%2F(2x---2)-%2B-3%2F(x-%5E-2---1)---(x-%2B-3)%2F(2x-%2B-2))-*-(4x-%5E-2---4)%2F5.jpeg "((x + 1)/(2x - 2) + 3/(x ^ 2 - 1) - (x + 3)/(2x + 2)) * (4x ^ 2 - 4)/5")
Simplifying Algebraic Expressions: A Step-by-Step Guide
In this article, we will simplify the algebraic expression:
$\left(\frac{x + 1}{2x - 2} + \frac{3}{x^2 - 1} - \frac{x + 3}{2x + 2}\right) \cdot \frac{4x^2 - 4}{5}$
Step 1: Simplify the Fractions
Let's start by simplifying each fraction individually.
1. Simplify $\frac{x + 1}{2x - 2}$
We can factor the numerator and denominator:
$\frac{x + 1}{2x - 2} = \frac{(x + 1)}{2(x - 1)}$
2. Simplify $\frac{3}{x^2 - 1}$
We can factor the denominator:
$\frac{3}{x^2 - 1} = \frac{3}{(x + 1)(x - 1)}$
3. Simplify $\frac{x + 3}{2x + 2}$
We can factor the numerator and denominator:
$\frac{x + 3}{2x + 2} = \frac{(x + 3)}{2(x + 1)}$
Step 2: Combine the Fractions
Now, let's combine the fractions:
$\left(\frac{x + 1}{2(x - 1)} + \frac{3}{(x + 1)(x - 1)} - \frac{x + 3}{2(x + 1)}\right)$
Step 3: Simplify the Overall Expression
Finally, let's multiply the combined fraction by $\frac{4x^2 - 4}{5}$:
$\left(\frac{x + 1}{2(x - 1)} + \frac{3}{(x + 1)(x - 1)} - \frac{x + 3}{2(x + 1)}\right) \cdot \frac{4x^2 - 4}{5}$
After simplifying, we get:
$\boxed{\frac{4(x^2 + 1)}{5(x - 1)(x + 1)}}$
And that's the final simplified form of the algebraic expression!
