Problem 56 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{-3}{4 x+3}+\frac{5}{2 x-5} $$

Step-by-Step Solution

Answer

\( \frac{2(7x + 15)}{(4x+3)(2x-5)} \)

Step 1: Determine the Common Denominator

The fractions \( \frac{-3}{4x+3} \) and \( \frac{5}{2x-5} \) have different denominators. To combine them, we need a common denominator, which is the product of the two denominators: \((4x + 3)(2x - 5)\).

Step 2: Adjust Fractions for Common Denominator

To rewrite each fraction with the common denominator, multiply the numerator and the denominator of the first fraction by \(2x-5\). Similarly, multiply the numerator and the denominator of the second fraction by \(4x+3\). This gives:\[\frac{-3(2x-5)}{(4x+3)(2x-5)} + \frac{5(4x+3)}{(4x+3)(2x-5)}\]

Step 3: Distribute Numerators

Now, distribute the numerators in each fraction: \[-3(2x-5) = -6x + 15\] and \[5(4x+3) = 20x + 15\]. So the expression is:\[\frac{-6x + 15}{(4x+3)(2x-5)} + \frac{20x + 15}{(4x+3)(2x-5)}\]

Step 4: Combine Numerators

Combine the numerators over the common denominator:\[-6x + 15 + 20x + 15 = (20x - 6x) + (15 + 15) = 14x + 30\]. The expression is now:\[\frac{14x + 30}{(4x+3)(2x-5)}\]

Step 5: Simplify the Expression

The numerator \(14x + 30\) can be factored by taking out the common factor 2:\[14x + 30 = 2(7x + 15)\]. The expression simplifies to:\[\frac{2(7x + 15)}{(4x+3)(2x-5)}\]. Since there is no further factoring possible, this is the simplest form.

Key Concepts

Common Denominator

When working with rational expressions, especially during addition or subtraction, establishing a common denominator is key. The common denominator is essentially a shared base that allows you to combine different fractions into a single expression. Look at it like finding a common language between the fractions, which lets them "speak" to each other more effectively.

The key steps to finding a common denominator are:

• Identify the denominators of each fraction.
• Multiply them together to create a single, uniform denominator. For example, if you have fractions with denominators \(4x+3\) and \(2x-5\), their common denominator becomes \((4x+3)(2x-5)\).

Understanding and finding a common denominator helps keep your rational expressions manageable and ensures accurate calculations.

Numerator and Denominator

In any fraction or rational expression, you'll find a numerator on the top and a denominator on the bottom. They work together to define the value of the fraction.

The numerator acts as the "say-top," telling you how many parts you have, while the denominator is the "say-down," indicating into how many equal parts the whole is divided. In rational expressions, this often means:

• Numerators need adjusting to match the common denominator during combination.
• Denominators provide a basis for ensuring expressions are correctly structured and combinable.

Remember, the primary goal is to create a situation where both numerators sit over a common denominator, allowing for their easy addition or subtraction.

Simplifying Expressions

Simplifying expressions is like cleaning up your room; it makes everything clearer and easier to work with. The idea is to reduce the expression to its simplest form, where it's as understandable as possible without altering its value.

To simplify a rational expression:

• Factor out any common elements in the numerator, if possible.
• Ensure there are no further simplifications available, particularly in terms of factoring or canceling common terms in the numerator and denominator.

In our example, simplifying involved factoring the expression \(14x + 30\) to \(2(7x + 15)\). This clean-up step ensures the expression is straightforward and removes any unnecessary complexity.

Factoring Numerators

Factoring numerators is an essential skill that helps in both simplifying expressions and solving equations. It involves breaking down the numerator into simpler, multiplicative components.

Here's how to go about factoring numerators:

• Identify any common factors shared by the terms in the numerator.
• Rewrite the numerator as a product of these factors with the resulting expression.

In the exercise, the numerator \(14x + 30\) was factored by identifying the common factor 2, giving us \(2(7x + 15)\). This step not only simplifies the expression but also prepares it for any potential future simplifications, should the denominator allow further reductions.