Solving the Inequality (x - ax - bx - c > 0): A Comprehensive Guide

Understanding and solving algebraic inequalities, such as (x - ax - bx - c > 0), requires a systematic approach. This article will guide you through each step of the process, ensuring clarity and accuracy in your solutions. Let's begin by breaking down the inequality and exploring the techniques needed to solve it effectively.

Step 1: Identify the Roots

The first step in solving the inequality (x - ax - bx - c > 0) is to identify the roots of the equation (x - ax - bx - c 0). The roots are given as:

x a x b x c

These roots will divide the number line into several intervals based on the order of the roots.

Step 2: Determine the Intervals

The critical points (a), (b), and (c) divide the number line into four intervals:

From (-infty) to the smallest root (let's assume it's a for simplicity). From the smallest root to the largest root, excluding the ends. From the largest root to (infty).

Step 3: Test Each Interval

To determine the sign of the expression (x - ax - bx - c) in each interval, choose a test point from each interval:

Interval 1: Choose a test point less than the smallest root, say, (x x - a, x - b, x - c will be negative, making the product negative.

Interval 2: Choose a test point between the smallest and largest root, such as (a

Interval 3: Choose a test point greater than the largest root, say, (x > c). All factors x - a, x - b, x - c will be positive, making the product positive.

Step 4: Analyze the Results

Based on the test points, we can determine:

The product is positive in the intervals where x is greater than the largest root and less than the smallest root, depending on the arrangement of a, b, c.

Step 5: Write the Solution

Depending on the specific values of a, b, c, the solution to the inequality (x - ax - bx - c > 0) will vary. The general solution can be expressed as:

If (a (-∞, a) ∪ (c, ∞)

If the roots are in a different order, adjust accordingly by replacing the roots with their respective positions.

For a visual aid, a simple sketch graph is indeed helpful. The graph will clearly show the intervals where the expression (x - ax - bx - c) is positive. By plotting the roots and the sign changes in the intervals, you can easily determine the solution set.

Conclusion

Solving algebraic inequalities like (x - ax - bx - c > 0) requires a structured approach involving identifying roots, determining intervals, and testing each interval. By following these steps, you can systematically solve such inequalities and determine the correct solution set.