Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ \frac{2}{(x-4)(x-2)}=\frac{1}{x-4}+\frac{2}{x-2} $$

Step-by-Step Solution

Answer

The equation has no valid solutions due to the denominator resulting in zero when a solution is substituted back into the original equation.

Step 1: Simplify the Right Side

Merge two fractions on the right side into one fraction by having a common denominator. Since the denominators \(x-4\) and \(x-2\) are already the denominator of the fraction on the left side, combine the right side: \(\frac{1(x-2)}{(x-4)(x-2)} + \frac{2(x-4)}{(x-4)(x-2)}\). After simplifying you get \(\frac{x-2+2x-8}{(x-4)(x-2)}\) which results in \(\frac{3x-10}{(x-4)(x-2)}\)

Step 2: Set Left Fraction Equal to Right Fraction and Solve.

The left side fraction is already structured to be equal to the right-hand side fraction. Therefore, set \(\frac{2}{(x-4)(x-2)} = \frac{3x-10}{(x-4)(x-2)}\). After cross multiplying, this gives \(2 = 3x - 10\). Solving, \(x = 4\).

Step 3: Check the Solution

Replace x in the original equation but remember if any replacement results in a denominator of zero you must exclude it as a solution. When \(x = 4\), the fraction \(\frac{1}{x-4}\) on the right side makes denominator zero. Hence, \(x = 4\) is not a valid solution and can be discarded.

Key Concepts

Cross Multiplication

Cross multiplication is a technique often used to solve rational equations that are set up as proportions. While dealing with fractions, it helps eliminate denominators and simplify the problem. Let's break down how this works:

• Imagine a fraction equation like \( \frac{a}{b} = \frac{c}{d} \).
• Cross multiplication involves multiplying the terms diagonally across the equal sign, giving us \( a \cdot d = b \cdot c \).

You use cross multiplication when both sides of the equation are fractions composed of numerators and denominators. This approach simplifies the equation to a more manageable, linear form that you can solve like a simple algebraic equation.
In the given exercise, once the fractions were set equal, cross multiplication simplified the equation from fractions to a single linear equation: \( 2 = 3x - 10 \). This step is crucial as it reduces the equation's complexity, making it easier to solve.

Fraction Simplification

Simplifying fractions is essential to solving rational equations efficiently. Let's explore what simplifying means and how it was applied in the exercise.

• Simplification involves combining terms and reducing them to their simplest form.
• This process eliminates redundant expressions, allowing for a clearer view of the equation's structure.

In the exercise, the fractions on the right side of the equation were combined using a common denominator. This step was key to transforming two distinct terms into one, making the equation more straightforward.
The fraction with \( \frac{1}{x-4} + \frac{2}{x-2} \) was given a common denominator \((x-4)(x-2)\), allowing the fractions to merge: \( \frac{x-2 + 2x - 8}{(x-4)(x-2)} = \frac{3x - 10}{(x-4)(x-2)} \).This simplification made it possible to directly compare both sides of the equation and apply cross multiplication.

Common Denominator

When working with rational equations involving fractions, finding a common denominator is crucial. Here's why it matters and how it was applied in the given exercise:

• A common denominator allows for the addition of fractions by ensuring that all terms share the same base beneath the numerator.
• In a rational equation, aligning denominators is the first step toward combination and simplification.

In the provided problem, the denominators \(x-4\) and \(x-2\) on the right side needed to be the same. This was achieved by multiplying terms appropriately so that each fraction had the denominator \((x-4)(x-2)\).
Matching the denominators this way ensured that the fractions could be combined into a single expression. The next logical step was direct comparison to the left fraction, as they now shared the same denominator space. This strategy not only simplifies the arithmetic but streamlines further operations, like cross multiplication.