Problem 17 - Precalculus

Question

Solve each equation. $$\frac{1}{x-3}-\frac{2}{x+3}=\frac{1}{x^{2}-9}$$

Step-by-Step Solution

Answer

The solution is \(x = 8\).

Step 1: Identify the Denominator

The equation involves fractions with denominators of \(x - 3\), \(x + 3\), and \(x^2 - 9\). Notice that \(x^2 - 9\) factors into \((x - 3)(x + 3)\). We'll use this to find a common denominator.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator of the fractions is \((x - 3)(x + 3)\) because \(x^2 - 9 = (x - 3)(x + 3)\). We will rewrite each fraction in the equation to have this common denominator.

Step 3: Rewrite Fractions with the LCD

Rewrite each term of the equation with the common denominator: - Multiply \(\frac{1}{x-3}\) by \(\frac{x+3}{x+3}\), resulting in \(\frac{x+3}{(x-3)(x+3)}\).- Multiply \(\frac{2}{x+3}\) by \(\frac{x-3}{x-3}\), resulting in \(\frac{2(x-3)}{(x-3)(x+3)}\).- The third fraction \(\frac{1}{x^2 - 9}\) already has the common denominator.

Step 4: Combine Fractions

Now that all fractions have the same denominator, combine them:\[\frac{x+3 - 2(x-3)}{(x-3)(x+3)} = \frac{1}{(x-3)(x+3)}\].Simplify the numerator: \[x + 3 - 2x + 6 = -x + 9\], leading to: \[\frac{-x + 9}{(x-3)(x+3)} = \frac{1}{(x-3)(x+3)}\].

Step 5: Solve the Numerator Equation

Set the numerators equal to each other, since the denominators are identical:\(-x + 9 = 1\).Solve for \(x\):Subtract 9 from both sides: \(-x = -8\),Multiply both sides by -1: \(x = 8\).

Step 6: Verify the Solution

Check if \(x = 8\) is a valid solution. Substitute \(x = 8\) into the original denominators:- \(x - 3 = 5\)- \(x + 3 = 11\)- \((x - 3)(x + 3) = 55\)Since none of these are zero, \(x = 8\) is a valid solution.

Key Concepts

Common Denominator

To solve rational equations, finding a common denominator is essential. A common denominator is simply a shared denominator among all the fractions involved in an equation. This allows the fractions to be combined or compared easily. In our example, the denominators are \(x - 3\), \(x + 3\), and \(x^2 - 9\). The expression \(x^2 - 9\) is particularly interesting because it can be factored into \((x - 3)(x + 3)\). Thus, the least common denominator (LCD) for all the terms is \((x - 3)(x + 3)\). By identifying this common denominator, we can adjust each fraction so they can be combined into a single equation for easier solving.
Common denominators are a backbone to solving equations with multiple fractions and are key to simplifying complex rational expressions.

Fraction Operations

Once we identify the common denominator, the next step involves performing fraction operations. This means adjusting each fraction in the equation so they all have the common denominator \((x - 3)(x + 3)\).
To do this, we multiply each fraction by a form of 1 that will give them the needed denominator:

• For \(\frac{1}{x-3}\), multiply by \(\frac{x+3}{x+3}\), resulting in \(\frac{x+3}{(x-3)(x+3)}\).
• For \(\frac{2}{x+3}\), multiply by \(\frac{x-3}{x-3}\), resulting in \(\frac{2(x-3)}{(x-3)(x+3)}\).
• The fraction \(\frac{1}{x^2 - 9}\) is already in terms of our common denominator.

These operations are crucial as they transform distinct fractions into comparable forms and allow us to move toward solving the equation effectively.

Factoring

Factoring is a vital skill in working with rational equations. It involves breaking down an expression into simpler components—or factors—that are multiplied together. In our exercise, recognizing that \(x^2 - 9\) can be factored into \((x - 3)(x + 3)\) was pivotal.
This type of expression is known as a difference of squares, which has a general form of \(a^2 - b^2 = (a - b)(a + b)\).
By factoring, we simplify complex expressions and find common components among terms, which helps in matching denominators. This not only simplifies the process of finding a common denominator but also aids in solving equations by setting the stage for possible cancellation or simplification of terms.

Verification

Verification is the final crucial step in ensuring our solution is correct and valid. After finding a solution for the variable, here \(x = 8\), we substitute it back into the original equation to confirm it satisfies all parts of the equation.
Here's how:

• Substitute \(x = 8\) into the expressions \(x - 3\), \(x + 3\), and \((x - 3)(x + 3)\) to make sure none equals zero, since a zero denominator is undefined.
• Check that substituting \(x = 8\) returns equal values on both sides of the equation.

Performing these checks guarantees the solution doesn't produce a mathematical contradiction or invalid scenario. This step is vital in not only ensuring the mathematical correctness but also confirming the solution's practical viability. Verification solidifies that our journey through finding a solution was accurate and meaningful.