 'defined' values for our rational expression lie in the set of real numbers, excluding \(x = \frac{1}{3}\). Understanding this exclusion ensures clarity in both computation and the overall interpretation of the expression's domain.