Problem 53 - Elementary Algebra

Question

For Problems \(52-54\), solve each equation. $$ \frac{x-2}{x^{2}-1}+\frac{3}{x+1}=\frac{-5}{x-1} $$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Identify the Common Denominator

The expression includes three distinct denominators: \(x^2 - 1\), \(x + 1\), and \(x - 1\). Notice that \(x^2 - 1\) can be factored to \((x - 1)(x + 1)\). Thus, the common denominator for these fractions is \((x - 1)(x + 1)\), as it encompasses all parts of the other denominators.

Step 2: Rewrite Each Term With the Common Denominator

To combine the fractions, we need to rewrite each fraction with the common denominator \((x-1)(x+1)\): 1. Multiply the first fraction by \(\frac{1}{1}\), so it remains the same: \(\frac{x-2}{(x-1)(x+1)}\).2. Multiply the second fraction by \(\frac{x-1}{x-1}\): \(\frac{3(x - 1)}{(x - 1)(x + 1)}\).3. Multiply the right side by \(\frac{x+1}{x+1}\): \(\frac{-5(x + 1)}{(x - 1)(x + 1)}\).

Step 3: Combine Fractions

Now, with a common denominator, you can combine the fractions on the left-hand side:\[\frac{(x-2) + 3(x-1)}{(x-1)(x+1)} = \frac{-5(x+1)}{(x-1)(x+1)}\]

Step 4: Simplify Numerators and Set Equations Equal

Expand the numerators:For the left:\(x - 2 + 3x - 3 = 4x - 5\) For the right:\(-5(x+1) = -5x - 5\)This gives us the equation:\[\frac{4x - 5}{(x-1)(x+1)} = \frac{-5x - 5}{(x-1)(x+1)}\]

Step 5: Solve the Equation by Equating Numerators

Now, equate the numerators since the denominators are the same:\[4x - 5 = -5x - 5\]Add \(5x\) to both sides:\[9x - 5 = -5\]Add \(5\) to both sides:\[9x = 0\]Divide by \(9\):\[x = 0\]

Step 6: Verify Solution Constraints

Check the constraints of the equation:\(x^2 - 1 = 0\) implies \((x - 1)(x + 1) = 0\), leading to \(x eq 1, -1\). Since \(x = 0\) does not violate any restrictions, it is a valid solution.

Key Concepts

Common Denominator

When solving rational equations, identifying a common denominator is a pivotal step. In rational equations, the common denominator helps combine fractions involving different denominators into a single equation, making it easier to solve.

In the problem given, the denominators are \(x^2 - 1\), \(x + 1\), and \(x - 1\). Recognizing that \(x^2 - 1\) can be factored into \((x - 1)(x + 1)\) is crucial. This factorization allows us to see that the common denominator is

• \( (x - 1)(x + 1) \)

This common denominator embraces all parts of the other denominators, allowing each fraction to be rewritten under the same denominator, which facilitates adding or subtracting them later. Always look for opportunities to factor complicated denominators to find the common denominator efficiently.

Remember, having a common denominator across an equation simplifies the solution process by bringing the equation to a single coherent expression.

Factoring Quadratic Expressions

Factoring is a technique to rewrite quadratic expressions into a product of simpler expressions. This method is often utilized in solving equations involving quadratic terms because it helps in identifying roots or simplifying expressions.

In our problem, we see the quadratic expression \(x^2 - 1\). This expression can be factored as:

• \(x^2 - 1 = (x - 1)(x + 1)\)

This factorization is an example of a "difference of squares" because \(x^2\) and \(1\) are both perfect squares and they are subtracted from each other.

Whenever you encounter a quadratic expression, look to factor it as early as possible. This helps in simplifying the problem and often makes it easier to find the solution to the equation. Factoring reveals the roots of a quadratic equation and is a handy tool that underpins many algebraic solutions including finding common denominators and simplifying complex equations.

Simplifying Fractions

Simplifying fractions is a cornerstone of algebra, especially when dealing with rational equations. It involves reducing fractions to their simplest form, making it easier to solve equations and interpret results.

Once we have a common denominator, as demonstrated in our exercise, simplifying the numerators becomes necessary. Here are the simplified steps:

• For the left side of the equation:
  Combine like terms: \( x - 2 + 3(x - 1) \) simplifies to \( 4x - 5 \).
• For the right side:
  Expand the term: \(-5(x + 1)\) simplifies to \(-5x - 5 \).

The equation then becomes:
\[\frac{4x - 5}{(x-1)(x+1)} = \frac{-5x - 5}{(x-1)(x+1)}\]
Both sides are now unified in terms of their denominator, allowing us to equate the numerators directly.

Simplifying fractions reduces complexity and often reveals the straightforward path to solving the equation, ensuring every term is dealt with efficiently and accurately in pursuit of the solution. Keeping track of simplified terms and expressions helps prevent mistakes and clarifies the process of solving.