Solve: [tex]\frac{x^2-1}{x^2+1} \leq 0[/tex]
Answer (2)
Analyze the inequality x 2 + 1 x 2 − 1 ≤ 0 .
Recognize that x 2 + 1 is always positive.
Solve x 2 − 1 ≤ 0 , which simplifies to x 2 ≤ 1 .
The solution is − 1 ≤ x ≤ 1 , thus the final answer is − 1 ≤ x ≤ 1 .
Explanation
Understanding the Problem We are given the inequality x 2 + 1 x 2 − 1 ≤ 0 . We need to find the range of x that satisfies this inequality. Note that x 2 + 1 is always positive for real x, since x 2 ≥ 0 , so 0"> x 2 + 1 ≥ 1 > 0 .
Simplifying the Inequality Since 0"> x 2 + 1 > 0 for all real x , the sign of the fraction x 2 + 1 x 2 − 1 is determined by the sign of the numerator x 2 − 1 . Therefore, we need to solve the inequality x 2 − 1 ≤ 0 .
Rewriting the Inequality Rewrite the inequality as x 2 ≤ 1 .
Solving for x Taking the square root of both sides, we have ∣ x ∣ ≤ 1 , which means − 1 ≤ x ≤ 1 .
Final Answer Therefore, the solution to the inequality x 2 + 1 x 2 − 1 ≤ 0 is − 1 ≤ x ≤ 1 .
Examples
Imagine you're designing a sound system where the ratio of signal power to total power must be non-positive to avoid distortion. This problem helps determine the range of input signal values that ensure the system operates within acceptable limits. Understanding inequalities and ratios can guide you in scenarios such as signal processing, control systems, or any situation where maintaining a specific ratio is crucial for optimal performance.
The solution to the inequality x 2 + 1 x 2 − 1 ≤ 0 is found by analyzing the numerator, leading to − 1 ≤ x ≤ 1 . Since the denominator is always positive, the critical points come from the numerator. Thus, the final solution is − 1 ≤ x ≤ 1 .
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