Problem 49 - Algebra and Trigonometry

Question

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 a Common Denominator

To add or subtract fractions with different denominators, we first need to find a common denominator. The denominators here are \(x+1\) and \(x+2\), so the common denominator is their product: \((x+1)(x+2)\).

Step 2: Rewrite Each Fraction

Rewrite each fraction with the common denominator \((x+1)(x+2)\). For \(\frac{1}{x+1}\), multiply numerator and denominator by \(x+2\) to get \(\frac{x+2}{(x+1)(x+2)}\). For \(\frac{1}{x+2}\), multiply numerator and denominator by \(x+1\) to get \(\frac{x+1}{(x+1)(x+2)}\).

Step 3: Perform the Subtraction

Now that the fractions have a common denominator, subtract the numerators: \(\frac{x+2}{(x+1)(x+2)} - \frac{x+1}{(x+1)(x+2)} = \frac{(x+2) - (x+1)}{(x+1)(x+2)}\). Simplify the numerator: \(x+2 - x - 1 = 1\). This gives us \(\frac{1}{(x+1)(x+2)}\).

Step 4: Simplify the Expression

The expression \(\frac{1}{(x+1)(x+2)}\) cannot be simplified further in terms of the polynomial, so this is the simplified form of the original problem.

Key Concepts

Common Denominator

When working with algebraic fractions, finding a common denominator is crucial for adding or subtracting them. The common denominator is a number (or expression) that is a multiple of each of the fractions’ denominators. For algebraic fractions, this often involves multiplying the given denominators together.
To demonstrate using the exercise, consider the fractions \( \frac{1}{x+1} \) and \( \frac{1}{x+2} \). The denominators are \( x+1 \) and \( x+2 \) respectively. A simple method to find a common denominator here is by multiplying these two expressions together, which results in \( (x+1)(x+2) \).
This common denominator enables us to rewrite each fraction such that they can be compared or combined. Using the common denominator ensures that both fractions express parts of the whole in the same terms, which is key for accurate addition or subtraction.

Fraction Addition and Subtraction

Algebraic fractions can be tricky, especially because they involve variables. Like regular fractions, the addition or subtraction of algebraic fractions requires them to have the same denominator. Once a common denominator is found, each fraction is rewritten with this unified denominator.

• In our exercise, \( \frac{1}{x+1} \) is modified to \( \frac{x+2}{(x+1)(x+2)} \). The numerator here is obtained by multiplying \(1\) by the other denominator \((x+2)\).
• Similarly, \( \frac{1}{x+2} \) becomes \( \frac{x+1}{(x+1)(x+2)} \), where the numerator is \(x+1\), multiplied by the original fraction denominator \((x+1)\).

With identical denominators in hand, you can proceed to add or subtract the numerators effectively. In this case, subtract the numerators: \( (x+2) - (x+1) \), which simplifies to \(1\). This results in \( \frac{1}{(x+1)(x+2)} \).
The tricky part is often in the first rewrite phase, ensuring that the operation translates just like with numerical fractions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of reducing them to the most basic form without changing their value. This involves reducing complex fractions, eliminating unnecessary terms, or recombining like terms. The main goal is clarity and simplicity.
After performing the operations involved in a problem, you might end up with a more complex fraction or expression. Such was the final step in our example problem, where the result of the subtraction \( \frac{1}{(x+1)(x+2)} \) is already in its simplest form.
Unfortunately, not all expressions simplify straightforwardly. You should always check if a numerator or denominator can be factored further, though in this particular solution, \((x+1)(x+2)\) cannot reduce any more. Understanding when an expression is fully simplified helps in verifying your answers and gives you confidence in the accuracy of your solution.