Problem 101 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{6}{5 d^{2}-5 d}-\frac{3}{5 d-5}$$

Step-by-Step Solution

Answer

\(\frac{6 - 3d}{5d(d-1)}\)

Step 1: Factor Denominators

Begin by factoring the denominators of each fraction. For the first fraction, the denominator is \(5d^2 - 5d = 5d(d-1)\). For the second fraction, the denominator is \(5d - 5 = 5(d-1)\).

Step 2: Find a Common Denominator

The common denominator for both fractions is the least common multiple of the denominators, which is \(5d(d-1)\).

Step 3: Rewrite Fractions with Common Denominator

Rewrite each fraction so that they share the common denominator. For the first fraction, it's already \(\frac{6}{5d(d-1)}\). For the second fraction, multiply both the numerator and the denominator by \(d\) to obtain: \[ \frac{3 \cdot d}{5d(d-1)} = \frac{3d}{5d(d-1)} \]

Step 4: Subtract the Fractions

Both fractions now have the same denominator, so subtract the numerators:\[ \frac{6 - 3d}{5d(d-1)} \]

Step 5: Simplify the Result

Since the numerator \(6 - 3d\) cannot be factored further and does not share any common factors with the denominator, this is the simplified form of the expression: \[ \frac{6 - 3d}{5d(d-1)} \]

Key Concepts

Factoring Denominators

When dealing with rational expressions, one of the first steps in many problems is to factor the denominators. This process makes it easier to manipulate the expressions later on, especially when finding common denominators or simplifying the expression. For example, if you have a denominator like \(5d^2 - 5d\), you can factor it into \(5d(d-1)\). This is because both terms in the expression \(5d^2\) and \(-5d\) share common factors.
This step is crucial as it sets the stage for combining fractions with different denominators. Remember, factoring transforms complex polynomials into simpler components. It reveals the structure of the expression and makes further steps possible.

Common Denominator

To perform addition or subtraction between fractions, one must find a common denominator. This common denominator is typically the least common multiple (LCM) of the individual denominators.
For instance, with denominators like \(5d(d-1)\) and \(5(d-1)\), the LCM is \(5d(d-1)\). This means the denominators must be multiplied appropriately to match this common term.

• Identify the LCM of the denominators.
• Rewrite each fraction so they both have this common denominator.

This step ensures the fractions can be aligned properly for the next operation, facilitating either addition or subtraction.

Subtracting Fractions

Once the common denominator is established, the next step is straightforward yet crucial: subtracting the fractions. This process involves subtracting the numerators while maintaining the denominator constant. For example, if you have two fractions \(\frac{6}{5d(d-1)}\) and \(\frac{3d}{5d(d-1)}\), you can subtract the numerators directly:

• Subtract the numerators: \(6 - 3d\).
• Keep the common denominator: \(5d(d-1)\).

This step essentially combines the fractions into a single expression, smoothing out the equation with shared parts. It requires accuracy; getting the numerators wrong can lead to an incorrect final expression.

Simplifying Expressions

After combining the fractions by subtracting, the final task is to simplify the expression. Simplification means reducing the expression to its simplest form, where no further reduction is possible. For our example, the resulting expression is \(\frac{6 - 3d}{5d(d-1)}\).
Check for any common factors between the numerator and the denominator that could be reduced. If none exist, as in this case where \(6 - 3d\) shares no factors with \(5d(d-1)\), your fraction is fully simplified.
This step is important to ensure clarity and enable easier handling of the expression in further calculations or analyses. It marks the completion of manipulating the expression into its most concise form.