(x-a)(x-b) 0 Inequality

5 min read Jun 17, 2024

Understanding the Inequality (x - a)(x - b) > 0

The inequality (x - a)(x - b) > 0 represents a fundamental concept in algebra and is used in various applications like finding the solution to quadratic equations, analyzing functions, and determining the intervals where a function is positive or negative. Let's break down this inequality and explore its solution.

Key Idea: Sign Analysis

The core of solving this inequality lies in sign analysis. We need to figure out when the expression (x - a)(x - b) is positive. To do this, we consider the signs of each factor (x - a) and (x - b) for different values of x.

Steps to Solve the Inequality

Find the Critical Points: The critical points are the values of x where the expression becomes zero. These are x = a and x = b. These points divide the number line into three intervals:
- x < a
- a < x < b
- x > b
Sign Table: Create a sign table to track the signs of (x - a), (x - b), and their product (x - a)(x - b) in each interval:

Interval x - a x - b (x - a)(x - b)

x < a - - +

a < x < b + - -

x > b + + +
- Explanation: For example, when x < a, both (x - a) and (x - b) are negative, making their product positive.
Solution: The inequality (x - a)(x - b) > 0 holds true when the product is positive. Therefore, the solution is:
- x < a or x > b

Interval	x - a	x - b	(x - a)(x - b)
x < a	-	-	+
a < x < b	+	-	-
x > b	+	+	+

Visual Representation

The solution can be visualized on a number line:

     <------(a)--------(b)----->
        -----     -----

The shaded regions represent the intervals where the inequality is satisfied.

Example

Let's solve the inequality (x - 2)(x + 3) > 0:

Critical Points: x = 2 and x = -3
Sign Table:

Interval x - 2 x + 3 (x - 2)(x + 3)

x < -3 - - +

-3 < x < 2 - + -

x > 2 + + +
Solution: x < -3 or x > 2

Interval	x - 2	x + 3	(x - 2)(x + 3)
x < -3	-	-	+
-3 < x < 2	-	+	-
x > 2	+	+	+

Applications

Quadratic Inequalities: The inequality (x - a)(x - b) > 0 is directly related to solving quadratic inequalities in the form ax² + bx + c > 0. By factoring the quadratic, we can apply the same sign analysis method.
Function Analysis: By determining the intervals where a function is positive or negative, we can understand its behavior and identify key features like its zeros, extrema, and intervals of increase and decrease.
Problem Solving: This inequality is crucial in solving various real-world problems involving optimization, maximization, and minimization.

Understanding and applying the concepts related to the inequality (x - a)(x - b) > 0 is essential for mastering algebra and its applications in various fields.