1. Solve the rational equation by following the given steps.
2/x²-1  -   1/x-1= 1/2

Step 1. Find the Least Common Denominator (LCD).
Step 2. Multiply both sides of the equation by its the
LCD.
Step 3. Apply the Distributive Property and then
simplify.
Step 4. Find all the possible values of x.
Step 5. Check each value by substituting into
original equation and reject any extraneous root/s.

Let's solve the given rational equation step by step.

### Given equation:
\[
\frac{2}{x^2 - 1} - \frac{1}{x - 1} = \frac{1}{2}
\]

### Step 1: Find the Least Common Denominator (LCD)

The denominators in the equation are:
1. \(x^2 - 1\)
2. \(x - 1\)
3. 2

First, factor \(x^2 - 1\):
\[
x^2 - 1 = (x - 1)(x + 1)
\]
So, the LCD is the product of all distinct factors:
\[
\text{LCD} = 2(x - 1)(x + 1)
\]

### Step 2: Multiply both sides by the LCD

Multiply the entire equation by the LCD \(2(x - 1)(x + 1)\):
\[
2(x - 1)(x + 1) \left( \frac{2}{(x - 1)(x + 1)} - \frac{1}{x - 1} \right) = 2(x - 1)(x + 1) \left( \frac{1}{2} \right)
\]

### Step 3: Apply the Distributive Property and simplify

Simplifying each term:
- First term:
\[
2(x - 1)(x + 1) \cdot \frac{2}{(x - 1)(x + 1)} = 4
\]
- Second term:
\[
2(x - 1)(x + 1) \cdot \frac{1}{x - 1} = 2(x + 1)
\]
- Right side:
\[
2(x - 1)(x + 1) \cdot \frac{1}{2} = (x - 1)(x + 1) = x^2 - 1
\]

So, the equation becomes:
\[
4 - 2(x + 1) = x^2 - 1
\]

### Step 4: Find all possible values of \(x\)

Simplify the equation:
\[
4 - 2x - 2 = x^2 - 1
\]
\[
2 - 2x = x^2 - 1
\]
Rearrange all terms to one side:
\[
0 = x^2 + 2x - 3
\]
Solve the quadratic equation:
\[
x^2 + 2x - 3 = 0
\]
Factor the quadratic:
\[
(x + 3)(x - 1) = 0
\]
Thus, the possible solutions are:
\[
x = -3 \quad \text{or} \quad x = 1
\]

### Step 5: Check for extraneous roots

Substitute \(x = -3\) into the original equation:
\[
\frac{2}{(-3)^2 - 1} - \frac{1}{-3 - 1} = \frac{1}{2}
\]
\[
\frac{2}{9 - 1} - \frac{1}{-4} = \frac{1}{2}
\]
\[
\frac{2}{8} + \frac{1}{4} = \frac{1}{2}
\]
\[
\frac{1}{4} + \frac{1}{4} = \frac{1}{2}
\]
True. So, \(x = -3\) is a valid solution.

Now, substitute \(x = 1\):
\[
\frac{2}{1^2 - 1} - \frac{1}{1 - 1} = \frac{1}{2}
\]
The second term involves division by zero, which is undefined. So, \(x = 1\) is an extraneous solution and must be rejected.

### Final Answer:
The only solution is:
\[
x = -3
\]

Would you like to go over any details, or have further questions? 

---

Here are 5 related questions to extend your understanding:
1. How do we find the LCD of more complex rational expressions?
2. What are the steps to solving equations with multiple extraneous solutions?
3. How can we factor quadratics like \(x^2 + 2x - 3\)?
4. What role do restrictions on domain play in rational equations?
5. How do we handle equations where all potential solutions are extraneous?

**Tip:** Always check for extraneous solutions by substituting them back into the original equation, especially when there are restrictions like division by zero!

Solve the rational equation by following the given steps. 2/x²-1  -   1/x-1 = 1/2. Step 1. Find the Least Common Denominator (LCD). Step 2. Multiply both sides of the equation by its LCD. Step 3. Apply the Distributive Property and then simplify. Step 4. Find all the possible values of x. Step 5. Check each value by substituting into original equation and reject any extraneous root/s.

This page provides a step-by-step solution to the rational equation 2/x²-1 - 1/x-1 = 1/2. Learn how to find the Least Common Denominator (LCD), apply the Distributive Property, solve quadratic equations, and check for extraneous solutions. Suitable for students in Grades 8-10.

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