Jasonbradley
Home

(x+9)(x-5)^2(x-18) 0

4 min read Jun 17, 2024
(x+9)(x-5)^2(x-18) 0

Solving the Inequality: (x+9)(x-5)^2(x-18) > 0

This article will guide you through the process of solving the inequality (x+9)(x-5)^2(x-18) > 0. We'll use a combination of techniques to understand the solution set.

1. Finding the Critical Points

The critical points are the values of x where the expression equals zero. To find these points, set each factor equal to zero and solve:

  • x + 9 = 0 => x = -9
  • (x - 5)^2 = 0 => x = 5
  • x - 18 = 0 => x = 18

Therefore, the critical points are x = -9, x = 5, and x = 18.

2. Creating a Sign Chart

A sign chart helps visualize the sign of the expression in different intervals defined by the critical points.

  1. Divide the Number Line: Draw a number line and mark the critical points (-9, 5, and 18). This divides the number line into four intervals:

    • x < -9
    • -9 < x < 5
    • 5 < x < 18
    • x > 18

  2. Choose Test Points: Select a test point within each interval. For example:

    • x < -9: Choose x = -10
    • -9 < x < 5: Choose x = 0
    • 5 < x < 18: Choose x = 10
    • x > 18: Choose x = 20

  3. Evaluate the Expression: Substitute each test point into the original expression (x+9)(x-5)^2(x-18) and determine the sign:

    • x = -10: (-1)(-15)^2(-28) = Positive
    • x = 0: (9)(-5)^2(-18) = Negative
    • x = 10: (19)(5)^2(-8) = Negative
    • x = 20: (29)(15)^2(2) = Positive

  4. Fill the Sign Chart: Mark the sign of the expression in each interval on the number line:

   -----(-9)-----(5)-----(18)-----
   +     -       -        +

3. Identifying the Solution

We want to find the intervals where (x+9)(x-5)^2(x-18) > 0. This means we're looking for the intervals where the expression is positive.

From the sign chart, we see that the expression is positive in the following intervals:

  • x < -9
  • x > 18

4. Accounting for Multiplicity

Notice that the factor (x-5)^2 has a multiplicity of 2. This means that the expression doesn't change sign at x = 5. It remains negative in the interval -9 < x < 5 and also in the interval 5 < x < 18.

5. The Final Solution

The solution to the inequality (x+9)(x-5)^2(x-18) > 0 is:

x < -9 or x > 18

Related Post

Featured Posts