X^2-3x+2/x^2-5x+6 +x^2-12x+20/ x^2-13x+30  >o

The given expression is:

\[
\frac{x^2 - 3x + 2}{x^2 - 5x + 6} + \frac{x^2 - 12x + 20}{x^2 - 13x + 30}
\]

We need to determine when this expression is greater than zero.

### Step 1: Factorize each quadratic expression

#### Numerators:
1. \( x^2 - 3x + 2 \) factors as \( (x - 1)(x - 2) \).
2. \( x^2 - 12x + 20 \) factors as \( (x - 2)(x - 10) \).

#### Denominators:
1. \( x^2 - 5x + 6 \) factors as \( (x - 2)(x - 3) \).
2. \( x^2 - 13x + 30 \) factors as \( (x - 3)(x - 10) \).

So, the expression becomes:

\[
\frac{(x - 1)(x - 2)}{(x - 2)(x - 3)} + \frac{(x - 2)(x - 10)}{(x - 3)(x - 10)}
\]

### Step 2: Simplify the expression
1. In the first fraction, cancel out the common factor \((x - 2)\):

   \[
   \frac{x - 1}{x - 3}
   \]

2. In the second fraction, cancel out the common factor \((x - 10)\):

   \[
   \frac{x - 2}{x - 3}
   \]

So the expression simplifies to:

\[
\frac{x - 1}{x - 3} + \frac{x - 2}{x - 3}
\]

### Step 3: Combine the fractions

Since the denominators are the same, combine the numerators:

\[
\frac{(x - 1) + (x - 2)}{x - 3} = \frac{2x - 3}{x - 3}
\]

### Step 4: Determine when the expression is greater than zero
We need to solve the inequality:

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

The sign of a rational expression like this depends on the signs of the numerator and denominator:

1. **Numerator:** \(2x - 3 > 0 \) implies \( x > \frac{3}{2} \).
2. **Denominator:** \(x - 3 > 0 \) implies \( x > 3 \).

So, the expression is positive when \( x > 3 \).

However, we also need to consider the points where the expression is undefined or zero. 

- The expression is undefined at \(x = 3\) because the denominator is zero.
- The expression equals zero when the numerator \(2x - 3 = 0\), which gives \(x = \frac{3}{2}\).

### Conclusion:
The inequality \( \frac{2x - 3}{x - 3} > 0 \) holds true when \(x > 3\). At \(x = 3\), the expression is undefined. For \(x = \frac{3}{2}\), the expression is zero.

Would you like any additional details or have any questions?

### Relative Questions:
1. What happens when \(x = 3\)?
2. How do the factors influence the sign of the expression?
3. What if the inequality was \( \frac{2x - 3}{x - 3} \geq 0 \)?
4. How would the solution change if the inequality was less than zero?
5. Can you graph the function \( \frac{2x - 3}{x - 3} \) to visualize the solution?

### Tip:
Always consider the points where the expression is undefined and where the numerator equals zero when solving rational inequalities.

Learn how to solve the rational inequality (2x - 3)/(x - 3) > 0 with a detailed step-by-step solution. This problem covers rational expressions, inequalities, and quadratic equations. Suitable for advanced high school students.

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