Understanding Inequalities Involving Roots and Absolute Values: A Comprehensive Guide

In this detailed guide, we will delve into the intricacies of solving inequalities that involve roots and absolute values, using a practical and mathematical approach. We will explore the conditions and solutions related to inequalities such as sqrt{ax^2} – x, and demonstrate how to apply mathematical principles to arrive at the correct conclusions.

Introduction to the Problem

We start with a fundamental mathematical expression: sqrt{1x^2}sqrt{x^2} |x|. This identity forms the basis for understanding more complex expressions and inequalities. Let's explore the example provided to gain a clearer understanding.

Step-by-Step Solution

Let's consider the inequality: sqrt{1x^2} x 0.

First, we recognize that sqrt{1x^2} |x|. Therefore, the inequality transforms into: |x| x 0.

Next, we factorize the left-hand side of the inequality: |x| – x sqrt{1x^2 – x}

We then analyze the inequality: sqrt{1x^2} – x sqrt{1x^2 – x} sqrt{1 – x^2} lesqrt{2}.

Further Analysis

We further simplify and explore the conditions for equality:

left(sqrt{1x^2} – xright) sqrt{1 – x^2} lesqrt{2} color{darkblue}{Longleftrightarrow} left(sqrt{1x^2} xright) left(sqrt{1x^2} – xright) sqrt{1 – x^2} lesqrt{2} left(sqrt{1x^2} xright)

sqrt{1 – x^2} lesqrt{2} left(sqrt{1x^2} xright)

1 – x^2 lesqrt{2}^2 left(sqrt{1x^2} xright)^2

1 – x^2 lesqrt{2} (1 2x^2 2x sqrt{1x^2})

1 – 5x^2 – 4x sqrt{1x^2} ge 0

(2x sqrt{1x^2} 1) ge 0

2x sqrt{1x^2} 1 ge 0 color{darkblue}{Longleftrightarrow} x –1 / sqrt{3} (x 0 is not a solution)

Generalized Form

From the specific case, we can generalize the inequality:

left(sqrt{ax^2} – xright) sqrt{1 – x^2} lesqrt{1a} a ge 0

with equality if and only if x –1 / (sqrt{a^2}), where x is a non-negative value.

Conclusion

In summary, understanding and solving inequalities involving roots and absolute values requires a solid grasp of mathematical principles. By following a systematic approach, we can simplify complex expressions and find the correct solutions. The generalized form provides a useful framework for solving similar inequalities.

Key Takeaways

The relationship sqrt{1x^2} |x| is fundamental in solving such inequalities. The conditions for equality in such inequalities are specific and can be determined through algebraic manipulation. The generalized form is a powerful tool for solving a broad range of similar problems.

References

For more detailed information and additional examples, refer to the following resources:

Square Roots on MathIsFun Inequalities Involving Roots Absolute Value on Wikipedia