(x-2)(x-5) 0

Jasonbradley

2 min read

·

Jun 17, 2024

12

Share

Solving the Inequality: (x-2)(x-5) > 0

This article will guide you through solving the inequality (x-2)(x-5) > 0.

Understanding the Inequality

The inequality (x-2)(x-5) > 0 asks us to find the values of x for which the product of (x-2) and (x-5) is positive.

Solving the Inequality

1. Find the critical points: The critical points are the values of x where the expression equals zero. In this case:

   • x - 2 = 0 => x = 2
   • x - 5 = 0 => x = 5

2. Create a sign chart: Divide the number line into intervals using the critical points:

   • Interval 1: x < 2
   • Interval 2: 2 < x < 5
   • Interval 3: x > 5

   Interval	x-2	x-5	(x-2)(x-5)
   x < 2	Negative	Negative	Positive
   2 < x < 5	Positive	Negative	Negative
   x > 5	Positive	Positive	Positive

3. Determine the solution: We are looking for the intervals where (x-2)(x-5) > 0 (i.e., positive). From the sign chart, we see that:

   • x < 2 or x > 5

Solution in Interval Notation

The solution to the inequality (x-2)(x-5) > 0 is: (-∞, 2) ∪ (5, ∞)

Conclusion

By using a sign chart, we effectively determined the values of x that satisfy the inequality (x-2)(x-5) > 0. This approach allows for a clear and organized way to solve similar inequalities.