Math Problem Statement

Which expression is equivalent to (3x)/((x+2)(x+3)) - 2/(x+2)?

Solution

It appears that the image you uploaded contains a math problem related to simplifying a rational expression. The expression is:

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

The question asks which expression is equivalent to the given problem. To solve this, we need to find a common denominator, combine the fractions, and simplify the result.

Let's break it down:

  1. The first term, 3x(x+2)(x+3)\frac{3x}{(x+2)(x+3)}, already has the denominator (x+2)(x+3)(x+2)(x+3).
  2. The second term, 2(x+2)\frac{2}{(x+2)}, needs to be rewritten with the common denominator (x+2)(x+3)(x+2)(x+3). To do this, we multiply the numerator and denominator of the second term by (x+3)(x+3), giving us 2(x+3)(x+2)(x+3)\frac{2(x+3)}{(x+2)(x+3)}.

Now, the expression becomes:

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

Since the denominators are now the same, we can subtract the numerators:

3x2(x+3)(x+2)(x+3)\frac{3x - 2(x+3)}{(x+2)(x+3)}

Next, expand the numerator:

3x2(x+3)=3x2x6=x63x - 2(x+3) = 3x - 2x - 6 = x - 6

So the expression simplifies to:

x6(x+2)(x+3)\frac{x-6}{(x+2)(x+3)}

This matches option B, which is:

x6(x+2)(x+3)\frac{x-6}{(x+2)(x+3)}

Thus, the correct answer is B.

Would you like more details or have any questions about this solution?


Here are some related questions you might find interesting:

  1. How do you simplify a rational expression with different denominators?
  2. What does it mean to find a common denominator in algebra?
  3. How do you perform polynomial expansion in the numerator when simplifying rational expressions?
  4. Can you explain how to deal with higher degree polynomials in rational expressions?
  5. What other strategies can be used to combine rational expressions besides finding a common denominator?

Tip: When combining rational expressions, always look for a common denominator first, as this allows you to combine the terms more easily.

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Math Problem Analysis

Mathematical Concepts

Rational Expressions
Algebra
Simplifying Fractions

Formulas

(a/b) - (c/d) = (ad - bc)/(bd)

Theorems

Common Denominator

Suitable Grade Level

Grades 8-10