Problem 64 - Intermediate Algebra

Question

Solve equation. If a solution is extraneous, so indicate. \(\frac{3}{4 x-8}=\frac{1}{36}-\frac{2}{6-3 x}\)

Step-by-Step Solution

Answer

The solution is \(x = 12\); no extraneous solutions.

Step 1: Identify the Domain

Consider the denominators to identify the values of \(x\) that would make them zero. The denominators are \(4x - 8\) and \(6 - 3x\). Solve \(4x - 8 = 0\) to find \(x = 2\) and \(6 - 3x = 0\) to find \(x = 2\). Therefore, \(x = 2\) is not in the domain of the solution.

Step 2: Eliminate the Fractions by Finding a Common Denominator

Identify the least common denominator (LCD) of all the fractions, which is \((4x-8)(36)(6-3x)\). Multiply each term in the equation by this LCD to eliminate the fractions.

Step 3: Simplify the Equation

Multiply each term by the LCD to get the equation: \[36(6 - 3x) \, (3) = (4x - 8)(6 - 3x)(1) - (2)(36)(4x - 8)\]. Simplify each term to get rid of the denominators.

Step 4: Solve for x

Expand all terms in the equation:\[108(6 - 3x) = (4x - 8)(6 - 3x) - 72(4x - 8)\].Solve for \(x\) by simplifying and combining like terms. This involves expanding and combining all terms to one side.

Step 5: Identify Solutions and Check for Extraneous Solutions

Once \(x\) is found, check whether this solution causes any of the initial denominators to become zero.For this problem:Substitute back the solutions into the denominators to check if they zero any denominator.

Key Concepts

Extraneous Solutions

When solving rational equations, it's essential to check for extraneous solutions. These are solutions that appear during the algebraic process but do not satisfy the original equation. They often arise when we multiply both sides of the equation by an expression containing the variable, which can introduce solutions that don't actually work in the real-world context of the problem.
To manage extraneous solutions effectively:

• Always substitute your solutions back into the original equation to verify whether they make any denominator zero.
• If a solution does make a denominator zero, it's considered extraneous and must be excluded from the final set of solutions.

Checking for extraneous solutions ensures that your final answer is valid under the original constraints of the problem.

Common Denominator

Finding a common denominator in rational equations allows you to eliminate fractions, making it simpler to solve the equation. A common denominator is simply a multiple that each individual denominator divides into evenly. This process is crucial as it transforms the equation into a form without fractions, making further simplification and solving possible.
Here's how to find and use a common denominator:

• Identify the least common denominator (LCD) among all fractions. In the example equation \( \frac{3}{4x-8} = \frac{1}{36} - \frac{2}{6-3x} \), the LCD is \((4x-8)(36)(6-3x)\).
• Multiply each term by the LCD to form a new equation without fractions.
• Simplify the result to further combine terms and solve the equation.

Using an LCD streamlines the process of handling complex rational equations, focusing entirely on the variable terms without the complication of fractions.

Domain Restrictions

Domain restrictions are critical in identifying the values of the variable that would make the original equation or expression undefined. Understanding domain restrictions prevents division by zero, a fundamental rule in mathematics where division by zero is undefined.
Here’s how to address domain restrictions:

• Look at each denominator in the equation. Consider what value of the variable would make these expressions equal to zero.
• In the example equation, solve \(4x - 8 = 0\) and \(6 - 3x = 0\) to find that \(x = 2\) makes both denominators zero.
• Exclude these values from your solution set to prevent undefined expressions.

Applying domain restrictions ensures your solutions are valid and respects the limits of mathematical operations, allowing only meaningful answers to be considered.