Problem 69 - Algebra and Trigonometry

Question

Simplify the compound fractional expression. $$ \frac{\frac{x+2}{x-1}-\frac{x-3}{x-2}}{x+2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4x-7}{(x-1)(x-2)(x+2)}\).

Step 1: Simplify the Expression in the Numerator

We have the expression \(\frac{x+2}{x-1} - \frac{x-3}{x-2}\) in the numerator. To simplify, we need a common denominator. The common denominator of \(x-1\) and \(x-2\) is \((x-1)(x-2)\). Thus, rewrite each fraction with this common denominator: \[\frac{(x+2)(x-2)-(x-3)(x-1)}{(x-1)(x-2)}\] Now, expand the numerators: \( (x+2)(x-2) = x^2 - 4 \) \( (x-3)(x-1) = x^2 - 4x + 3 \). Substitute back into the fraction: \[\frac{x^2 - 4 - (x^2 - 4x + 3)}{(x-1)(x-2)}\] This simplifies to: \[ \frac{x^2 - 4 - x^2 + 4x - 3}{(x-1)(x-2)} = \frac{4x - 7}{(x-1)(x-2)} \].

Step 2: Combine the Numerator and Denominator of the Original Expression

The original expression is now \(\frac{\frac{4x-7}{(x-1)(x-2)}}{x+2}\). This is a compound fraction that can be simplified by multiplying the numerator by the reciprocal of the denominator: \[\frac{4x-7}{(x-1)(x-2)} \times \frac{1}{x+2} = \frac{4x-7}{(x-1)(x-2)(x+2)}\].

Step 3: Final Simplified Expression

After multiplying and eliminating common factors, we are left with the expression \(\frac{4x-7}{(x-1)(x-2)(x+2)}\). Since there are no common factors between the numerator and the denominator, this is the simplest form of the expression.

Key Concepts

Simplifying Fractional Expressions

Simplifying fractional expressions is a vital skill in algebra that involves reducing fractions to their simplest form. Imagine a scenario where you have a complex fraction, filled with polynomials, and you're tasked with making it more manageable. To achieve this, you must first understand that simplifying means finding a simpler, yet mathematically equivalent, version of the expression.

Start by identifying common factors within the numerator and the denominator. You can eliminate any shared factors to reach the simplest form of the fraction. Being keen on mathematical rules is essential here; only zero-exceptional values can be factored out.

Steps to Simplify Fractional Expressions:

• Find common factors in both the numerator and the denominator.
• Cancel them out to simplify the expression.
• Make sure the value remains the same, focusing on an equivalent expression."
• Avoid illegal division by zero.

Algebraic Fractions

Algebraic fractions work much like numerical fractions, but instead of numbers, they contain variables. They represent the division of one algebraic expression by another, such as \( \frac{x+2}{x-1}\). Handling algebraic fractions requires understanding variable properties and operations such as addition, subtraction, multiplication, and division.

To manipulate algebraic fractions, you apply the same principles as with regular fractions. Find a common denominator for addition or subtraction, and factor where possible to simplify. This can often mean making what's complex into something more manageable.

Key Points for Handling Algebraic Fractions:

• Understand basic operations: multiplication, division, addition, and subtraction.
• Find a common denominator for adding or subtracting algebraic fractions.
• Factor expressions to simplify fractions where possible.
• Keep an eye out for and avoid division by zero, which can occur when a denominator is zero.

Compound Fractions

Compound fractions, or complex fractions, involve fractions within a fraction. They can seem intimidating, but with practice, they become much more approachable. To tackle compound fractions, you typically convert them into a simpler form by multiplying by the reciprocal or using the least common denominator method.

Consider a problem like \( \frac{\frac{a}{b}}{c} \). You can solve this by rewriting it as \( \frac{a}{b \cdot c} \), transforming a seemingly complex fraction into a much simpler one. It's a methodical approach that requires attention to mathematical operations and factor relationships.

Steps to Simplify Compound Fractions:

• Identify inner and outer fractions in the problem.
• Multiply the outer numerator by the reciprocal of the outer denominator.
• Simplify further by factoring to reduce the fraction.
• Always check that no division by zero occurs at any point in solving.

Recognizing the structure and applying these methods makes compound fractions far less daunting.