Solve `(1)/(x-2)+(1)/(x-1) gt (1)/(x)`.

Explanation

To solve the inequality \(\frac{1}{x-2} + \frac{1}{x-1} > \frac{1}{x}\), we can follow these steps:

### Step 1: Rearranging the Inequality
First, we will move \(\frac{1}{x}\) to the left side of the inequality:
\[ \frac{1}{x-2} + \frac{1}{x-1} - \frac{1}{x} > 0 \]

### Step 2: Finding a Common Denominator
The common denominator for the fractions is \((x-2)(x-1)x\). We can rewrite the left side:
\[ \frac{(x)(1) + (x)(1) - ((x-2)(x-1))}{(x-2)(x-1)x} > 0 \]
This simplifies to:
\[ \frac{x + x - (x^2 - 3x + 2)}{(x-2)(x-1)x} > 0 \]

### Step 3: Simplifying the Numerator
Now, simplify the numerator:
\[ 2x - (x^2 - 3x + 2) = 2x - x^2 + 3x - 2 = -x^2 + 5x - 2 \]
So the inequality becomes:
\[ \frac{-x^2 + 5x - 2}{(x-2)(x-1)x} > 0 \]

### Step 4: Factoring the Numerator
Next, we can factor the quadratic \(-x^2 + 5x - 2\):
\[ -x^2 + 5x - 2 = -(x^2 - 5x + 2) \]
To factor \(x^2 - 5x + 2\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{5 \pm \sqrt{25 - 8}}{2} = \frac{5 \pm \sqrt{17}}{2} \]
Let \(r_1 = \frac{5 - \sqrt{17}}{2}\) and \(r_2 = \frac{5 + \sqrt{17}}{2}\).

### Step 5: Setting Up the Number Line
Now we have critical points at \(0\), \(1\), \(2\), \(r_1\), and \(r_2\). We will test intervals around these points to determine where the expression is positive.

### Step 6: Testing Intervals
1. **Interval \((-∞, 0)\)**: Choose \(x = -1\):
\[ \frac{-(-1)^2 + 5(-1) - 2}{(-1-2)(-1-1)(-1)} = \frac{-1 - 5 - 2}{(-3)(-2)(-1)} < 0 \]

2. **Interval \((0, r_1)\)**: Choose \(x = 1\):
\[ \frac{-(1)^2 + 5(1) - 2}{(1-2)(1-1)(1)} = \text{undefined} \]

3. **Interval \((r_1, 1)\)**: Choose \(x = 0.5\):
\[ \frac{-0.25 + 2.5 - 2}{(-1.5)(-0.5)(0.5)} > 0 \]

4. **Interval \((1, r_2)\)**: Choose \(x = 1.5\):
\[ \frac{-(1.5)^2 + 5(1.5) - 2}{(1.5-2)(1.5-1)(1.5)} < 0 \]

5. **Interval \((r_2, 2)\)**: Choose \(x = 1.8\):
\[ \frac{-(1.8)^2 + 5(1.8) - 2}{(1.8-2)(1.8-1)(1.8)} > 0 \]

6. **Interval \((2, ∞)\)**: Choose \(x = 3\):
\[ \frac{-(3)^2 + 5(3) - 2}{(3-2)(3-1)(3)} > 0 \]

### Step 7: Conclusion
The solution to the inequality is:
\[ x \in \left(0, \frac{5 - \sqrt{17}}{2}\right) \cup \left(2, \infty\right) \]