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

3 min read Jun 17, 2024

Solving the Inequality (x-1)(x-4)(x-9) > 0

This inequality involves a product of three linear factors, and we need to find the values of x that make the entire product positive. Here's a step-by-step approach to solving this:

1. Find the Critical Points

The critical points are the values of x that make each factor equal to zero. In this case, the critical points are:

x = 1 (from x - 1 = 0)
x = 4 (from x - 4 = 0)
x = 9 (from x - 9 = 0)

2. Divide the Number Line into Intervals

The critical points divide the number line into four intervals:

x < 1
1 < x < 4
4 < x < 9
x > 9

3. Test a Value in Each Interval

We'll choose a test value within each interval and plug it into the inequality to see if it makes the inequality true or false.

Interval 1: x < 1

Test value: x = 0
(0 - 1)(0 - 4)(0 - 9) = 36 > 0 True

Interval 2: 1 < x < 4

Test value: x = 2
(2 - 1)(2 - 4)(2 - 9) = -14 < 0 False

Interval 3: 4 < x < 9

Test value: x = 5
(5 - 1)(5 - 4)(5 - 9) = -16 < 0 False

Interval 4: x > 9

Test value: x = 10
(10 - 1)(10 - 4)(10 - 9) = 54 > 0 True

4. Solution

The inequality (x-1)(x-4)(x-9) > 0 is true for the intervals where the test value resulted in a positive product. Therefore, the solution is:

x < 1 or x > 9

Graphing the Solution

We can represent the solution graphically on a number line. The open circles indicate that the endpoints are not included in the solution.

     <----|----|----|----|----|----|----->
     0     1     4     9
      -----       -----

Conclusion

The inequality (x-1)(x-4)(x-9) > 0 is satisfied by all values of x less than 1 or greater than 9.