(x-1)(x-4)(x-9)>0

To solve the inequality \((x-1)(x-4)(x-9) > 0\), we need to determine where the product is positive.

Step 1: Find the roots

The roots of the inequality occur when each factor equals zero:

1. \(x - 1 = 0 \Rightarrow x = 1\)
2. \(x - 4 = 0 \Rightarrow x = 4\)
3. \(x - 9 = 0 \Rightarrow x = 9\)

Thus, the roots are \(x = 1\), \(x = 4\), and \(x = 9\). These points will divide the number line into intervals.

Step 2: Analyze the sign in each interval

The intervals to consider (created by the roots) are:

1. \((-∞, 1)\)
2. \((1, 4)\)
3. \((4, 9)\)
4. \((9, ∞)\)

We will test a point from each interval to determine if the product \((x-1)(x-4)(x-9)\) is positive or negative.

• Interval \((-∞, 1)\): Choose \(x = 0\) \[ (0-1)(0-4)(0-9) = (-1)(-4)(-9) = -36 \quad (\text{negative}) \]

• Interval \((1, 4)\): Choose \(x = 2\) \[ (2-1)(2-4)(2-9) = (1)(-2)(-7) = 14 \quad (\text{positive}) \]

• Interval \((4, 9)\): Choose \(x = 5\) \[ (5-1)(5-4)(5-9) = (4)(1)(-4) = -16 \quad (\text{negative}) \]

• Interval \((9, ∞)\): Choose \(x = 10\) \[ (10-1)(10-4)(10-9) = (9)(6)(1) = 54 \quad (\text{positive}) \]

Step 3: Collect the results

From the tests, we find:

• The expression is negative on \((-∞, 1)\).
• The expression is positive on \((1, 4)\).
• The expression is negative on \((4, 9)\).
• The expression is positive on \((9, ∞)\).

Step 4: Construct the solution

We need the intervals where \((x-1)(x-4)(x-9) > 0\):

• From \((1, 4)\)
• From \((9, ∞)\)

Final Solution

The solution to the inequality \((x-1)(x-4)(x-9) > 0\) is: \[ \boxed{(1, 4) \cup (9, ∞)} \]