Problem 43 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{3}{x+1}-\frac{3}{x} $$

Step-by-Step Solution

Answer

The answer is \(-\frac{3}{x(x+1)}\).

Step 1: Identify the Least Common Denominator

In this case, the least common denominator (LCD) is the product of the distinct denominators \(x+1\) and \(x\). Therefore, the LCD is \(x(x+1)\).

Step 2: Convert Each Fraction to Have the LCD as the Denominator

Multiply each fraction by a form of 1 that will convert each fraction's denominator to the LCD. \(\frac{3}{x+1} * \frac{x}{x} = \frac{3x}{x(x+1)}\), \(\frac{3}{x} * \frac{x+1}{x+1} = \frac{3x+3}{x(x+1)}\).

Step 3: Combine the Fractions

They can now be combined since they have the same denominator: \(\frac{3x}{x(x+1)} - \frac{3x+3}{x(x+1)} = \frac{3x - 3x - 3}{x(x+1)}\).

Step 4: Simplify the Fraction

The fraction simplifies to \(-\frac{3}{x(x+1)}\).

Key Concepts

Least Common Denominator

In operations involving fractions, especially when adding or subtracting, finding the Least Common Denominator (LCD) is crucial.
The LCD is the smallest expression that all denominators can divide into, without a remainder. This is important as it allows us to combine the fractions under a common denominator.
In mathematical terms, when you have denominators like \(x+1\) and \(x\), finding the LCD involves computing the product of these distinct denominators. In our example, this means the LCD is \(x(x+1)\).
By equating the denominators, you ensure that your fractions can be added or subtracted effortlessly.

• Picking the correct denominators ensures accurate simplification later.
• It forms the basis for converting fractions to work with each other.

Computing the correct LCD is a foundational skill in working with rational expressions, setting you up for success in more complex algebraic manipulations.

Simplifying Fractions

Once fractions are rewritten with a common denominator, the task is to combine and then simplify them.
Simplifying a fraction means to express it in its simplest form, where the numerator and denominator share no common factors other than 1.
For instance, when you encounter \(\frac{3x - 3x - 3}{x(x+1)}\), simplifying involves performing the subtraction in the numerator first.
Here, you will simplify it further by recognizing any common factors in both the numerator and the denominator.
In this particular example, the subtraction reduces to \(-3\). Thus, you ultimately arrive at \(-\frac{3}{x(x+1)}\).

• Simplification increases clarity and makes further calculations easier.
• Always look for and cancel common factors to simplify fully.

Knowing how to simplify ensures you're proficient in deriving the most reduced form available, a critical step in solving fraction-related problems.

Rational Expressions

Rational expressions, much like fractions, consist of a numerator and a denominator.
They are defined as the quotient of two polynomials. Handling operations with rational expressions follows a similar approach to fractions: ensuring denominators are the same and simplifying wherever possible.
Rational expressions show up frequently in equations and need careful handling.

• Know how to manipulate these expressions, like finding the LCD and simplifying.
• Practice is crucial as it deepens understanding, especially in algebra.

When you are comfortable with the arithmetic of rational expressions, step-by-step operations like addition, subtraction, and simplification become second nature.
These skills are part of solidifying your mathematical foundation. It's not only about arriving at the correct answer but understanding the logical steps that bring you there.