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

To solve the inequality \(\frac{(x - 1)(x - 2)(x - 3)^2}{(x - 4)^2(x - 5)^3} < 0\), we need to determine where the expression is negative. This involves identifying the critical points and analyzing the sign of the expression in each interval.

Step 1: Identify Critical Points

The critical points occur where the numerator or denominator is zero:

• Numerator: \(x - 1 = 0 \Rightarrow x = 1\)
• Numerator: \(x - 2 = 0 \Rightarrow x = 2\)
• Numerator: \(x - 3 = 0 \Rightarrow x = 3\) (note: this is squared, so it does not change the sign)
• Denominator: \(x - 4 = 0 \Rightarrow x = 4\) (squared, so it does not change the sign)
• Denominator: \(x - 5 = 0 \Rightarrow x = 5\) (cubed, so it changes the sign)

Step 2: Sign Analysis

Now we analyze the sign of the expression in the intervals defined by these critical points:

• Interval: \((-∞, 1)\)
• Interval: \((1, 2)\)
• Interval: \((2, 3)\)
• Interval: \((3, 4)\)
• Interval: \((4, 5)\)
• Interval: \((5, ∞)\)

Testing Each Interval

Choose a test point from each interval to determine the sign of the expression:

• For \((-∞, 1)\), test \(x = 0\): \(\frac{(-)(-)(+)}{(+)(-)} < 0\) (negative)
• For \((1, 2)\), test \(x = 1.5\): \(\frac{(+)(-)(+)}{(+)(-)} > 0\) (positive)
• For \((2, 3)\), test \(x = 2.5\): \(\frac{(+)(+)(+)}{(+)(-)} < 0\) (negative)
• For \((3, 4)\), test \(x = 3.5\): \(\frac{(+)(+)(+)}{(+)(-)} < 0\) (negative)
• For \((4, 5)\), test \(x = 4.5\): \(\frac{(+)(+)(+)}{(-)(-)} > 0\) (positive)
• For \((5, ∞)\), test \(x = 6\): \(\frac{(+)(+)(+)}{(+)(+)} > 0\) (positive)

Step 3: Combine Results

The expression is negative in the intervals \((-∞, 1)\), \((2, 3)\), and \((3, 4)\). We also note that the expression is undefined at \(x = 4\) and \(x = 5\) and does not include these points.

Final Solution Set

The solution set for the inequality is:

• \((-∞, 1)\)
• \((2, 3)\)
• \((3, 4)\)

In interval notation, the solution set is: \((-∞, 1) \cup (2, 3) \cup (3, 4)\).