Problem 106 - Intermediate Algebra

Question

Write a rational equation that has an extraneous solution of \(3 .\)

Step-by-Step Solution

Answer

The rational equation \(\frac{x + 1}{x - 3} = 2 \) has an extraneous solution of 3.

Step 1: Understanding the Definition

A rational equation is an equation that involves at least one rational expression, meaning it can be expressed as a fraction with a polynomial in both the numerator and the denominator. An extraneous solution is a solution obtained during the process of solving the equation, but it does not satisfy the original equation.

Step 2: Rewriting the Rational Equation

To create a rational equation with an extraneous solution of 3, we need to ensure that when 3 is substituted into the original equation, it does not satisfy it. A common way to achieve this is to include a factor in the denominator that becomes zero at the value 3. Consider the equation: \[ \frac{x + 1}{x - 3} = 2 \].

Step 3: Ensuring the Extraneous Solution

We can check that 3 is an extraneous solution for our equation \( \frac{x + 1}{x - 3} = 2 \). The extraneous nature prevents 3 from being a valid solution due to the denominator \(x - 3\), which becomes zero when \(x = 3\).

Step 4: Solving the Equation

To find if there are any valid solutions, solve: \( \frac{x + 1}{x - 3} = 2 \). Multiply both sides by \(x - 3\) to eliminate the fraction:\[ x + 1 = 2(x - 3) \]Simplify:\[ x + 1 = 2x - 6 \]Re-arrange the terms:\[ 1 + 6 = 2x - x \]\[ 7 = x \].

Step 5: Verifying the Solution

Confirm if \( x = 7 \) is valid. Substitute back into the original equation:\[ \frac{7 + 1}{7 - 3} = \frac{8}{4} = 2 \], which holds true. Thus, 7 is a valid solution to the equation, confirming the extraneous nature of 3.

Key Concepts

Extraneous Solution

When tackling rational equations, you may encounter what is known as an extraneous solution. This term might sound complicated, but it's actually simple to understand. During the process of solving an equation, especially those involving fractions, you might transform or manipulate the equation. However, some of the solutions you find may not fit the original equation.

These are what we call "extraneous solutions." They occur because the manipulations can introduce solutions that don't satisfy the original equation.

Here are some key takeaways:

• An extraneous solution arises during equation solving, but it's not valid in the original equation.
• It is crucial to always substitute your solution back into the original equation to confirm its validity.
• In rational equations, these often arise due to a zero denominator issue.

Understanding extraneous solutions helps avoid assuming wrong answers are correct, ultimately leading to a more rigorous and accurate problem-solving approach.

Polynomial

Rational equations often feature polynomials in their structure. A polynomial is a mathematical expression that consists of variables raised to whole-number exponents and coefficients.

In simpler terms, they can look like this:

• Linear: \(x + 2\)
• Quadratic: \(x^2 - 4x + 4\)
• Cubic: \(x^3 + 3x^2 - x + 1\)

Polynomials are essential in rational equations because they often form the building blocks of the numerator and denominator. Each polynomial shows a different kind of relationship depending on its degree. The degree is simply the highest power of the variable.

Why do polynomials matter in rational equations?

• They dictate the equation's behavior and solutions.
• Equations solved through factoring involve breaking down these polynomials effectively.
• Understanding their structure aids in analyzing the core of a rational expression.

Familiarity with polynomials is crucial for solving equations and finding valid solutions amidst potential extraneous ones.

Denominator

In rational equations, the denominator plays a pivotal role. The denominator is the part of a fraction below the line. In rational expressions, you often find polynomials as denominators.

Understanding the denominator is essential because:

• If it becomes zero, the entire expression becomes undefined.
• It affects the solutions and can lead to extraneous ones.
• Special attention is necessary to ensure each solution keeps the denominator from becoming zero.

For instance, in our example equation \(\frac{x + 1}{x - 3} = 2\), the denominator \(x - 3\) becomes zero if \(x = 3\), making 3 an unacceptable solution.

Key notes on denominators include:

• Always consider restrictions imposed by the denominator when solving rational equations.
• Check that no solution renders the denominator zero apart from any extraneous identifiers.

This vigilance ensures all solved solutions are valid in the context of the original equation, improving accuracy in results.