Solving the Modulus Equation: x^2 - x - 6 |x^2 - x - 6|

In this article, we will explore the modulus equation x^2 - x - 6 |x^2 - x - 6| and understand how to solve it. We will break down the problem step by step using algebraic methods and analyze its solutions.

Introduction to Modulus and Absolute Value

Before diving into the solution, let's quickly review the concept of modulus and absolute value. The absolute value of a number is its distance from zero on the number line, without considering direction. Mathematically, it is represented as |x|.

The equation x^2 - x - 6 |x^2 - x - 6| means that the expression on the right side, which is the absolute value of x^2 - x - 6, is equal to the left side. This can be simplified using the definition of the absolute value function:

|A| A if A ≥ 0 and |A| -A if A

Step-by-Step Solution

Let's denote the expression x^2 - x - 6 as Y.

Part 1: Analyzing the Case Y ≥ 0

Given: Y x^2 - x - 6.

When Y ≥ 0, we have: x^2 - x - 6 x^2 - x - 6.

This is always true, so any value of x that makes x^2 - x - 6 ≥ 0 will satisfy the equation.

Part 2: Analyzing the Case Y

When Y , we have: x^2 - x - 6 -(x^2 - x - 6).

Simplifying the right side: x^2 - x - 6 -x^2 x 6.

Rearranging the terms: 2x^2 - 2x - 12 0.

Dividing by 2: x^2 - x - 6 0.

Factoring the quadratic equation: (x - 3)(x 2) 0.

Solving for x: x 3 or x -2.

Checking the validity of these solutions in the original context:

For x 3, Y 3^2 - 3 - 6 9 - 3 - 6 0, which is not less than 0.

For x -2, Y (-2)^2 - (-2) - 6 4 2 - 6 0, which is not less than 0.

Thus, none of these solutions are valid for the case Y .

Final Solution

From the analysis, we conclude that the equation x^2 - x - 6 |x^2 - x - 6| is valid when x^2 - x - 6 ≥ 0.

Graphical Interpretation

The inequality x^2 - x - 6 ≥ 0 can be visualized using the graph of the quadratic function y x^2 - x - 6. The roots of the quadratic equation are x 3 and x -2. The quadratic function is non-negative outside the interval between these roots.

Therefore, the solution to the inequality is:

x ∈ (-∞, -2] ∪ [3, ∞)

Conclusion

In summary, solving the modulus equation x^2 - x - 6 |x^2 - x - 6| involves breaking it down into cases and analyzing each case carefully. The final solution is that the equation holds true for values of x in the intervals (-∞, -2] and [3, ∞). Understanding the nature of modulus equations and the behavior of quadratic functions is crucial in finding the solution.