your answers are incomplete, there should be x=1 and x=7/2

Thank you for pointing that out. Let's go through the solution again, step by step, making sure to find the correct critical points, including \(x = 1\) and \(x = \frac{7}{2}\).

### Given Inequality
The inequality is:

\[
\frac{x - 3}{x - 2} > 3 - \frac{2}{x^2 - 5x + 6}
\]

### Step 1: Factor the Quadratic Expression
The quadratic expression \(x^2 - 5x + 6\) factors as:

\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

So, the inequality becomes:

\[
\frac{x - 3}{x - 2} > 3 - \frac{2}{(x - 2)(x - 3)}
\]

### Step 2: Combine the Expressions
To combine the fractions, express both sides with a common denominator:

\[
\frac{x - 3}{x - 2} - \frac{3(x - 2)(x - 3) - 2}{(x - 2)(x - 3)} > 0
\]

### Step 3: Simplify the Right-Hand Side
Expand the expression on the right:

\[
3(x - 2)(x - 3) = 3(x^2 - 5x + 6) = 3x^2 - 15x + 18
\]

So the right-hand side becomes:

\[
\frac{3x^2 - 15x + 18 - 2}{(x - 2)(x - 3)} = \frac{3x^2 - 15x + 16}{(x - 2)(x - 3)}
\]

### Step 4: Simplify the Inequality
Now, subtract the left-hand side from the right-hand side:

\[
\frac{(x - 3)(x - 3) - (3x^2 - 15x + 16)}{(x - 2)(x - 3)} > 0
\]

The numerator simplifies as follows:

\[
(x - 3)^2 = x^2 - 6x + 9
\]

So the numerator becomes:

\[
x^2 - 6x + 9 - 3x^2 + 15x - 16 = -2x^2 + 9x - 7
\]

### Step 5: Factor the Quadratic Numerator
Factor the quadratic expression \(-2x^2 + 9x - 7\):

\[
-2x^2 + 9x - 7 = -(2x^2 - 9x + 7)
\]

To factor \(2x^2 - 9x + 7\), find the roots using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 2\), \(b = -9\), and \(c = 7\):

\[
x = \frac{9 \pm \sqrt{(-9)^2 - 4(2)(7)}}{2(2)} = \frac{9 \pm \sqrt{81 - 56}}{4} = \frac{9 \pm \sqrt{25}}{4} = \frac{9 \pm 5}{4}
\]

This gives:

\[
x = \frac{14}{4} = \frac{7}{2} \quad \text{and} \quad x = \frac{4}{4} = 1
\]

So the quadratic factors as:

\[
-2x^2 + 9x - 7 = -2(x - 1)(x - \frac{7}{2})
\]

### Step 6: Solve the Inequality
The inequality is now:

\[
\frac{-2(x - 1)(x - \frac{7}{2})}{(x - 2)(x - 3)} > 0
\]

### Step 7: Determine the Sign of the Expression
Determine where the expression is positive by testing the sign in the intervals defined by the critical points \(x = 1\), \(x = \frac{7}{2}\), \(x = 2\), and \(x = 3\):

- **Interval \( (-\infty, 1) \)**: Negative
- **Interval \( (1, 2) \)**: Positive
- **Interval \( (2, 3) \)**: Negative (Note: \(x = 2\) and \(x = 3\) are excluded as they make the denominator zero)
- **Interval \( (3, \frac{7}{2}) \)**: Positive
- **Interval \( (\frac{7}{2}, \infty) \)**: Negative

### Step 8: Write the Final Solution
The solution to the inequality is:

\[
1 < x < 2 \quad \text{and} \quad 3 < x < \frac{7}{2}
\]

This excludes the points \(x = 2\) and \(x = 3\) because they make the denominator zero.

Would you like any further details or explanations?

Here are five related questions you might explore:

1. How can the quadratic formula be applied to find roots efficiently?
2. Why is it necessary to consider the sign of the expression in each interval?
3. How does factoring help simplify solving inequalities?
4. What are the implications of undefined points in rational inequalities?
5. How can you use interval testing to verify the solutions of an inequality?

**Tip:** When solving inequalities involving rational expressions, always factor both the numerator and denominator to find critical points where the expression could change sign.

Learn how to solve a rational inequality involving quadratic expressions step-by-step. Understand the concept of critical points and interval testing. This problem covers key algebraic concepts, including inequalities and rational expressions, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation