Problem 98 - Algebra for College Students

Question

Discussion. For each equation, find the values for \(x\) that cannot be solutions to the equation. Do not solve the equations. a) \(\frac{1}{x}+\frac{1}{x-1}=\frac{1}{2}\) b) \(\frac{x}{x-1}=\frac{1}{2}\) c) \(\frac{1}{x^{2}+1}=\frac{1}{x+1}\)

Step-by-Step Solution

Answer

For a), x = 0, 1. For b), x = 1. For c), no values.

Step 1: Identify the Denominators

For each equation, identify the denominators that could possibly make the equation undefined if they equal zero.

Step 2: Set Each Denominator to Zero

To find the values that make the equation undefined, set each denominator equal to zero and solve for x.

Step 3: Determine the Forbidden Values for Each Equation

For each possible problematic denominator, solve the equations you set up in Step 2 to find the x values that make the equation undefined.

Key Concepts

Denominators

When solving equations involving fractions, the denominators play a crucial role. The denominator is the number or expression below the fraction line. For instance, in the fraction \(\frac{1}{x}\), the denominator is \(x\).
It's important to check the denominators because they determine whether a fraction is undefined.
Any denominator equal to zero makes the fraction undefined.
Zero in the denominator results in an invalid calculation because division by zero is undefined.
Therefore, the first step in many math problems involving fractions is to identify all the denominators.

Equation Solving

Equation solving generally involves finding the value of the variable that makes the equation true.
But in problems like these, we focus on the values that should be avoided.
Here, starting by identifying and understanding the denominators in the equation helps highlight any potential pitfalls. For example, in the equation \(\frac{1}{x} + \frac{1}{x-1} = \frac{1}{2}\), identify \(x\) and \(x-1\) as denominators.
Setting each of these to zero and solving tells us which values of \(x\) will render the equation undefined or invalid.
Hence, solving equations isn't just about finding solutions but also about recognizing and avoiding errors.

Forbidden Values

Forbidden values are values that make the equation invalid.
These values typically lead to division by zero or other undefined operations within the given equation.
To find forbidden values, we set each denominator to zero and solve for the variable.
For example, in \(\frac{x}{x-1} = \frac{1}{2}\), setting the denominator \(x-1\) to zero, we solve for \(x\) and get \(x = 1\).
This value is forbidden because substituting \(x = 1\) back into the equation makes the denominator zero, which is undefined.
Always check for such forbidden values before moving forward with solving the equation.
Ignoring forbidden values can lead to incorrect solutions.

Undefined Expressions

Undefined expressions occur when you have a division by zero or similar invalid operations.
In mathematical terms, operations like \(\frac{1}{0}\) are undefined.
When working with equations that have fractions, always consider where undefined expressions might appear.
Check the denominators and set them equal to zero to identify potential undefined expressions.
Solving for those values and recognizing them as forbidden ensures you avoid undefined operations.
For instance, in the equation \(\frac{1}{x^2 + 1} = \frac{1}{x+1}\), both \(x^2 + 1\) and \(x + 1\) must be checked. If solving shows that any value leads to zero in the denominator, then that value must be avoided since it would create an undefined expression.
Understanding this concept helps in making fewer mistakes and ensures the mathematical validity of your solutions.