Problem 34 - Precalculus Mathematics for Calculus

Question

Solve the equation both algebraically and graphically. $$\frac{4}{x+2}-\frac{6}{2 x}=\frac{5}{2 x+4}$$

Step-by-Step Solution

Answer

There is no valid solution for the equation, as solved algebraically and confirmed graphically.

Step 1: Identify the Equation

We have the equation \( \frac{4}{x+2} - \frac{6}{2x} = \frac{5}{2x+4} \). We will first aim to solve this algebraically.

Step 2: Simplify the Denominators

Notice that the denominator \(2x+4\) can be factored as \(2(x+2)\). This will help us in finding a common denominator for the left-hand side fractions.

Step 3: Find a Common Denominator

To eliminate fractions, find a common denominator. The least common denominator (LCD) for the terms is \(2x(x+2)\). Rewrite each fraction with this common denominator.

Step 4: Rewrite Each Term

The first fraction becomes \( \frac{4 \cdot 2x}{2x(x+2)} = \frac{8x}{2x(x+2)} \). The second fraction becomes \( \frac{6(x+2)}{2x(x+2)} \). The right-hand side becomes \( \frac{5x}{2x(x+2)} \).

Step 5: Combine the Left-hand Side

Combine the fractions on the left-hand side: \( \frac{8x}{2x(x+2)} - \frac{6(x+2)}{2x(x+2)} = \frac{8x - 6(x+2)}{2x(x+2)} \).

Step 6: Simplify the Numerator

Simplify the numerator: \( 8x - 6(x+2) = 8x - 6x - 12 = 2x - 12 \). Thus, the left-hand side becomes \( \frac{2x-12}{2x(x+2)} \).

Step 7: Set Equations Equal

Set the simplified left-hand side equal to the right-hand side: \( \frac{2x-12}{2x(x+2)} = \frac{5x}{2x(x+2)} \).

Step 8: Cross Multiply

Since the denominators are the same, equate the numerators: \( 2x - 12 = 5x \).

Step 9: Solve for x

Rearrange the equation: \( 2x - 5x = 12 \), which simplifies to \( -3x = 12 \). Divide by -3 to find \( x = -4 \).

Step 10: Check the Solution

Substitute \( x = -4 \) back into the original denominators to ensure they do not produce zero. For \( x=-4 \), the term \( x+2 = -2 \), thus this solution leads to division by zero, so \( x = -4 \) is not valid.

Step 11: Graphical Approach

Graph the functions \( y_1 = \frac{4}{x+2} - \frac{6}{2x} \) and \( y_2 = \frac{5}{2x+4} \). Look for intersections of these graphs.

Step 12: Analyze the Graph

Plotting these graphs, they seem not to intersect due to an undefined gap at \( x=-4 \). Hence, no valid solutions exist within the real numbers.

Key Concepts

Common Denominator

When solving equations involving fractions, finding a common denominator can be very helpful. It simplifies the process of combining fractions. The common denominator is the least common multiple of all the denominators involved.

In the given equation, we have denominators like \( x+2 \), \( 2x \), and \( 2x+4 \). By factoring, we can simplify these into compatible terms. For example, \( 2x+4 \) can be factored into \( 2(x + 2) \). This step is essential because it aligns the denominators, allowing us to rewrite each term with the common denominator \( 2x(x+2) \).

By doing this, you can effectively add or subtract the rational expressions since they share the same denominator. This process keeps calculations manageable and paves the way for further algebraic simplifications. Remember, accurately finding and using the common denominator is crucial in any fractions-related algebra problem.

Algebraic Simplification

Algebraic simplification is an essential skill in solving equations. It involves reducing complex expressions into simpler forms, making them easier to solve or understand.

In our exercise, once a common denominator is established, the next step is to simplify the expressions. Each term of the equation is rewritten with the common denominator \( 2x(x+2) \). This process entails factoring and distributing terms to simplify expressions. For instance, \( \frac{8x - 6(x+2)}{2x(x+2)} \) can be simplified by expanding the numerator to \( 8x - 6x - 12 \), which further simplifies to \( 2x - 12 \).

Simplifying algebraic expressions can involve combining like terms, factoring, or applying distributive properties. These steps clear the path to solving the equation by making complex fractions into straightforward numerators and denominators, which can then be solved algebraically.

Graphical Solutions

Graphical solutions offer a visual interpretation of equations, assisting in identifying solutions by plotting graphs. This method involves creating graph functions for the terms of an equation and looking for their intersection points.

For the given equation, the graphical solution involves plotting \( y_1 = \frac{4}{x+2} - \frac{6}{2x} \) versus \( y_2 = \frac{5}{2x+4} \). If these graphs intersect, the \( x \)-coordinate of the intersection is a solution to the equation.

Graphically, some solutions may be invalid (such as division by zero areas) and can be identified through undefined points or gaps in the graph. For instance, in this exercise, while attempting to graph, it becomes evident that there’s a gap at \( x=-4 \), indicating an issue with the solution. This approach not only provides a solution when intersection points exist but also highlights the constraints of the equation.

Rational Expressions

Rational expressions involve ratios of polynomials, similar to fractions but with polynomial numerators and denominators. Solving equations with rational expressions requires careful handling of these terms.

In the exercise, each side of the equation holds rational expressions. The key lies in ensuring the expressions are simplified and have common denominators before applying algebraic principles to solve the equation.

Managing rational expressions involves understanding the polynomial structure and carefully manipulating the fractions. Always check for restrictions, such as values of \( x \) that make the denominator zero, to avoid undefined expressions. This was evident in the failed solution attempt with \( x = -4 \). Conclusively, well-managed rational expressions can be solved using both algebraic and graphical methods, provided their constraints are well understood.