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(x+2)(x-4)(x-5) 0

3 min read Jun 16, 2024
(x+2)(x-4)(x-5) 0

Solving the Inequality (x+2)(x-4)(x-5) < 0

This article explores how to solve the inequality (x+2)(x-4)(x-5) < 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)(x-4)(x-5) < 0 asks us to find all the values of 'x' that make the product of the three factors negative.

Steps to Solve

  1. 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

  2. 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

  3. Test Intervals: Choose a test value within each interval and substitute it into the original inequality (x+2)(x-4)(x-5) < 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)

  4. Solution: The inequality is true only in the interval 4 < x < 5.

Conclusion

Therefore, the solution set to the inequality (x+2)(x-4)(x-5) < 0 is 4 < x < 5. This means all values of 'x' between 4 and 5 (excluding 4 and 5) make the inequality true.

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