Problem 34 - Algebra and Trigonometry

Question

\(31-42\) . 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

The solution to the equation is \( x = 8.5 \).

Step 1: Simplify the fractions

Rewrite the fraction \( \frac{5}{2x+4} \) as \( \frac{5}{2(x+2)} \), which is equivalent since they represent the same expression.

Step 2: Find a common denominator

The denominators are \( x+2 \), \( 2x \), and \( 2(x+2) \). To find a common denominator, determine the least common multiple, which is \( 2x(x+2) \).

Step 3: Re-express each term with the common denominator

Multiply each term by the necessary factor to have the common denominator:- First term: \( \frac{4}{x+2} = \frac{4 \cdot 2x}{2x(x+2)} = \frac{8x}{2x(x+2)} \)- Second term: \( \frac{6}{2x} = \frac{6(x+2)}{2x(x+2)} = \frac{6x+12}{2x(x+2)} \)- Third term: \( \frac{5}{2(x+2)} = \frac{5}{2(x+2)} \) is already in terms of the common denominator.

Step 4: Set up the equation

Combine the transformations into a single equation using the common denominator:\[ \frac{8x}{2x(x+2)} - \frac{6x + 12}{2x(x+2)} = \frac{5}{2(x+2)} \]

Step 5: Eliminate the denominators

Since all fractions have the same denominator, set only the numerators equal to each other:\[ 8x - (6x + 12) = 5 \]

Step 6: Simplify the equation

Simplify the equation:1. Distribute the negative sign: \( 8x - 6x - 12 = 5 \).2. Combine like terms: \( 2x - 12 = 5 \).

Step 7: Solve for \( x \)

Add 12 to both sides to isolate \( 2x \):\[ 2x = 17 \]Finally, divide by 2 to solve for \( x \):\[ x = \frac{17}{2} \] or \( x = 8.5 \).

Step 8: Graphical method

Graph each side of the equation as separate functions: \( f(x) = \frac{4}{x+2} - \frac{6}{2x} \) and \( g(x) = \frac{5}{2x+4} \). Locate the x-coordinate of the intersection point of these functions on the graph, which should also be at \( x = \frac{17}{2} \) or \( x = 8.5 \). This confirms the algebraic solution.

Key Concepts

Common Denominator

When solving rational equations, finding a common denominator is crucial. This allows us to combine fractions easily. Let’s explore this concept in detail. In a rational equation like \( \frac{4}{x+2} - \frac{6}{2x} = \frac{5}{2x+4} \), each term is a fraction with different denominators.To solve it, we first need a common denominator, so we can manipulate the fractions correctly. A common denominator is essentially the least common multiple (LCM) of all the fraction's denominators. It’s like finding a common ground where each term can meet for comparison or combination.

• Analyze each denominator: \(x+2\), \(2x\), and \(2(x+2)\).
• Determine the LCM: here, it is \(2x(x+2)\). It includes each different factor from the denominators, ensuring any number that divides one will divide the LCM.

Using the common denominator \(2x(x+2)\), each fraction is re-expressed so they can be unified and simplified. This is the foundation for solving the equation algebraically.

Fraction Simplification

Simplification of fractions is an essential process in solving equations effectively. It allows us to reduce complex expressions into simpler forms, making them easier to work with. This can lead to more straightforward calculations and better understanding of the problem at hand.In our exercise, after determining a common denominator, each term \( \left(\frac{4}{x+2}, \frac{6}{2x}, \frac{5}{2(x+2)}\right) \) was transformed to have this uniform denominator. Here's how it works:

• First term: Multiply \(\frac{4}{x+2} \) by \(\frac{2x}{2x}\) to get \(\frac{8x}{2x(x+2)}\).
• Second term: Multiply \(\frac{6}{2x}\) by \(\frac{x+2}{x+2}\) to obtain \(\frac{6x + 12}{2x(x+2)}\).
• Third term is already expressed as needed: \(\frac{5}{2(x+2)}\).

This method of fraction simplification ensures all terms are comparable, allowing us to eliminate the denominators thoroughly by focusing solely on the numerators. It’s a powerful technique in algebra that transforms the equation into an easier solve.

Graphical Solution

Solving equations graphically can provide a visual perspective which is often very insightful and helps verify algebraic solutions. Let's see how this method applies to the presented exercise.For this graphing approach, consider each part of the equation as separate functions:

• Function for the left side: \( f(x) = \frac{4}{x+2} - \frac{6}{2x} \)
• Function for the right side: \( g(x) = \frac{5}{2x+4} \)

By plotting these functions on a coordinate plane, you look for points where they intersect, which represents the solution to the equation. In our example, these functions intersect at \( x = 8.5 \), visually verifying our algebraic result.Graphical solutions are especially useful when dealing with complex equations, as they offer an intuitive way to grasp where and how solutions emerge. They serve both as a verification tool and an exploratory technique in mathematics.