Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{2}{2 x+1}-\frac{3}{2 x-1}=\frac{-2 x+7}{4 x^{2}-1}$$

Step-by-Step Solution

Answer

No solution exists for the given equation.

Step 1: Identify the common denominator

The common denominator for the left side of the equation is the product of the denominators: \( (2x+1)(2x-1)\). Notice that \(4x^2-1 = (2x+1)(2x-1)\), which means it's already a common denominator for both sides.

Step 2: Rewrite the equation with common denominator

Since \(4x^2-1 = (2x+1)(2x-1)\), rewrite the equation with the commondenominator \( (2x+1)(2x-1)\): \[\frac{2(2x-1)-3(2x+1)}{(2x+1)(2x-1)} = \frac{-2x+7}{(2x+1)(2x-1)}.\]

Step 3: Simplify the left side of the equation

Distribute and simplify the left side: \[ \frac{2(2x-1) - 3(2x+1)}{(2x+1)(2x-1)} = \frac{4x-2 - 6x -3}{(2x+1)(2x-1)} = \frac{-2x - 5}{(2x+1)(2x-1)}. \]

Step 4: Equate the numerators

With both sides having the same denominator, equate the numerators: \( -2x - 5 = -2x + 7\).

Step 5: Solve for x

Add \(2x\) to both sides, resulting in \(-5 = 7\). However, this is inconsistent, indicating no solution exists.

Key Concepts

Algebra

Algebra is the mathematics of operations and relations. It uses symbols and letters to represent numbers and quantities in equations and formulas. In rational equations, like the one given in the problem, algebra helps us understand the relationship between different terms and apply operations systematically.

We often use algebra to simplify expressions, manipulate equations, and find unknown values. Using algebraic techniques, such as distribution and combining like terms, we can transform complex expressions into simpler forms, aiding in understanding and solving equations.

Common Denominator

A common denominator in fractions is a number or an expression that is a multiple of each of the denominators involved. It allows us to combine fractions by rewriting them with the same denominator.

In rational equations, finding a common denominator is crucial for simplifying and solving the equation. With the problem at hand, notice that each term can be expressed with the same denominator \( (2x+1)(2x-1) \). This allows us to easily subtract the fractions on the left side of the equation and equate the numerators.

• Identify denominators
• Find the least common multiple
• Rewrite each fraction with the common denominator

Equation Solving

Solving an equation involves finding the value of the variable that makes the equation true. In the context of rational equations, solving usually consists of several steps: simplifying the equation, eliminating denominators, and isolating the variable.

For rational equations, it is important to first ensure all terms share a common denominator. This lets us focus on the numerators by equating them, transforming a potentially complex rational equation into a simpler linear equation. However, one must check if the obtained solution is valid within the constraints of the original equation.

No Solution

Sometimes, an algebraic equation may have no solution. This occurs when simplifications and solving steps lead to a contradiction or when the variable's result is not permissible within the equation's domain.

In the given example, after equating the numerators, we reach an inconsistent statement \( -5 = 7 \). Such contradictions highlight that no value of \( x \) can satisfy the original equation.

• Acknowledge contradictions often signal no solution
• Check the feasibility of potential solutions against the context of the problem
• Recognize the domain restrictions