Problem 38 - College Algebra

Question

Add and subtract the rational expressions, and then simplify. $$ \frac{x-1}{x+1}-\frac{2 x+3}{2 x+1} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-6x-4}{(x+1)(2x+1)} \).

Step 1: Identify the Least Common Denominator (LCD)

First, we identify the least common denominator (LCD) for the rational expressions. The denominators are \(x+1\) and \(2x+1\). The LCD of \(x+1\) and \(2x+1\) is their product, \((x+1)(2x+1)\), since they have no common factors other than 1.

Step 2: Rewrite Each Rational Expression with the LCD

Next, we rewrite each fraction with the LCD: \(\frac{x-1}{x+1}\) and \(\frac{2x+3}{2x+1}\). Multiply the first fraction by \((2x+1)/(2x+1)\) and the second by \((x+1)/(x+1)\): \[\frac{(x-1)(2x+1)}{(x+1)(2x+1)} - \frac{(2x+3)(x+1)}{(2x+1)(x+1)}\]

Step 3: Expand the Numerators

Now expand the numerators of both expressions:For \((x-1)(2x+1)\), expand: \((x)(2x)+(x)(1)-(1)(2x)-(1)(1) = 2x^2+x-2x-1 = 2x^2-x-1\).For \((2x+3)(x+1)\), expand: \((2x)(x)+(2x)(1)+(3)(x)+(3)(1) = 2x^2+2x+3x+3 = 2x^2+5x+3\).

Step 4: Substitute and Combine the Numerators

Substitute the expanded numerators back into the equation:\[\frac{2x^2-x-1}{(x+1)(2x+1)} - \frac{2x^2+5x+3}{(x+1)(2x+1)}\]Combine the numerators over the common denominator:\[\frac{(2x^2-x-1) - (2x^2+5x+3)}{(x+1)(2x+1)}\]

Step 5: Simplify the Numerator

Simplify the numerator by distributing the subtraction and combining like terms: \[2x^2-x-1-2x^2-5x-3 = 2x^2-2x^2-x-5x-1-3 = -6x-4\]

Step 6: Write the Final Simplified Expression

The final simplified rational expression after combining the numerators and simplification is:\[\frac{-6x-4}{(x+1)(2x+1)}\]

Key Concepts

Least Common Denominator

When working with rational expressions, much like those with regular fractions, finding the least common denominator (LCD) is crucial for operations such as addition and subtraction. The LCD is the smallest expression that each of the denominators can divide into without a remainder. The goal is to transform all denominators involved into this common denominator to allow the rational expressions to be combined.

In our exercise, the denominators were \(x+1\) and \(2x+1\). They share no common factors other than 1, making their product \((x+1)(2x+1)\) the LCD. Understanding how to find and use the LCD is key:

• If the denominators have a common factor, include it once in the LCD.
• Multiply all other unique factors to get the LCD.
• Use the LCD to rewrite each fraction, which we'll discuss in the next step.

A clear knowledge of the LCD simplifies adding and subtracting fractions by turning them into like denominators.

Numerator Expansion

Once you have identified and applied the least common denominator, the next step in managing rational expressions is to adjust each fraction so that it reflects this common base. This requires expanding the numerators so they remain equivalent to the original fractions but use the LCD.

For any fraction \(\frac{a}{b}\), rewritten with an LCD, you multiply both the numerator \(a\) and the denominator \(b\) by whatever makes \(b\) become the LCD. For example, applying this to our problem:

• Multiply \(\frac{x-1}{x+1}\) by \(\frac{2x+1}{2x+1}\).
• Multiply \(\frac{2x+3}{2x+1}\) by \(\frac{x+1}{x+1}\).

In the solutions, these multiplied results led to the expressions \(\frac{(x-1)(2x+1)}{(x+1)(2x+1)}\) and \(\frac{(2x+3)(x+1)}{(2x+1)(x+1)}\). Now, it’s necessary to expand these numerators by applying the distributive property.

• For \((x-1)(2x+1)\), use: \(x \cdot 2x + x \cdot 1 - 1 \cdot 2x - 1 \cdot 1\).
• This resolves to \(2x^2-x-1\).
• For \((2x+3)(x+1)\), use: \(2x \cdot x + 2x \cdot 1 + 3 \cdot x + 3 \cdot 1\).
• This resolves to \(2x^2 + 5x + 3\).

By expanding and rewriting each component, you prepare both numerators for subtraction or addition, preserving the equality of the expressions.

Rational Expression Simplification

After creating a common base using the least common denominator and handling the numerators through expansion, the final step in the process involves simplifying the rational expression. This means reducing it to its simplest form, making it more manageable and clearer.

Once the numerators are expanded and combined under the LCD, you subtract or add them as dictated by the problem. For our exercise, we have:

• Subtract the expanded numerators: \((2x^2 - x - 1) - (2x^2 + 5x + 3)\)
• Apply distribution: \(2x^2 - x - 1 - 2x^2 - 5x - 3\)
• Combine like terms: \(-6x - 4\)

The expression \(\frac{-6x-4}{(x+1)(2x+1)}\) is the result of simplifying each step in the rational expression's journey.

When simplifying, always ensure to combine terms accurately:

• Double-check algebraic signs when distributing multiplication or subtraction.
• Try factoring if possible, for further simplification.

Simplification is essential, as it reveals the simplest form of the original problem, making interpretations and solutions more visually comprehensible.