solve (3x divided by x^2-4) - ( 4 divided by (2-x))

Let's solve the given expression:

\[
\frac{3x}{x^2 - 4} - \frac{4}{2 - x}
\]

### Step 1: Factor the denominator \( x^2 - 4 \)
The term \( x^2 - 4 \) is a difference of squares and can be factored as:
\[
x^2 - 4 = (x - 2)(x + 2)
\]
So the expression becomes:
\[
\frac{3x}{(x - 2)(x + 2)} - \frac{4}{2 - x}
\]

### Step 2: Simplify the second fraction
Notice that \( 2 - x \) is the same as \( -(x - 2) \). Therefore:
\[
\frac{4}{2 - x} = -\frac{4}{x - 2}
\]

### Step 3: Combine the fractions
Now the expression becomes:
\[
\frac{3x}{(x - 2)(x + 2)} + \frac{4}{x - 2}
\]
To combine these, we need a common denominator, which is \( (x - 2)(x + 2) \). Therefore, rewrite the second fraction:
\[
\frac{4}{x - 2} = \frac{4(x + 2)}{(x - 2)(x + 2)}
\]

So, the expression becomes:
\[
\frac{3x + 4(x + 2)}{(x - 2)(x + 2)}
\]

### Step 4: Simplify the numerator
Expand and combine like terms in the numerator:
\[
3x + 4(x + 2) = 3x + 4x + 8 = 7x + 8
\]

So the final expression is:
\[
\frac{7x + 8}{(x - 2)(x + 2)}
\]

### Final Answer:
\[
\frac{7x + 8}{x^2 - 4}
\]

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Would you like further explanation or details on any step?

Here are five related questions:
1. How do you factor differences of squares?
2. What is the process for finding a common denominator when adding or subtracting fractions?
3. How do you simplify expressions involving variables?
4. Can you explain why \( 2 - x \) equals \( -(x - 2) \)?
5. What are some common pitfalls when simplifying algebraic fractions?

**Tip:** Always look for opportunities to factor and simplify before performing operations with algebraic fractions. It can make the process much easier!

Learn how to solve the algebraic expression 3x/(x^2 - 4) - 4/(2 - x) step-by-step with detailed explanations and examples. This problem involves algebraic fractions, factoring differences of squares, and simplifying expressions. Suitable for students in Grades 9-12.

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