Solving the Inequality (x+2)(x4)(x5) < 0
This article explores how to solve the inequality (x+2)(x4)(x5) < 0. This type of inequality involves a polynomial function and requires a methodical approach to find the solution set.
Understanding the Problem
The inequality (x+2)(x4)(x5) < 0 asks us to find all the values of 'x' that make the product of the three factors negative.
Steps to Solve

Find the Critical Points: The critical points are the values of 'x' where the expression equals zero.
 Set each factor to zero and solve:
 x + 2 = 0 => x = 2
 x  4 = 0 => x = 4
 x  5 = 0 => x = 5
 Set each factor to zero and solve:

Number Line: Draw a number line and mark the critical points (2, 4, and 5) on it. These points divide the number line into four intervals:
 x < 2
 2 < x < 4
 4 < x < 5
 x > 5

Test Intervals: Choose a test value within each interval and substitute it into the original inequality (x+2)(x4)(x5) < 0. Determine if the inequality is true or false.

x < 2: Let x = 3. Then (3 + 2)(3  4)(3  5) = (1)(7)(8) < 0 (False)

2 < x < 4: Let x = 0. Then (0 + 2)(0  4)(0  5) = (2)(4)(5) < 0 (False)

4 < x < 5: Let x = 4.5. Then (4.5 + 2)(4.5  4)(4.5  5) = (6.5)(0.5)(0.5) < 0 (True)

x > 5: Let x = 6. Then (6 + 2)(6  4)(6  5) = (8)(2)(1) < 0 (False)


Solution: The inequality is true only in the interval 4 < x < 5.
Conclusion
Therefore, the solution set to the inequality (x+2)(x4)(x5) < 0 is 4 < x < 5. This means all values of 'x' between 4 and 5 (excluding 4 and 5) make the inequality true.