Problem 5 - Intermediate Algebra

Question

Consider the rational equation: \(\frac{x}{x-3}=\frac{1}{x}+\frac{2}{x-3}\). a. What values of \(x\) make a denominator \(0 ?\) b. What values of \(x\) make a rational expression undefined? c. What numbers can't be solutions of the equation?

Step-by-Step Solution

Answer

a) \( x = 3 \) and \( x = 0 \); b) \( x = 3 \) and \( x = 0 \); c) \( x = 3 \) and \( x = 0 \) can't be solutions.

Step 1: Identify Denominators

First, identify the denominators in the equation: \( x - 3 \), \( x \), and again \( x - 3 \). These are the expressions in the denominators that can potentially equal zero.

Step 2: Set Denominators to Zero

To find when the denominators are zero, solve each denominator for zero: \[ x - 3 = 0 \text{ and } x = 0. \]

Step 3: Solve for Zero Denominator

Solve each equation: \( x - 3 = 0 \) gives \( x = 3 \) and \( x = 0 \) is already given. These values make the denominators zero.

Step 4: Determine Undefined Values

The rational expressions become undefined at the values found in Step 3. Therefore, \( x = 3 \) and \( x = 0 \) make the rational expression undefined.

Step 5: List Non-solution Values

Since division by zero is undefined, any solution to the equation cannot be \( x = 3 \) or \( x = 0 \). These are called extraneous solutions.

Key Concepts

Understanding Undefined Expressions

In any mathematical expression or equation, an undefined expression occurs when you try to perform a mathematical operation that is not valid. In the case of rational equations, this happens when you divide by zero. This is because division by zero is undefined in mathematics.

Consider the rational equation from the exercise:

• You have the denominators \( x - 3 \) and \( x \).
• If either \( x - 3 \) or \( x \) equals zero, the expression becomes undefined.

So, it's crucial to determine these points because any value of \( x \) that results in a denominator of zero will cause the expression to be undefined. Always ensure that denominators are non-zero to keep expressions valid.

The Role of a Zero Denominator

A zero denominator is the main cause of undefined rational expressions. It's like trying to find a total if you're dividing by nothing, which doesn't make sense.

• In the given equation, you identify the denominators: \( x - 3 \) and \( x \).
• To avoid undefined expressions, set each denominator equal to zero and solve.
• For \( x - 3 = 0 \), \( x \) must not equal 3.
• For \( x = 0 \), obviously, \( x \) must not be zero.

When these denominators are set to zero, the expressions themselves become invalid. This tells us which values of \( x \) we cannot use because they would make the equation mathematically incorrect.

Explaining Extraneous Solutions

Extraneous solutions are potential solutions that arise during the solving process of an equation, but do not satisfy the original equation.

• After solving rational equations, you must check whether these solutions make the original equation undefined.
• Often in rational equations, solving can lead to possible solutions that were not solutions to start with.
• In the given exercise, solutions included \( x = 3 \) and \( x = 0 \).
• However, these are not valid because they make the original equation undefined by causing division by zero.

Always verify your solutions by substituting back into the original equation to ensure they do not result in undefined expressions.Therefore, knowing how to identify and exclude extraneous solutions is key to correctly solving rational equations.