Prove the inequality x2 + (1)/(x2) 2 for any non-zero real number x.

March 28, 2026Mathematics

ScanSolve AI Answer

March 28, 2026

Answer

$x^2 + \frac{1}{x^2} \ge 2$

Step 1: Start with a known true inequality. For any real number $x$ (where $x \ne 0$ ), the square of the difference between $x$ and $\frac{1}{x}$ is always non-negative: $\left(x - \frac{1}{x}\right)^2 \ge 0$

Step 2: Expand the left side of the inequality. Using the formula $(a-b)^2 = a^2 - 2ab + b^2$ : $x^2 - 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 \ge 0$

Step 3: Simplify the expanded expression. $x^2 - 2 + \frac{1}{x^2} \ge 0$

Step 4: Add 2 to both sides of the inequality. $x^2 + \frac{1}{x^2} \ge 2$ This proves the inequality for $x \ne 0$ .

The final answer is $\boxed{x^2 + \frac{1}{x^2} \ge 2}$ .

Was this helpful?

Need help with your own homework?

Get instant step-by-step solutions to any question. Free to start.

Ask Your Question

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp

More Mathematics Questions

QFactorize 12r2 - 27y2, Factorize 1 - (a - b)2, Factorize 16x2 - (1 - y)2, Factorize Completely 3 - 1…QCan u explain how do u add 3 to both sides QDetermine the gradient below 1.1. y = 2x - 3 x intercept let y = 0 0 = 2x - 3 2x + 3 = 0 -2/2 = 1.2.…QDetermine the gradient below 1.1. y: 2x - 3 x intercept let y = 0 0 = 2x - 3 2x + 3 = 0 -2/2 = 1.2.…QIn 2025 sixty students were asked how they travelled to college. Pie chart A below shows the results…

See all Mathematics questions →

Still have questions?

Quick Answer

Start with a known true inequality. For any real number x (where x 0), the square of the difference between x and (1)/(x) is always non-negative: (x - (1)/(x))^2 0 Step 2: Expand the left side of the inequality.

Asked on March 28, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on March 28, 2026

Step 1: Start with a known true inequality. For any real number x (where x 0), the square of the difference between x and (1)/(x) is always non-negative: (x - (1)/(x))^2 0 Step 2: Expand the left side of the inequality. Using the formula (a-b)^2 = a^2 - 2ab + b^2: x^2 - 2 · x · (1)/(x) + ((1)/(x))^2 0 Step 3: Simplify the expanded expression. x^2 - 2 + (1)/(x^2) 0 Step 4: Add 2 to both sides of the inequality. x^2 + (1)/(x^2) 2 This proves the inequality for x 0. The final answer is x^2 + (1)/(x^2) 2.