Problem 9 - Algebra 1

Question

Simplify the expression. $$\frac{x-2}{2 x-10}+\frac{x+3}{x-5}$$

Step-by-Step Solution

Answer

\(\frac{3x}{2} - 1\)

Step 1: Identify the common denominator

The denominator of the first term is \(2x-10\), which can be written as \(2*(x-5)\). Now, it's clear to see the common denominator of both fractions, which is \(x-5\).

Step 2: Rewrite the fractions with the common denominator

Now, rewrite the first fraction as follows: \(\frac{(x-2)/2}{(2x-10)/2} = \frac{x-2}{2*(x-5)}\). Now, both fractions are \(\frac{x-2}{2*(x-5)}\) and \(\frac{x+3}{x-5}\).

Step 3: Simplify the first fraction

Now, simplify the first fraction to get: \(\frac{x-2}{2*(x-5)} = \frac{x-2}{2} * \frac{1}{x-5}\). This fractions simplify to: \(\frac{x-2}{2} * \frac{1}{x-5} = \frac{x-2}{2} * \frac{1}{x-5} = \frac{x}{2} - \frac{2}{2} = \frac{x}{2} - 1\).

Step 4: Add the two fractions

Now we can add the two simplified fractions because they have the common denominator: \(\frac{x}{2} - 1 + \frac{x+3}{x-5} = \frac{3x}{2} - 1\).

Key Concepts

Common Denominator

When you are adding algebraic fractions, finding a common denominator is an essential step. Think about it like uniting different fractions under a single roof, where they can be easily combined.

• Both expressions need to have the same denominator to be added or subtracted effectively.
• For rational expressions, finding a common denominator helps to manage variables systematically.

In our exercise, the fractions \(\frac{x-2}{2x-10}\) and \(\frac{x+3}{x-5}\) needed a common base. By breaking down \(2x-10\) into \(2*(x-5)\), you can see both terms involve the factor \(x-5\), showing that it is a shared factor. This shared factor becomes the common denominator.
This step allows us to phrase the terms in such a way where they can be easily combined.

Simplifying Fractions

Before diving into adding fractions together, it is crucial to simplify each one as much as possible. Simplifying makes the math easier and helps you to see the relationships between parts more clearly.

• Simplification involves reducing the fraction to its smallest form.
• This sometimes means factoring out common terms or reducing coefficients.

Focusing back on our algebraic fraction \(\frac{x-2}{2x-10}\), once you noted \(2x-10 = 2*(x-5)\), simplifying further gave \(\frac{x-2}{2(x-5)}\) which is broken down to make future calculations smoother.
Each term is now manageable and ready to be combined with others.

Adding Fractions

Adding fractions, especially with algebraic terms, requires that all parts are lined up perfectly, just like tuning instruments before a symphony. When you achieve a common denominator, you pave the way for smooth addition.

• Combine fractions only after ensuring they share the same denominator.
• Add the numerators, keeping the denominator as it is.

In our problem, both simplified fractions share the common denominator of \(x-5\). This commonality allows for the numerators to be combined directly.
The result for the expression \(\frac{x-2}{2} * \frac{1}{x-5} + \frac{x+3}{x-5}\) simplifies elegantly into a single expression because of this groundwork.
Thus, understanding the mechanics of adding algebraic fractions proves invaluable for conquering algebra challenges.