
Solving Rational Equations and Inequalities

Before you proceed on the lesson proper try to answer the rational equations and inequality using guided procedure. You can synthesize yotir own steps in solving the problem. You can refer to previous activities if you are having differuiry prog arithmetic properties. Hope you enjoy answering before you continue to the next past of the discussion.

1. Solve example of the rational equation by following the oven steps

1. Find the Least Common Denominator (LCD).


2. Multiply both sides of the equation by its the LCD
3.solve
        2/x²+1 - 1/x-1 = 1/2

Let's break down and solve the 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 are \( x^2 + 1 \), \( x - 1 \), and 2. The LCD is simply the product of these distinct denominators:

\[
\text{LCD} = 2(x^2 + 1)(x - 1)
\]

### Step 2: Multiply both sides of the equation by the LCD.

Multiply each term by the LCD to eliminate the fractions:

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

This simplifies to:

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

### Step 3: Solve the equation.

Expand and simplify each side:

Left side:
\[
4x - 4 - 2x^2 - 2
\]
Which becomes:
\[
-2x^2 + 4x - 6
\]

Right side:
First, expand \( (x^2 + 1)(x - 1) \):
\[
(x^2 + 1)(x - 1) = x^3 - x^2 + x - 1
\]

Now the equation is:
\[
-2x^2 + 4x - 6 = x^3 - x^2 + x - 1
\]

Bring all terms to one side:
\[
0 = x^3 + x^2 - 3x + 5
\]

### Step 4: Solve the cubic equation.

To solve this cubic equation, we would normally use methods like factoring, synthetic division, or numerical techniques. However, this specific cubic does not factor nicely, so we might need to rely on numerical approximation or graphing to find the roots.

### Final Answer:

The exact roots of the equation are complex and not easily factored, so you would typically approximate them using numerical methods or a calculator.

Would you like more details on the steps or further explanation on how to approximate the roots?

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Here are five related questions for further practice:

1. Solve the rational equation: \( \frac{3}{x + 2} - \frac{1}{x - 3} = \frac{2}{x^2 - x - 6} \).
2. What are the steps to determine extraneous solutions in a rational equation?
3. How can you solve rational inequalities using a sign chart?
4. Why is it important to find the LCD when solving rational equations?
5. What is the difference between solving rational equations and solving rational inequalities?

**Tip:** Always check your solutions in the original equation to ensure they don't make any denominator zero.

Learn how to solve rational equations and inequalities step-by-step. Includes finding LCD, multiplying by LCD, solving cubic equations, and more. Suitable for high school students studying advanced algebra.

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