(3-x)(x-8)^2 0

3 min read Jun 16, 2024

Solving the Inequality: (3-x)(x-8)^2 > 0

This article will guide you through solving the inequality (3-x)(x-8)^2 > 0. This type of inequality involves finding the values of x that make the expression greater than zero.

Understanding the Problem

The expression (3-x)(x-8)^2 represents a product of two factors: (3-x) and (x-8)^2. To understand when this product is positive, we need to analyze the sign of each factor.

(3-x): This factor is positive when x < 3 and negative when x > 3.
(x-8)^2: This factor is always non-negative (greater than or equal to zero) because it is squared. It equals zero when x = 8.

Analyzing the Signs

Now, we can analyze the sign of the product (3-x)(x-8)^2 based on the sign of each factor:

x < 3: (3-x) is positive, and (x-8)^2 is positive. Therefore, the product is positive.
3 < x < 8: (3-x) is negative, and (x-8)^2 is positive. Therefore, the product is negative.
x > 8: (3-x) is negative, and (x-8)^2 is positive. Therefore, the product is negative.
x = 8: (3-x) is negative, and (x-8)^2 is zero. Therefore, the product is zero.

Solution

From the analysis above, we can conclude that the product (3-x)(x-8)^2 is greater than zero when x < 3. This is our solution to the inequality.

Visual Representation

You can visualize the solution on a number line:

     <------------------------>
     0   3   8 
     |   |   |
     +   -   -

The plus sign represents where the product is positive, and the minus sign represents where the product is negative.

Conclusion

The solution to the inequality (3-x)(x-8)^2 > 0 is x < 3. This means that all values of x less than 3 will make the expression greater than zero.