Simplify Fractions: A Step-by-Step Algebraic Guide

Hey everyone! Let's dive into a crucial skill in algebra: performing operations with fractions, especially when they involve variables. We're going to break down a specific problem step-by-step, but the techniques we'll use can be applied to a wide range of similar problems. So, buckle up and let's get started!

The Problem at Hand

Our mission, should we choose to accept it, is to simplify the following expression:

$$\frac{2}{x-2} + \frac{x}{x+9} - \frac{x+20}{x^2+7x-18}$$

This looks a bit daunting at first, but don't worry! We'll tackle it methodically. The key to handling such expressions lies in finding a common denominator. Think of it like adding regular fractions – you need the denominators to match before you can combine the numerators. The same principle applies here, but with the added twist of algebraic expressions.

Step 1: Factoring the Denominators

Factoring denominators is our first key step. To find a common denominator, we need to factor each denominator completely. This will help us identify the least common denominator (LCD), which is the smallest expression that all the denominators divide into evenly. Let's start by looking at the denominators individually:

• The first denominator, x - 2, is already in its simplest form. It's a linear expression and cannot be factored further.
• The second denominator, x + 9, is also a linear expression and is already factored.
• The third denominator, x² + 7x - 18, is a quadratic expression. This is where we need to put our factoring skills to work. We're looking for two numbers that multiply to -18 and add up to 7. Those numbers are 9 and -2. Therefore, we can factor the quadratic as (x + 9)(x - 2). This step is crucial because it reveals a common factor between the third denominator and the first two denominators.

Now, let's rewrite the original expression with the factored denominator:

$$\frac{2}{x-2} + \frac{x}{x+9} - \frac{x+20}{(x+9)(x-2)}$$

See how much clearer things are now? We've identified the building blocks of our denominators, which will make finding the LCD much easier.

Step 2: Finding the Least Common Denominator (LCD)

Now that we've factored the denominators, we can find the least common denominator (LCD). The LCD is the smallest expression that is divisible by all the denominators. In our case, the denominators are (x - 2), (x + 9), and (x + 9)(x - 2). To construct the LCD, we need to include each unique factor the greatest number of times it appears in any one denominator.

• The factor (x - 2) appears once in the first denominator and once in the third denominator.
• The factor (x + 9) appears once in the second denominator and once in the third denominator.

Therefore, the LCD is simply the product of these factors: (x - 2)(x + 9). This is the expression we'll use to rewrite each fraction with a common denominator.

Step 3: Rewriting Fractions with the LCD

This is where the magic happens! We're going to rewrite each fraction with the LCD as its denominator. To do this, we'll multiply the numerator and denominator of each fraction by the factors needed to obtain the LCD. Remember, multiplying the numerator and denominator by the same expression is equivalent to multiplying by 1, so we're not changing the value of the fraction, just its form.

• For the first fraction, 2/(x - 2), we need to multiply the numerator and denominator by (x + 9) to get the LCD:

  $$\frac{2}{x-2} \cdot \frac{x+9}{x+9} = \frac{2(x+9)}{(x-2)(x+9)}$$

• For the second fraction, x/(x + 9), we need to multiply the numerator and denominator by (x - 2) to get the LCD:

  $$\frac{x}{x+9} \cdot \frac{x-2}{x-2} = \frac{x(x-2)}{(x-2)(x+9)}$$

• The third fraction, (x + 20)/((x + 9)(x - 2)), already has the LCD as its denominator, so we don't need to do anything to it.

Now our expression looks like this:

$$\frac{2(x+9)}{(x-2)(x+9)} + \frac{x(x-2)}{(x-2)(x+9)} - \frac{x+20}{(x+9)(x-2)}$$

See how all the denominators are the same? We're ready to combine the numerators!

Step 4: Combining the Numerators

Now for the fun part: combining the numerators! Since all the fractions have the same denominator, we can simply add and subtract the numerators, keeping the denominator the same. Let's write it all out:

$$\frac{2(x+9) + x(x-2) - (x+20)}{(x-2)(x+9)}$$

Now, we need to simplify the numerator by expanding the products and combining like terms. This is where careful algebra is essential.

• First, distribute the 2 in the first term: 2(x + 9) = 2x + 18
• Next, distribute the x in the second term: x(x - 2) = x² - 2x
• Remember to distribute the negative sign in the third term: -(x + 20) = -x - 20

Now our numerator looks like this:

$$2x + 18 + x^2 - 2x - x - 20$$

Let's combine the like terms:

• The 2x and -2x terms cancel each other out.
• Combining the -x term, we have -x.
• Combining the constant terms, 18 - 20 = -2

So, the simplified numerator is x² - x - 2. Our expression now looks like this:

$$\frac{x^2 - x - 2}{(x-2)(x+9)}$$

Step 5: Simplifying the Result

We're almost there! The final step is to simplify the result as much as possible. This often involves factoring the numerator and seeing if any factors cancel with factors in the denominator. Let's factor the numerator, x² - x - 2. We need to find two numbers that multiply to -2 and add up to -1. Those numbers are -2 and 1. So, we can factor the numerator as (x - 2)(x + 1). Our expression now becomes:

$$\frac{(x-2)(x+1)}{(x-2)(x+9)}$$

Now we can see a common factor of (x - 2) in both the numerator and the denominator. We can cancel these factors out, as long as x ≠ 2 (because we can't divide by zero). This gives us our final simplified expression:

$$\frac{x+1}{x+9}$$

Final Answer

Therefore, the simplified form of the expression $\frac{2}{x-2} + \frac{x}{x+9} - \frac{x+20}{x^2+7x-18}$ is $\frac{x+1}{x+9}$, where $x \ne 2$.

Key Takeaways

• Factoring is your friend: Factoring the denominators is crucial for finding the LCD.
• LCD is the key: The least common denominator allows you to combine fractions.
• Simplify, simplify, simplify: Always simplify your final answer by canceling common factors.
• Watch out for restrictions: Remember to note any values of x that would make the denominator zero.

By following these steps, you can confidently tackle complex algebraic fractions. Keep practicing, and you'll become a pro in no time!