Problem 5 - College Algebra with Corequisite Support

Question

When solving the following equation: \(\frac{2}{x-5}=\frac{4}{x+1}\) explain why we must exclude \(x=5\) and \(x=-1\) as possible solutions from the solution set.

Step-by-Step Solution

Answer

Exclude \(x = 5\) and \(x = -1\) because they make the denominators zero, leading to undefined expressions.

Step 1: Understand the equation structure

The given equation \( \frac{2}{x-5}=\frac{4}{x+1} \) is a rational equation, which means it involves fractions with variables in the denominators. It's crucial to remember that division by zero is undefined in mathematics.

Step 2: Identify when denominators are zero

To identify the values of \(x\) that would make the denominator zero, set each denominator equal to zero and solve. For the term \( \frac{2}{x-5} \), the denominator is zero when \( x - 5 = 0 \), so \( x = 5 \). For the term \( \frac{4}{x+1} \), the denominator is zero when \( x + 1 = 0 \), so \( x = -1 \).

Step 3: Exclude non-permissible values

Because division by zero is not defined, any potential solution that makes a denominator zero must be excluded from the solution set. Therefore, \(x = 5\) and \(x = -1\) have to be excluded since they make the equation undefined, ensuring that all solutions are valid.

Key Concepts

Understanding Denominators in Rational Equations

In rational equations, denominators play a critical role since they contain variables that can change the equation's behavior. A rational equation is an equation that involves at least one fraction where the numerator and/or the denominator are polynomials. The value in the denominator is essential because:

• It dictates the conditions for which the equation is defined or undefined.
• Any values that make the denominator zero are non-permissible, as division by zero isn't possible.

To determine these non-permissible values, you set the denominator to zero and solve for the variable. By doing this, you identify values that would result in zero in the denominator, leading to an undefined equation.
A useful tip is to always consider the denominators separately when inspecting a rational equation, as they set the limits for valid solutions.

The Challenge of Undefined Division

Division by zero is one of the fundamental things we need to avoid in rational equations. Any time you encounter a variable in a denominator, it's crucial to identify when the denominator would become zero. If the denominator is zero, it's like trying to break something into pieces where the number of pieces is zero—it simply doesn't work.
For example, in the equation \( \frac{2}{x-5}=\frac{4}{x+1} \), we're vigilant about values of \(x\) like 5 and -1. Plug either of these values into the denominators, and you will end up dividing by zero:

• For \( x = 5 \), the denominator \( x - 5 \) becomes zero, leading to division by zero.
• For \( x = -1 \), the denominator \( x + 1 \) becomes zero, also resulting in division by zero.

Therefore, recognizing and eliminating these values is essential in order to find a valid solution.

Solving Rational Equations

Solving rational equations involves a few key steps that ensure accurate results. Firstly, you should identify any restrictions due to the denominators, which involves solving where each denominator equals zero. Next, solve the equation as you would any other equation.
Here’s a simple outline for tackling these equations:

• **Identify non-permissible values** - Check each denominator and find solutions that make them zero; these are to be excluded from potential solutions.
• **Cross-multiply** - Assuming no non-permissible values are encountered, you can cross-multiply to simplify the equation if dealing with one fraction on each side.
• **Solve the simplified equation** - Use basic algebra to solve it, treating it as if it had no fractions.
• **Verify the solutions** - Ensure none of the solutions make the original denominators zero, excluding any that do.

This method keeps your solution valid and free from undefined zones.