Find the solution set of `((x-1)(x-2)(x-3)^(2))/((x-4)^(2)(x-5)^(3)) lt 0`

Answer

To solve the inequality \(\frac{(x-1)(x-2)(x-3)^2}{(x-4)^2(x-5)^3} < 0\), we will follow these steps:

### Step 1: Identify the critical points
The critical points occur when the numerator or denominator is equal to zero. We set each factor in the numerator and denominator to zero:

- \(x - 1 = 0 \Rightarrow x = 1\)
- \(x - 2 = 0 \Rightarrow x = 2\)
- \(x - 3 = 0 \Rightarrow x = 3\) (note that this factor is squared)
- \(x - 4 = 0 \Rightarrow x = 4\) (note that this factor is squared)
- \(x - 5 = 0 \Rightarrow x = 5\) (note that this factor is cubed)

Thus, the critical points are \(x = 1, 2, 3, 4, 5\).

### Step 2: Determine the intervals
The critical points divide the number line into the following intervals:
- \((-∞, 1)\)
- \((1, 2)\)
- \((2, 3)\)
- \((3, 4)\)
- \((4, 5)\)
- \((5, ∞)\)

### Step 3: Test each interval
We will test a point from each interval to determine the sign of the expression in that interval.

1. **Interval \((-∞, 1)\)**: Choose \(x = 0\)
\[ \frac{(0-1)(0-2)(0-3)^2}{(0-4)^2(0-5)^3} = \frac{(-1)(-2)(9)}{(16)(-125)} = \frac{18}{-2000} < 0 \]

2. **Interval \((1, 2)\)**: Choose \(x = 1.5\)
\[ \frac{(1.5-1)(1.5-2)(1.5-3)^2}{(1.5-4)^2(1.5-5)^3} = \frac{(0.5)(-0.5)(2.25)}{(6.25)(-27)} = \frac{-2.8125}{-168.75} > 0 \]

3. **Interval \((2, 3)\)**: Choose \(x = 2.5\)
\[ \frac{(2.5-1)(2.5-2)(2.5-3)^2}{(2.5-4)^2(2.5-5)^3} = \frac{(1.5)(0.5)(0.25)}{(2.25)(-15.625)} = \frac{0.1875}{-35.15625} < 0 \]

4. **Interval \((3, 4)\)**: Choose \(x = 3.5\)
\[ \frac{(3.5-1)(3.5-2)(3.5-3)^2}{(3.5-4)^2(3.5-5)^3} = \frac{(2.5)(1.5)(0.25)}{(0.25)(-11.390625)} = \frac{0.9375}{-2.84765625} < 0 \]

5. **Interval \((4, 5)\)**: Choose \(x = 4.5\)
\[ \frac{(4.5-1)(4.5-2)(4.5-3)^2}{(4.5-4)^2(4.5-5)^3} = \frac{(3.5)(2.5)(1.5^2)}{(0.25)(-0.125)} = \frac{(3.5)(2.5)(2.25)}{-0.03125} > 0 \]

6. **Interval \((5, ∞)\)**: Choose \(x = 6\)
\[ \frac{(6-1)(6-2)(6-3)^2}{(6-4)^2(6-5)^3} = \frac{(5)(4)(9)}{(4)(1)} = 45 > 0 \]

### Step 4: Combine the results
From our tests, we found that the expression is negative in the intervals:
- \((-∞, 1)\)
- \((2, 3)\)
- \((3, 4)\)

### Step 5: Write the solution set
The solution set for the inequality \(\frac{(x-1)(x-2)(x-3)^2}{(x-4)^2(x-5)^3} < 0\) is:
\[ (-\infty, 1) \cup (2, 3) \cup (3, 4) \]