Problem 59 - Algebra and Trigonometry

Question

Perform the addition or subtraction and simplify. $$ \frac{1}{x+3}+\frac{1}{x^{2}-9} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x-2}{(x+3)(x-3)}\).

Step 1: Identify the LCD

First, identify the least common denominator (LCD) for the fractions. The denominators are \(x+3\) and \(x^2-9\). Notice that \(x^2-9\) can be factored into \((x+3)(x-3)\). Therefore, the LCD is \((x+3)(x-3)\).

Step 2: Rewrite the Fractions with the LCD

Rewrite each fraction with the common denominator \((x+3)(x-3)\). For the first fraction \(\frac{1}{x+3}\), multiply the numerator and the denominator by \(x-3\) to get \(\frac{x-3}{(x+3)(x-3)}\). The second fraction \(\frac{1}{x^2-9}\) already has the common denominator so it remains the same: \(\frac{1}{(x+3)(x-3)}\).

Step 3: Combine the Fractions

Add the rewritten fractions: \(\frac{x-3}{(x+3)(x-3)} + \frac{1}{(x+3)(x-3)}\). Combine the numerators: \(x-3+1\). This simplifies to \(x-2\).

Step 4: Simplify the Expression

Your combined fraction is \(\frac{x-2}{(x+3)(x-3)}\). This expression cannot be simplified further because there are no common factors between the numerator and the denominator.

Key Concepts

Least Common Denominator

When adding or subtracting fractions, finding the least common denominator (LCD) is crucial. It allows you to rewrite each fraction so they have the same denominator, making it easier to perform the operation.

• The denominators of the fractions in the exercise are \(x + 3\) and \(x^2 - 9\).
• To find the LCD, consider factoring \(x^2 - 9\). This expression can be rewritten as \((x + 3)(x - 3)\), a process called factoring.
• The least common denominator, therefore, is \((x + 3)(x - 3)\). This ensures both fractions can be expressed with a common base.

Understanding the LCD simplifies the process, making addition or subtraction of fractions straightforward. By breaking complex expressions into factors, finding a common denominator becomes a manageable task.

Factoring

Factoring is the process of breaking down an expression into simpler, more manageable components, called factors. This skill is indispensable when working with rational expressions, particularly when finding the LCD.

• In the exercise given, the expression \(x^2 - 9\) is factored into \((x + 3)(x - 3)\).
• Recognizing the form \(a^2 - b^2 = (a + b)(a - b)\) helps to quickly factor differences of squares.
• Factoring reveals common components between different denominators, aiding in the identification of the LCD.

By efficiently factoring expressions, you simplify the complex task of managing rational expressions. Recognizing common factoring patterns, like differences of squares, can simplify your work significantly.

Simplifying Fractions

Simplifying a fraction involves making it as simple as possible by reducing it to its smallest terms.

• Once the fractions are rewritten with a common denominator, like the LCD \((x + 3)(x - 3)\), the numerators can be combined.
• After combining, the new fraction should be checked for any common factors between the numerator and the denominator.
• In this exercise, the expression \(\frac{x-2}{(x+3)(x-3)}\) needs inspection for common factors, but since \(x-2\) shares no factors with the denominator, the fraction is already in its simplest form.

Understanding what makes fractions simple ensures they are easier to interpret and work with. Simplifying fractions aids clarity and precision in mathematical expressions.