Problem 7 - Elementary and 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. The denominators equal zero when \(x = 0\) and \(x = 3\). b. The expression is undefined for \(x = 0\) and \(x = 3\). c. Thus, \(x = 0\) and \(x = 3\) can't be solutions.

Step 1: Identify Denominators

Examine the rational equation \( \frac{x}{x-3}=\frac{1}{x} + \frac{2}{x-3} \). The denominators are \(x\) and \(x-3\).

Step 2: Find Where Denominators Equal Zero

Set each denominator to zero to find values that make them undefined. For \(x = 0\), the denominator \(x\) becomes zero. For \(x - 3 = 0\), solve to get \(x = 3\).

Step 3: Determine Undefined Values for the Expression

The expression is undefined for any \(x\) that makes any denominator zero, which are \(x = 0\) and \(x = 3\).

Step 4: Check for Possible Solutions Constraints

The numbers \(x = 0\) and \(x = 3\) cannot be solutions of the equation because they make a denominator zero and the expressions undefined.

Key Concepts

Understanding Undefined Expressions

Undefined expressions are a crucial part of solving rational equations because they help identify the values of variables that make a mathematical expression invalid. In mathematics, a rational expression is deemed undefined when its denominator equals zero. Since division by zero is undefined in mathematics, any expression resulting in such a scenario cannot be evaluated. As a direct result, the values which cause this must be identified to exclude them from possible solutions of an equation.

For instance, in the equation \( \frac{x}{x-3}=\frac{1}{x}+\frac{2}{x-3} \), if \( x=0 \) or \( x=3 \), the denominators become zero and thus make the expression undefined. Recognizing undefined expressions is essential for understanding how rational equations behave and for finding valid solutions.

Examining Denominators Closely

Denominators play a vital role in rational expressions and equations. They are the part of a fraction that is below the line, which indicates division. In rational equations, denominators determine where an expression can become undefined. Therefore, it's crucial to consider the denominators when solving these equations.

In our example equation, the denominators are \( x \) and \( x-3 \). To find points where they become zero, you set each denominator equal to zero and solve for x:

• Setting \( x = 0 \) shows that the denominator \( x \) becomes zero.
• Setting \( x-3 = 0 \) gives \( x = 3 \), where the denominator \( x - 3 \) becomes zero.

By closely examining the denominators, you can easily determine the conditions under which the equation is undefined. This step is vital before proceeding to solve rational equations, as it informs us which values of \( x \) are invalid.

Approach to Solving Rational Equations

To solve rational equations, it is essential to follow a systematic approach, keeping in mind the undefined values determined from examining the denominators. Rational equations involve fractions that can be balanced by finding a common denominator or by eliminating denominators through multiplication.

Here’s how you can approach solving rational equations:

• First, identify all denominators and find values of \( x \) that make them zero. These values are not valid solutions and should be excluded.
• Subsequently, either multiply through by the least common denominator to eliminate the denominators or rearrange the equation to simplify it.
• Finally, solve the resulting equation. Verify that any solutions are within the domain, meaning they do not make any original denominators zero.

By following these steps, you ensure that only valid solutions are considered for a rational equation, thus accurately solving the problem.