Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Explain why a solution of an equation involving fractional expressions may be extraneous.

Step-by-Step Solution

Answer

An equation involving fractional expressions may have an extraneous (not valid) solution because during the process of clearing the fractions via multiplication, additional solutions may be introduced that are not valid for the original equation. Particularly, if the fraction's denominator contains a variable, a solution that makes the denominator zero would be extraneous as it would make the expression undefined. Therefore, always verify the solution by substituting it back into the original equation.

Step 1: Understanding what an extraneous solution is

An extraneous solution is a solution that emerges from the process of solving the equation but is not a valid solution to the original problem. These often arise when both sides of an equation are multiplied by a variable expression.

Step 2: Knowing how extraneous solutions appear in equations including fractional expressions

While solving equations involving fractional expressions, an extraneous solution may occur during the process of clearing the fractions by multiplication. If the fraction within the equation has a variable in the denominator, multiplication to remove the fraction can introduce additional solutions that are not valid for the original equation since original variables should not make the denominator become zero.

Step 3: Confirming the validity of a solution

To confirm whether the solutions obtained are valid or not, substitute them back into the original equation. If it makes a fraction's denominator zero, then it is an extraneous solution. This is because division by zero is undefined in mathematics.

Key Concepts

Fractional Expressions

When dealing with equations that include fractional expressions, it is essential to understand what fractional expressions actually are. A fractional expression consists of a numerator and a denominator, much like a regular fraction. However, they contain variables as part of their equations.

For instance, an equation might look like \( \frac{x+3}{x-2} \). Here, \( x-2 \) is the denominator, and because \( x \) is a variable, you must be very careful throughout the processes of solving, especially concerning the values that \( x \) can take. These variables, when solved incorrectly, can lead to extraneous solutions that don’t actually satisfy the original equation.

An important step when working with fractional expressions is ensuring you understand where variables exist in both numerators and denominators. Keep in mind that variables in the denominator are particularly key to watch during calculations as they directly relate to the concept of division by zero.

Division by Zero

The concept of division by zero is one of the critical aspects to consider when solving equations involving fractional expressions. Let's start with why division by zero is a significant concern.

In mathematics, division by zero is undefined. Why is that? Imagine you have a number and you want to divide it by zero, essentially asking, "How many times does zero fit into that number?" It doesn’t make any logical sense because zero times anything is still zero. Consequently, you can't reasonably state how many zeros fit into another number, making this operation undefined.

When solving equations, it’s crucial to ensure that no variable value causes the denominator of a fractional expression to become zero. If it does, the solution derived from the equation at that point is not valid, as it represents a division by zero situation.

• For instance, in the fractional expression \( \frac{x+3}{x-2} \), the value \( x=2 \) would result in division by zero because the denominator becomes zero when \( x=2 \).
• This leads to what's known as an extraneous solution, where a derived solution doesn’t satisfy the equation due to issues like division by zero.

Always remember to cross-check your solutions by substituting them back into the original equation to see if they result in any denominators equaling zero.

Valid Solutions

Determining whether a solution is valid is the last essential step after solving an equation involving fractional expressions. Not all solutions you calculate will necessarily be valid, especially when factoring in issues like division by zero.

A valid solution is one that satisfies the original equation completely. Meaning, when you plug it back into the equation, the equation holds true and, importantly, no fraction has a zero in the denominator.

To find out whether a solution is valid, perform a substitution test:

• After solving the equation, substitute your solutions back into the original equation.
• Check if any of those solutions make a denominator zero. If it does, then that solution is extraneous.
• Discard any extraneous solutions, keeping only those that fulfill the equation without making a denominator zero.

By ensuring these steps, you confirm the validity of your solutions, and thus, you correctly address the original problem. This practice is crucial for achieving accurate results in equations involving fractional expressions.