Problem 7 - Precalculus: Functions and Graphs

Question

Solve the equation. \(\frac{4}{x+2}+\frac{1}{x-2}=\frac{5 x-6}{x^{2}-4}\)

Step-by-Step Solution

Answer

The solutions are all real numbers except \(x = 2\) and \(x = -2\).

Step 1: Identify the Common Denominator

The common denominator of the fractions on both sides of the equation is \(x^2 - 4\), which can be factored as \((x+2)(x-2)\). We will use this common denominator to combine the terms.

Step 2: Clear the Denominator

Multiply each term by the common denominator \(x^2 - 4\) to eliminate the denominators. This yields: \[ 4(x-2) + 1(x+2) = (5x - 6) \]This removes the fractions from the equation.

Step 3: Simplify Both Sides

Expand and simplify each side of the equation:Left side: \[ 4(x-2) + 1(x+2) = 4x - 8 + x + 2 = 5x - 6 \]The right side remains:\((5x - 6)\).

Step 4: Solve the Simplified Equation

Since both sides simplify to \(5x - 6\), we can see that the expressions are identical, implying that any value of \(x\) that doesn't make the original denominators zero (\(x eq 2\), \(x eq -2\)) is a solution.

Key Concepts

Common Denominator

In solving rational equations, one crucial step is identifying a common denominator. A common denominator is a shared multiple of the denominators in your equation. It helps to simplify complex fractions and make dealing with them easier. For the given exercise, the expression \(x^2 - 4\) is the common denominator,and it can be factored into \((x+2)(x-2)\) to make it more manageable.

• Finding a common denominator allows you to combine fractions by rewriting each with the same denominator.
• Once all terms have the same denominator, you can manipulate the numerator directly, ignoring the denominator for calculation purposes.

In this particular case, recognizing that \(x^2 - 4\) factors into \((x+2)(x-2)\) brings clarity to how the denominators relate and streamlines the process of clearing fractions from the equation. The idea is to eliminate the fractions by multiplying through by this common denominator, which gets rid of any fractional expressions.
This is a pivotal step toward simplifying the problem and arriving at a solution.

Solving Rational Equations

Rational equations are equations featuring rational expressions, which are fractions with polynomials in the numerator and denominator. Solving these often involves clearing the denominators through multiplication. Once denominators are eliminated, the equation becomes easier to handle.

• Start by identifying a common denominator, as seen with \(x^2 - 4\).
• Multiply each term by this denominator to cancel out the fractions.
• Now focus on solving the simplified polynomial equation that emerges.

Clearing the denominator turns a potentially tricky situation involving fractions into a more straightforward polynomial equation. In our exercise, this results in both sides of the equation becoming \(5x - 6\), which reveals that any \(x\) not making the original denominators zero is a solution.
Remember always to check your solutions against the original equation to ensure no division by zero occurs with the denominators involved.

Factoring Expressions

Factoring plays a crucial role in understanding and solving equations involving polynomials. To factor an expression means breaking it down into simpler multiplied components. In our exercise, recognizing that \(x^2 - 4\) factors into \((x+2)(x-2)\) is key.

• Factoring identifies where an expression can simplify or cancel out.
• It reveals roots or solutions where polynomials either cross the x-axis or become zero.
• This makes working with both polynomials in equations and denominators in fractions much simpler.

The expression \(x^2 - 4\) is a "difference of squares," a specific type of polynomial that factors neatly into two conjugates. Recognizing these patterns enhances understanding and speeds up equation solving.
Always keep an eye out for opportunities to factor expressions further, as it can greatly streamline your approach and bring otherwise challenging problems within reach.