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Solve Rational Inequalities Problem step by step with a.determine the single rational, b.critical values, c.Number line, d.Fishball method, e.solution set, interval notation;Set notation and Graph.

Let's analyze the given equation step by step to solve the rational inequality.

The inequality provided in the image is:

\[
\frac{2}{x^2 - 1} - \frac{1}{x - 1} = \frac{1}{2}
\]

### Step 1: Determine the Single Rational Expression

First, we need to combine the rational expressions on the left-hand side of the equation. To do this, let's find a common denominator.

The common denominator for \(\frac{2}{x^2 - 1}\) and \(\frac{1}{x - 1}\) is \(x^2 - 1\). Notice that \(x^2 - 1\) can be factored as:

\[
x^2 - 1 = (x - 1)(x + 1)
\]

Thus, the expression becomes:

\[
\frac{2}{(x-1)(x+1)} - \frac{1 \cdot (x+1)}{(x-1)(x+1)} = \frac{1}{2}
\]

Simplify the left-hand side:

\[
\frac{2 - (x + 1)}{(x-1)(x+1)} = \frac{1}{2}
\]

Simplify the numerator:

\[
\frac{1 - x}{(x-1)(x+1)} = \frac{1}{2}
\]

We can rewrite \(1 - x\) as \(-(x - 1)\), so the expression becomes:

\[
\frac{-(x-1)}{(x-1)(x+1)} = \frac{1}{2}
\]

Cancel the \((x-1)\) in the numerator and denominator:

\[
\frac{-1}{x+1} = \frac{1}{2}
\]

### Step 2: Solving the Equation

Cross-multiply to solve for \(x\):

\[
-2 = x + 1
\]

Subtract 1 from both sides:

\[
x = -3
\]

### Step 3: Critical Values

Critical values are the values of \(x\) that make any denominator zero or the points where the expression is undefined. The critical values are:

\[
x = 1, \quad x = -1
\]

### Step 4: Number Line and Fishball Method

On the number line, we have critical points at \(x = -1\), \(x = 1\), and \(x = -3\). These critical points divide the number line into intervals:

- \((-\infty, -1)\)
- \((-1, -3)\)
- \((-3, 1)\)
- \((1, \infty)\)

### Step 5: Solution Set and Interval Notation

Given that the critical points divide the number line into these intervals, we need to check the sign of the expression in each interval to determine where the inequality holds.

The sign of the expression \( \frac{-1}{x+1} \) changes at the critical points \(x = -1\), \(x = 1\), and \(x = -3\).

For the interval that satisfies the inequality \( \frac{-1}{x+1} = \frac{1}{2} \):

The solution is \(x = -3\).

### Step 6: Set Notation and Graph

- **Set Notation**: \(\{x = -3\}\)
- **Interval Notation**: \(x = -3\) (single point, so no interval)

- **Graph**: This solution is a single point at \(x = -3\) on the number line.

Would you like more details on any step or another similar problem?

### Five Related Questions:
1. How do you determine when a rational expression is undefined?
2. How do you combine rational expressions with different denominators?
3. What is the significance of critical values in solving inequalities?
4. How do you graph solutions of rational inequalities on the number line?
5. How do you solve inequalities involving absolute value functions?

### Tip:
When solving rational inequalities, always check for undefined points and consider their impact on the solution set, as they can affect the interval notation and graph.

Solve the rational inequality: \frac{2}{x^2 - 1} - \frac{1}{x - 1} = \frac{1}{2}

This detailed solution walks you through solving the rational inequality \frac{2}{x^2 - 1} - \frac{1}{x - 1} = \frac{1}{2}. Learn about key algebraic concepts such as rational inequalities, critical values, and the fishball method, along with step-by-step explanations. Ideal for high school students in Grades 9-12.

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