Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, solve each rational equation for \(x .\) State all \(x\) -values that are excluded from the solution set. $$ \frac{1}{x}=\frac{1}{5}+\frac{3}{2 x} $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{5}{2}\), with excluded value 0.

Step 1: Identify the Least Common Denominator (LCD)

To solve the rational equation \(\frac{1}{x} = \frac{1}{5} + \frac{3}{2x}\), we first identify the least common denominator, which is important to eliminate the fractions. The denominators here are \(x\), \(5\), and \(2x\). The LCD of these denominators is \(10x\).

Step 2: Clear the Fractions

Multiply every term in the equation by the LCD, \(10x\), to clear the fractions.\[10x \cdot \frac{1}{x} = 10x \cdot \frac{1}{5} + 10x \cdot \frac{3}{2x}\]This simplifies to:\[10 = 2x + 15\]

Step 3: Simplify the Equation

After clearing the fractions, simplify the equation:\[10 = 2x + 15\]Subtract 15 from both sides to isolate the term with \(x\):\[-5 = 2x\].

Step 4: Solve for \(x\)

Divide both sides of the equation by \(2\) to solve for \(x\):\[x = -\frac{5}{2}\].

Step 5: State Excluded Values

The original denominators \(x\) and \(2x\) cannot be zero. Therefore, \(x\) cannot be \(0\). Excluded values for \(x\) are 0.

Key Concepts

Understanding the Least Common Denominator

When solving rational equations like \( \frac{1}{x} = \frac{1}{5} + \frac{3}{2x} \), one of the first steps is finding the Least Common Denominator (LCD). This step is vital because it helps us eliminate fractional parts of the equation. Why is the LCD Important?

• It gives us a common ground for comparison for different denominators.
• Helps simplify calculations by getting rid of fractions.

In the equation we're examining, the denominators are \( x \), \( 5 \), and \( 2x \). To find the LCD, we need to find a number that all these can divide into without leaving a remainder. Here, the LCD is \( 10x \), as it includes the factors of all denominators.Finding LCD

• List the factors of each denominator.
• Identify the highest powers of all factors.
• Multiply these factors together to get the LCD.

With the LCD found, we can proceed to clear the fractions in our equation.

Identifying Excluded Values

Before solving any rational equation, it's crucial to identify excluded values. These are values of \( x \) that would make any denominator in the equation equal to zero, which would render the expression undefined.Why Do We Exclude Values?

• A fraction with a zero denominator is not possible mathematically, as division by zero is undefined.
• Ensures that our solution is valid throughout the problem.

For the equation \( \frac{1}{x} = \frac{1}{5} + \frac{3}{2x} \), let's consider each denominator:

• \( x \) cannot be 0, since \( \frac{1}{x} \) would be undefined.
• Similarly, \( 2x \) cannot be 0. This also implies \( x \) cannot be 0.

Thus, the excluded value here is \( x = 0 \). Remembering to state this is a critical part of solving rational equations.

Clearing Fractions to Simplify the Equation

Once we have established the least common denominator and identified the excluded values, we move on to clearing the fractions. This is done by multiplying each term of the equation by the LCD we found earlier, converting our equation into a much simpler form without fractions.Steps to Clear Fractions

• Multiply each term in the equation by the LCD.
• Simplify by canceling out the denominators.

For our equation \( \frac{1}{x} = \frac{1}{5} + \frac{3}{2x} \), we use the LCD \( 10x \):\[10x \cdot \frac{1}{x} = 10x \cdot \frac{1}{5} + 10x \cdot \frac{3}{2x}\]After multiplying, the equation simplifies to:\[10 = 2x + 15\]This process removes the fractions, allowing us to work with a simpler linear equation. Simplification tasks like these are crucial, transforming complex algebraic equations into forms that are easier to solve.