Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The short answer is: \(\frac{x^2-8x-5}{(x+2)(x-1)}\).

Step 1: Find the Least Common Denominator (LCD)

The given fractions have denominators \(x+2\) and \(x-1\). Since these two expressions have no common factor, the least common denominator (LCD) will be the product of these two expressions: \((x+2)(x-1)\).

Step 2: Rewrite the fractions with the common denominator

We will rewrite both fractions with the common denominator \((x+2)(x-1)\). To do that, we multiply the numerator and denominator of the first fraction by \((x-1)\) and the numerator and denominator of the second fraction by \((x+2)\). \[\frac{2x-1}{x+2}\times\frac{x-1}{x-1} - \frac{x+3}{x-1}\times\frac{x+2}{x+2}\]

Step 3: Simplify the numerators

Now, simplify the numerators by expanding the brackets: \[\frac{(2x-1)(x-1)}{(x+2)(x-1)} - \frac{(x+3)(x+2)}{(x-1)(x+2)}\] \[\frac{2x^2-3x+1}{(x+2)(x-1)} - \frac{x^2+5x+6}{(x-1)(x+2)}\]

Step 4: Subtract the fractions

Now that the two fractions have the same denominator, we can subtract them by subtracting their numerators: \[\frac{2x^2-3x+1-(x^2+5x+6)}{(x+2)(x-1)}\]

Step 5: Simplify the result

Simplify the numerator of the fraction by combining the like terms: \[\frac{2x^2-3x+1-x^2-5x-6}{(x+2)(x-1)}\] \[\frac{x^2-8x-5}{(x+2)(x-1)}\] Thus, the simplified expression for the given problem is \(\frac{x^2-8x-5}{(x+2)(x-1)}\).

Key Concepts

Least Common Denominator

Understanding the concept of the least common denominator (LCD) is crucial when working with algebraic fractions. Essentially, the LCD is the smallest expression that can be used as a common denominator for all fractions involved in a calculation. In algebra, instead of just numbers, we deal with expressions. To find the LCD, we look for a common base that both denominators can divide into without leaving a remainder.

For instance, consider the denominators from our exercise, which are \(x+2\) and \(x-1\). These two expressions do not share any common factors, other than 1, which means we have to multiply them together to get our LCD. Thus, the LCD for the algebraic fractions \(\frac{2x-1}{x+2}\) and \(\frac{x+3}{x-1}\) is \(\text{(x+2)(x-1)}\).

In practice, finding the LCD allows us to combine fractions by creating a common ground for addition, subtraction, multiplication, or division.

Subtracting Fractions

When subtracting algebraic fractions with different denominators, like \(\frac{2x-1}{x+2}\) and \(\frac{x+3}{x-1}\), we must first rewrite them with a common denominator. Subtracting fractions is similar to subtracting whole numbers, but with an added step: aligning their denominators.

To subtract fractions correctly, follow these steps:

• Find the LCD of the fractions.
• Rewrite each fraction as an equivalent fraction with the LCD.
• Subtract the numerators of these new fractions.
• Simplify the resultant fraction if possible.

Our example problem demonstrates this by multiplying each fraction by a form of 1 (which is a fraction with equal numerator and denominator that does not change the fraction's value) that converts the denominators to our previously found LCD. Once both fractions are written with this common denominator, their numerators can be subtracted as shown in the provided solution.

Simplifying Expressions

Simplifying expressions, such as algebraic fractions, involves combining like terms and reducing the expression to its simplest form without changing its value. It is an essential skill in algebra, making complex expressions more manageable and easier to understand.

After subtracting algebraic fractions, we often get a compound fraction that requires further simplification. The process involves:

• Combining like terms in the numerator and denominator separately.
• Factoring when possible to further reduce the expression.
• Cancelling out any common factors between the numerator and denominator.

In our example, the step of combining like terms led to the simplified expression \(\frac{x^2-8x-5}{(x+2)(x-1)}\). There are no common factors to cancel between the numerator and denominator, which confirms that we’ve reached the simplest form of the expression. This simplification makes it easier to understand the properties and behavior of the algebraic expression, such as finding its zeros or asymptotes.