Solving Inequalities: Understanding the Values of x for (1 - x^2 leq 0)

When dealing with inequalities, it is essential to understand the behavior of different functions. In this article, we will explore the values of x for the inequality (1 - x^2 leq 0). By breaking down the equation and utilizing fundamental algebraic principles, we can determine the range of x that satisfies the inequality.

Understanding Square Terms and Their Properties

A square term, such as (1 - x^2), has unique properties. Specifically, a square term can never be less than zero. This concept can be proven as follows:

Proof: Square Terms are Always Non-Negative

If a is positive:

(a^2 a cdot a ) positive number multiplied by a positive number

(Rightarrow a^2) is a positive number

If a is negative:

(a^2 a cdot a ) negative number multiplied by a negative number

(Rightarrow a^2) is a positive number

Given these properties, let's analyze the inequality (1 - x^2 leq 0). To begin, solve for the exact points where the equality (1 - x^2 0) holds.

Solving the Equation (1 - x^2 0)

To find the values of x that satisfy (1 - x^2 0), we can proceed as follows:

(1 - x^2 0)

(Rightarrow 1 x^2)

(Rightarrow x^2 1)

(Rightarrow x pm 1)

Therefore, the specific values of x that satisfy the equation (1 - x^2 0) are (x 1) and (x -1).

Understanding the Inequality (1 - x^2 leq 0)

Now, let's consider the inequality (1 - x^2 leq 0). We need to determine the range of x that satisfies this inequality. Given that (1 - x^2) represents a square term, it is non-negative and can only equal zero at the points (x 1) and (x -1).

(1 - x^2 0) when (x 1) or (x -1)

(1 - x^2 > 0) for all values of x except (x 1) and (x -1)

To find the solution to the inequality (1 - x^2 leq 0), observe that (1 - x^2) is less than or equal to zero within the interval where the function touches or dips below the x-axis. This interval is exactly between (-1) and (1).

(Rightarrow -1 leq x leq 1)

Thus, the values of x that satisfy the inequality (1 - x^2 leq 0) are all x such that (-1 leq x leq 1).

Conclusion

In summary, the solution to the inequality (1 - x^2 leq 0) is the interval ([-1, 1]). This solves the inequality by understanding the properties of square terms and using algebraic techniques to isolate and solve the given quadratic expression.

Additional Notes on Square Terms and Inequalities

This article has covered the values of x for the inequality (1 - x^2 leq 0), but it is also important to explore the broader implications of square terms and their application in solving more complex inequalities and equations. Understanding the fundamental properties of square terms is crucial in various areas of mathematics, including algebra and calculus.