Problem 39 - Algebra and Trigonometry

Question

\(35-54\) . Perform the addition or subtraction and simplify. $$ \frac{1}{x+1}-\frac{1}{x+2} $$

Step-by-Step Solution

Answer

\(\frac{1}{(x+1)(x+2)}\)

Step 1: Identify the Problem

We need to subtract two rational expressions: \( \frac{1}{x+1} - \frac{1}{x+2} \). The goal is to find a common denominator and simplify the expression.

Step 2: Find the Least Common Denominator (LCD)

The denominators of the two fractions are \(x+1\) and \(x+2\). To combine them, we need the least common denominator, which is \((x+1)(x+2)\).

Step 3: Adjust the Fractions to Have the Same Denominator

Rewrite each fraction with the least common denominator. The first fraction becomes \(\frac{1 \cdot (x+2)}{(x+1)(x+2)} = \frac{x+2}{(x+1)(x+2)}\). The second fraction becomes \(\frac{1 \cdot (x+1)}{(x+2)(x+1)} = \frac{x+1}{(x+1)(x+2)}\).

Step 4: Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators: \(\frac{x+2}{(x+1)(x+2)} - \frac{x+1}{(x+1)(x+2)} = \frac{(x+2) - (x+1)}{(x+1)(x+2)}\).

Step 5: Simplify the Numerator

Subtract the terms in the numerator: \((x+2) - (x+1) = x + 2 - x - 1 = 1\).

Step 6: Write the Simplified Expression

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

Key Concepts

Understanding the Least Common Denominator (LCD)

When working with rational expressions, one crucial step is to identify the least common denominator (LCD). The LCD is the smallest expression that both denominators can divide into without a remainder. This is vital for adding or subtracting fractions because it makes the denominators the same, allowing us to combine the fractions smoothly.

Here's how to find the LCD:

• List all the denominators in the problem. In our example, they are \(x+1\) and \(x+2\).
• Determine the products of these denominators. The product in this case is \((x+1)(x+2)\).
• The LCD is this product because both \(x+1\) and \(x+2\) are factors of it.

The LCD of algebraic fractions might seem complex, but it's just about finding a common ground for denominators to help you carry out addition or subtraction.

Simplifying Rational Expressions

Once you have a common denominator, the next step is simplifying the rational expression. This involves rewriting each fraction using the least common denominator and then performing the required operation, addition or subtraction.

In simplification:

• First, ensure both expressions have the common denominator.For the exercise, \(\frac{1}{x+1}\) becomes \(\frac{x+2}{(x+1)(x+2)}\) and \(\frac{1}{x+2}\) becomes \(\frac{x+1}{(x+1)(x+2)}\).
• Next, perform the operation on the numerators, keeping the common denominator the same. Here, we subtract: \((x+2) - (x+1)\).
• Simplify the result to its simplest form. In this case, the simplified numerator is \(1\), leading to \(\frac{1}{(x+1)(x+2)}\).

Simplifying helps to clearly express the result without unnecessary complexity, making it easier to understand and use.

Exploring Algebraic Fractions

Algebraic fractions are similar to ordinary fractions, but their numerators and denominators contain algebraic expressions (e.g., variables like \(x\)). They play a crucial role in algebra because they help express relationships and solve for unknowns.

To work with algebraic fractions successfully:

• Understand that any fraction with a variable is an algebraic fraction.
• These can often be reduced or manipulated by finding a common denominator, just like with numeric fractions.
• Operations such as addition, subtraction, multiplication, and division are performed using similar rules, considering the variable expressions in the numerators and denominators.

Algebraic fractions require careful handling, particularly when simplifying, as keeping an eye on variable constraints is key to ensuring mathematical correctness.