Subtract And State The Difference In Simplest Form.$\frac{6}{x-3}-\frac{9}{x-2}$A. $\frac{-3x-39}{x^2-5x+6}$ B. $\frac{-3x-2}{x^2-5x+6}$ C. $\frac{-3}{x^2-5x+6}$ D. $\frac{-3x+15}{x^2-5x+6}$

Introduction

In mathematics, subtracting and simplifying fractions is a fundamental concept that requires a clear understanding of the rules and procedures involved. When dealing with fractions, it's essential to follow the correct order of operations and simplify the resulting expression to its simplest form. In this article, we will explore the process of subtracting and simplifying fractions, using the given problem as a case study.

The Problem

The problem we will be working with is:

$\frac{6}{x-3}-\frac{9}{x-2}$

Our goal is to subtract the two fractions and simplify the resulting expression to its simplest form.

Step 1: Find a Common Denominator

To subtract fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are $x-3$ and $x-2$. To find the LCM, we can list the multiples of each denominator:

Multiples of $x-3$: $x-3, 2(x-3), 3(x-3), ...$ Multiples of $x-2$: $x-2, 2(x-2), 3(x-2), ...$

The first multiple that appears in both lists is $2(x-3) = 2x-6$ and $2(x-2) = 2x-4$. However, we can see that $2x-6$ is not a multiple of $x-2$, but $2x-6$ is a multiple of $2(x-2)$. Therefore, the LCM of $x-3$ and $x-2$ is $2(x-2)(x-3)$.

However, we can simplify this by multiplying the two expressions by the least common multiple of the two denominators, which is $(x-2)(x-3)$.

Step 2: Rewrite the Fractions with the Common Denominator

Now that we have found the common denominator, we can rewrite each fraction with the common denominator:

$\frac{6}{x-3} = \frac{6(x-2)}{(x-2)(x-3)}$ $\frac{9}{x-2} = \frac{9(x-3)}{(x-2)(x-3)}$

Step 3: Subtract the Fractions

Now that we have rewritten the fractions with the common denominator, we can subtract them:

$\frac{6(x-2)}{(x-2)(x-3)} - \frac{9(x-3)}{(x-2)(x-3)}$

To subtract fractions, we need to subtract the numerators and keep the same denominator:

$\frac{6(x-2) - 9(x-3)}{(x-2)(x-3)}$

Step 4: Simplify the Numerator

Now that we have subtracted the fractions, we can simplify the numerator:

$6(x-2) - 9(x-3) = 6x-12-9x+27 = -3x+15$

So, the simplified expression is:

$\frac{-3x+15}{(x-2)(x-3)}$

Step 5: Factor the Denominator

The denominator can be factored as:

$(x-2)(x-3) = x^2-5x+6$

So, the final simplified expression is:

$\frac{-3x+15}{x^2-5x+6}$

Conclusion

In this article, we have walked through the process of subtracting and simplifying fractions, using the given problem as a case study. We have found a common denominator, rewritten the fractions with the common denominator, subtracted the fractions, simplified the numerator, and factored the denominator. The final simplified expression is:

$\frac{-3x+15}{x^2-5x+6}$

This is the correct answer, which is option D.

Discussion

This problem requires a clear understanding of the rules and procedures involved in subtracting and simplifying fractions. It's essential to follow the correct order of operations and simplify the resulting expression to its simplest form. The common denominator is a critical concept in this problem, and finding the LCM of the denominators is a crucial step in the process.

In addition, this problem requires the ability to factor the denominator, which is an essential skill in algebra. Factoring the denominator allows us to simplify the expression and write it in its simplest form.

Overall, this problem is a great example of the importance of following the correct order of operations and simplifying expressions to their simplest form. It requires a clear understanding of the rules and procedures involved in subtracting and simplifying fractions, as well as the ability to factor the denominator.

Answer Key

The correct answer is:

$\frac{-3x+15}{x^2-5x+6}$

This is option D.