Simplify: X + 3 X − 6 + 1 X − 2 \frac{x+3}{x-6}+\frac{1}{x-2} X − 6 X + 3 + X − 2 1 A) X ( X + 2 ) ( X − 6 ) ( X − 2 ) \frac{x(x+2)}{(x-6)(x-2)} ( X − 6 ) ( X − 2 ) X ( X + 2 ) B) 1 ( X − 6 ) ( X − 2 ) \frac{1}{(x-6)(x-2)} ( X − 6 ) ( X − 2 ) 1 C) X ( X − 6 ) ( X − 2 ) \frac{x}{(x-6)(x-2)} ( X − 6 ) ( X − 2 ) X D) X 2 + 2 X − 12 ( X − 6 ) ( X − 2 ) \frac{x^2+2x-12}{(x-6)(x-2)} ( X − 6 ) ( X − 2 ) X 2 + 2 X − 12

Mar 8, 2025 by ADMIN 427 views

$Simplify: $\frac{x+3}{x-6}+\frac{1}{x-2}$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including fractions, algebraic manipulation, and factoring. In this article, we will focus on simplifying a given algebraic expression, which involves adding two fractions with different denominators. We will use various techniques, including factoring, finding common denominators, and simplifying fractions, to arrive at the final simplified expression.

Step 1: Factor the Denominators

The given expression is $\frac{x+3}{x-6}+\frac{1}{x-2}$ . To simplify this expression, we need to factor the denominators. The first denominator, $x-6$ , can be factored as $(x-6)$ , and the second denominator, $x-2$ , can also be factored as $(x-2)$ .

Step 2: Find the Common Denominator

To add the two fractions, we need to find a common denominator. The common denominator is the product of the two denominators, which is $(x-6)(x-2)$ . We can rewrite each fraction with this common denominator by multiplying the numerator and denominator of each fraction by the necessary factors.

Step 3: Rewrite the Fractions with the Common Denominator

We can rewrite the first fraction as $\frac{(x+3)(x-2)}{(x-6)(x-2)}$ and the second fraction as $\frac{(x-6)}{(x-6)(x-2)}$ . Now that both fractions have the same denominator, we can add them together.

Step 4: Add the Fractions

To add the fractions, we need to add the numerators while keeping the common denominator the same. The numerator of the first fraction is $(x+3)(x-2)$ , and the numerator of the second fraction is $(x-6)$ . We can add these two numerators together to get $(x+3)(x-2) + (x-6)$ .

Step 5: Simplify the Numerator

To simplify the numerator, we need to expand the product $(x+3)(x-2)$ and then combine like terms. Expanding the product gives us $x^2 + x - 6$ . Now we can combine this with the other term, $(x-6)$ , to get $x^2 + x - 6 + x - 6$ .

Step 6: Combine Like Terms

Combining like terms in the numerator gives us $x^2 + 2x - 12$ . Now that we have simplified the numerator, we can rewrite the expression as $\frac{x^2 + 2x - 12}{(x-6)(x-2)}$ .

Conclusion

In this article, we simplified the given algebraic expression $\frac{x+3}{x-6}+\frac{1}{x-2}$ by factoring the denominators, finding the common denominator, rewriting the fractions with the common denominator, adding the fractions, simplifying the numerator, and combining like terms. The final simplified expression is $\frac{x^2 + 2x - 12}{(x-6)(x-2)}$ .

Discussion

The correct answer is D) $\frac{x^2+2x-12}{(x-6)(x-2)}$ . This is the final simplified expression that we arrived at after following the steps outlined in this article. The other options, A) $\frac{x(x+2)}{(x-6)(x-2)}$ , B) $\frac{1}{(x-6)(x-2)}$ , and C) $\frac{x}{(x-6)(x-2)}$ , are not correct and do not represent the simplified expression.

Final Answer

The final answer is D) $\frac{x^2+2x-12}{(x-6)(x-2)}$ .

Introduction

In our previous article, we simplified the given algebraic expression $\frac{x+3}{x-6}+\frac{1}{x-2}$ by factoring the denominators, finding the common denominator, rewriting the fractions with the common denominator, adding the fractions, simplifying the numerator, and combining like terms. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to factor the denominators. In this case, we need to factor the denominators $x-6$ and $x-2$ .

Q: Why do we need to find the common denominator?

A: We need to find the common denominator to add the two fractions together. The common denominator is the product of the two denominators, which is $(x-6)(x-2)$ .

Q: How do we rewrite the fractions with the common denominator?

A: We can rewrite the first fraction as $\frac{(x+3)(x-2)}{(x-6)(x-2)}$ and the second fraction as $\frac{(x-6)}{(x-6)(x-2)}$ . Now that both fractions have the same denominator, we can add them together.

Q: What is the next step after rewriting the fractions with the common denominator?

A: The next step is to add the fractions together. We need to add the numerators while keeping the common denominator the same.

Q: How do we simplify the numerator?

A: To simplify the numerator, we need to expand the product $(x+3)(x-2)$ and then combine like terms. Expanding the product gives us $x^2 + x - 6$ . Now we can combine this with the other term, $(x-6)$ , to get $x^2 + x - 6 + x - 6$ .

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{x^2 + 2x - 12}{(x-6)(x-2)}$ .

Q: Why is the final simplified expression important?

A: The final simplified expression is important because it represents the simplest form of the given algebraic expression. It can be used to solve equations and inequalities, and it can also be used to graph functions.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

Not factoring the denominators
Not finding the common denominator
Not rewriting the fractions with the common denominator
Not adding the fractions together
Not simplifying the numerator
Not combining like terms

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given algebraic expression $\frac{x+3}{x-6}+\frac{1}{x-2}$ . We covered topics such as factoring the denominators, finding the common denominator, rewriting the fractions with the common denominator, adding the fractions together, simplifying the numerator, and combining like terms. We also discussed the importance of the final simplified expression and some common mistakes to avoid when simplifying algebraic expressions.

Discussion

Final Answer

The final answer is D) $\frac{x^2+2x-12}{(x-6)(x-2)}$ .