Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. \(\frac{x}{1-x}+\frac{2 x+3}{x^{2}-1}\)

Step-by-Step Solution

Answer

\[-\frac{x^2-3x-3}{(1-x)(x+1)}\]

Step 1: Identify the common denominator

The given expression has two fractions with different denominators: \((1-x)\) and \((x^2-1)\). The common denominator between these two is the least common multiple (LCM) of both denominators. Since \((x^2-1)\) can be factored into \((x-1)(x+1)\), the LCM is \((1-x)(x+1)\).

Step 2: Transform the fractions to have the common denominator

Now we need to transform each fraction so that they have the common denominator \((1-x)(x+1)\). We do this by multiplying the numerator and the denominator of each fraction by the missing factor from the LCM. For the first fraction, we have: \[\frac{x}{1-x} \cdot \frac{x+1}{x+1} = \frac{x(x+1)}{(1-x)(x+1)}\] For the second fraction, we have: \[\frac{2x+3}{x^2-1} \cdot \frac{1-x}{1-x} = \frac{(2x+3)(1-x)}{(1-x)(x+1)}\]

Step 3: Add the two transformed fractions

Now that both fractions have a common denominator, we can add them together. \[\frac{x(x+1)}{(1-x)(x+1)} + \frac{(2x+3)(1-x)}{(1-x)(x+1)}= \frac{x(x+1) + (2x+3)(1-x)}{(1-x)(x+1)}\]

Step 4: Simplify the numerator

In this step, we will simplify the numerator by distributing and combining like terms. \[x(x+1) + (2x+3)(1-x) = x^2+x + (2x+3)(1-x)\] Now, distribute \((2x+3)\) to \((1-x)\): \[x^2+x+2x-2x^2+3-3x\] Combine the like terms: \[x^2+x+2x-2x^2+3-3x = -x^2+3x+3\]

Step 5: Substitute the simplified numerator

Now, substitute the simplified numerator back into the combined fraction. \[\frac{-x^2+3x+3}{(1-x)(x+1)}\] This is the simplified expression for the given problem. Final Answer: \[-\frac{x^2-3x-3}{(1-x)(x+1)}\]

Key Concepts

Simplifying Expressions

Simplifying expressions involving algebraic fractions is crucial for making calculations more manageable. Let’s explore the key steps to simplify them effectively.
Begin by analyzing the problem. Notice if fractions are involved, they often have different denominators. Simplification requires you to first convert them to have the same denominator.
Once you have a common denominator, the next step is to combine terms from the numerators and simplify as much as possible by canceling out terms. This process relies heavily on algebraic manipulation, including distributing terms across expressions and combining any like terms.
Always double-check your final expression to ensure no further simplification is possible. Remember, the end goal is to express the problem in the simplest form.

Common Denominator

When working with algebraic fractions, finding a common denominator is key to simplifying and combining them.
The common denominator is typically found by determining the least common multiple (LCM) of all the denominators in the expression. In many cases, this involves factoring the denominators into their simplest forms.
For example, if the denominators are \((1-x)\) and \(x^2-1\), we need to factor the latter to\((x-1)(x+1)\). Recognizing these factors helps pinpoint the LCM as \((1-x)(x+1)\).

• Multiply each fraction by any factor of the LCM that it is missing.
• Each fraction should now share a common denominator, allowing them to be combined.

This technique enables consistent and organized manipulation of expressions, which is crucial for maintaining accuracy in algebraic operations.

Combining Like Terms

After aligning algebraic fractions to have a common denominator, the next task is to combine the fractions through their numerators. This process involves distributing multiplication over addition or subtraction, often referred to as 'expanding'.
Once you complete the multiplication, "like terms" in the expanded expression must be combined. Like terms are expressions that share the same variable and power, such as \(-x^2\) and\(x^2\) or \(3x\) and \(x\). These terms can be combined by simply adding or subtracting their coefficients.

• Identify like terms within the expanded expression.
• Add or subtract the coefficients of these terms.

The result is a simplified expression, often much easier to work with in subsequent calculations. With practice, identifying and combining like terms becomes a swift and intuitive process.