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

3 min read Jun 17, 2024

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

This article will guide you through solving the inequality (x-5)(x+2)^2 > 0. We'll break down the process step by step, making it clear and easy to understand.

Understanding the Problem

The inequality (x-5)(x+2)^2 > 0 asks us to find all the values of 'x' that make the expression on the left side greater than zero.

Solving the Inequality

Find the critical points: These are the values of 'x' that make the expression equal to zero. To find them, set each factor equal to zero and solve:
- x - 5 = 0 => x = 5
- (x + 2)^2 = 0 => x = -2
Create a sign chart: This chart helps visualize the sign of the expression in different intervals.

Interval x - 5 (x + 2)^2 (x - 5)(x + 2)^2

x < -2 - + -

-2 < x < 5 - + -

x > 5 + + +
- x - 5: The factor (x - 5) is negative for x < 5 and positive for x > 5.
- (x + 2)^2: The factor (x + 2)^2 is always positive since it's squared.
- (x - 5)(x + 2)^2: The sign of the product is determined by the signs of each factor.
Identify the solution: We are looking for the intervals where the expression is greater than zero (positive). From the sign chart, we see that this occurs only when x > 5.

Interval	x - 5	(x + 2)^2	(x - 5)(x + 2)^2
x < -2	-	+	-
-2 < x < 5	-	+	-
x > 5	+	+	+

Solution

Therefore, the solution to the inequality (x-5)(x+2)^2 > 0 is x > 5.

Key Points

The critical points divide the number line into intervals.
A sign chart helps determine the sign of the expression in each interval.
We are interested in the intervals where the expression is positive (greater than zero).

This method provides a systematic approach to solving inequalities. By understanding the steps and the logic behind them, you can confidently tackle similar problems.