Problem 77 - Elementary Algebra

Question

For exercises \(75-78\), one part of simplifying a rational expression is completed. Problem: The first step in using Strategy 2 to simplify \(\frac{\frac{1}{x+1}-\frac{1}{x-2}}{\frac{2}{x+1}+\frac{3}{x-2}} \cdot \frac{(x+1)(x-2)}{(x+1)(x-2)}\) is to find the least common denominator of all the individual fractions, multiply by a fraction equal to 1 , use the distributive property, and simplify. Incorrect Answer: $$ \begin{aligned} & \frac{\frac{1}{x+1}-\frac{1}{x-2}}{\frac{2}{x+1}+\frac{3}{x-2}} \cdot \frac{(x+1)(x-2)}{(x+1)(x-2)} \\ =& \frac{\frac{1}{x+1} \cdot(x+1)(x-2)-\frac{1}{x-2}(x+1)(x-2)}{\frac{2}{x+1} \cdot(x+1)(x-2)+\frac{3}{x-2} \cdot(x+1)(x-2)} \\ =& \frac{x-2-x+1}{2 x-4+3 x+3} \end{aligned} $$

Step-by-Step Solution

Answer

\(\frac{-3}{5x-1}\)

Step 1: Rewrite the Expression

Rewrite the original expression as given: \[\frac{\frac{1}{x+1}-\frac{1}{x-2}}{\frac{2}{x+1}+\frac{3}{x-2}} \cdot \frac{(x+1)(x-2)}{(x+1)(x-2)} \]

Step 2: Identify the Least Common Denominator (LCD)

Identify the LCD of all the individual fractions. Here, the LCD of the denominators \(x+1\) and \(x-2\) is \((x+1)(x-2)\).

Step 3: Multiply by a Fraction Equal to 1

Multiply the given rational expression by \(\frac{(x+1)(x-2)}{(x+1)(x-2)}\). This fraction is equal to 1 and will help in simplifying the expression.

Step 4: Apply the Distributive Property

Apply the distributive property to both the numerator and the denominator: \[\frac{(x+1)(x-2)\left(\frac{1}{x+1} - \frac{1}{x-2}\right)}{(x+1)(x-2)\left(\frac{2}{x+1} + \frac{3}{x-2}\right)} \]

Step 5: Cancel and Simplify

Cancel the common terms in the numerator and denominator: \[\frac{\left((x-2) - (x+1)\right)}{\left(2(x-2) + 3(x+1)\right)} = \frac{(x-2) - (x+1)}{2(x+1) + 3(x-2)} \]

Step 6: Simplify Further

Simplify the numerator and denominator separately: \[\text{Numerator: } (x-2) - (x+1) = x-2 - x-1 = -3 \] \[\text{Denominator: } 2(x-1) + 3(x+1) = 2x-4 + 3x+3 = 5x-1 \]

Step 7: Final Expression

Combine the simplified results:\[\frac{-3}{5x-1} \]

Key Concepts

least common denominator

The least common denominator (LCD) is crucial when simplifying rational expressions involving fractions. The LCD is the smallest expression that each denominator can divide without a remainder. It allows you to combine fractions by giving them a common base.
In the given problem, the denominators are \(x+1\) and \(x-2\). The LCD for these denominators is calculated by multiplying the two distinct factors:

• \(x+1\)
• \(x-2\)

Combining these, the LCD is \((x+1)(x-2)\). This unified base makes it easier to simplify and combine the fractions in the rational expression.

distributive property

Using the distributive property simplifies expressions by expanding them. The distributive property states that \(a(b+c) = ab + ac\). In the given problem, after identifying the LCD, you multiply each term by \((x+1)(x-2)\) to clear the denominators:
\(\frac{(x+1)(x-2)\frac{1}{x+1} - (x+1)(x-2)\frac{1}{x-2}}{(x+1)(x-2)\frac{2}{x+1} + (x+1)(x-2)\frac{3}{x-2}}\)
This helps in combining fractions and making it possible to subtract and add them easily. Using the distributive property like this will untangle the complexities inherent in rational expressions, leading to a simpler form.

canceling terms

Canceling terms is a powerful technique that simplifies complex fractions by eliminating common factors. Once you've applied the distributive property, you'll notice terms in the numerator and denominator that can be canceled:

• In the numerator: \((x-2) - (x+1)\)
• In the denominator: \(2(x-1) + 3(x+1)\)

Here, you can simplify the expression further by canceling out identical terms.
Remember, canceling only works when both the numerator and denominator share common factors. It's like reducing fractions to their simplest form.

algebraic fractions

Algebraic fractions, also known as rational expressions, are ratios of polynomial expressions. Simplifying them often involves several steps:

• Finding the least common denominator (LCD)
• Applying the distributive property
• Canceling common terms

In the problem, you start with:\(\frac{\frac{1}{x+1} - \frac{1}{x-2}}{\frac{2}{x+1} + \frac{3}{x-2}} \times \frac{(x+1)(x-2)}{(x+1)(x-2)}\).
Each step simplifies the expression until you have a much simpler form: \(\frac{-3}{5x-1}\).
This transformation illustrates the effectiveness of these techniques in managing and simplifying algebraic fractions.