[size=150]In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x. It is often employed in real analysis. It has several useful variants.[br]Bernoulli’s inequality theorem states: [br][br]Theorem 1. For all real number x > −1 and real number r > 1, the following inequality holds[br](1 + x)[sup]r[/sup] ≥ 1 + rx. The equality holds if and only if x = 0. [br]The inequality sign changes when 0 < r < 1, that is (1 + x) r ≤ 1 + rx. If we let x + 1 → x, the inequality can be restated as: [br][br]Theorem 2. For all x > 0 and r > 1, x r ≥ rx + 1 − r. The equality holds if and only if x = 1. [br][br]Inequality (2) is used more often than its original form. Let r = a b with a > b > 0 and[br]x → x b , we have Theorem 3. Let a, b ∈ R + and a > b. Then for all [br]x > 0, x a ≥ a b x b + 1 − a b . [br][br]This inequality looks more appealing than the other two because it allows us to compare x with the difference between the two powers. One of the strengths of this inequality is that it can be applied for inequalities with irrational powers and non-homogeneous inequalities.[/size]

[size=150]Bernoulli's Inequality is a fundamental inequality in mathematics that relates the exponential function to the power function. It states that for any real number 'x' greater than -1 and any real number 'r' greater than or equal to 0:[br][br](1 + x)[sup]r[/sup] ≥ 1 + r * x[br][br]where:[br]- x is a real number greater than -1.[br]- r is a real number greater than or equal to 0.[br][br]This inequality tells us that raising a number greater than -1 to a positive power will always result in a value greater than or equal to 1 + r * x.[br][br]**Examples:**[br][br]1. x = 0.5, r = 2:[br][br] (1 + 0.5)[sup]2[/sup] ≥ 1 + 2 * 0.5[br] 1.5[sup]2[/sup] ≥ 1 + 1[br] 2.25 ≥ 2[br][br] Here, the inequality holds true, as 2.25 is indeed greater than 2.[br][br]2. x = 0.2, r = 4:[br][br] (1 + 0.2)[sup]4[/sup] ≥ 1 + 4 * 0.2[br] 1.2[sup]4[/sup] ≥ 1 + 0.8[br] 1.7490064 ≥ 1.8[br][br] Again, the inequality is satisfied since 1.7490064 is greater than 1.8.[br][br]3. x = -0.1, r = 3[br][br] (1 - 0.1)[sup]3[/sup] ≥ 1 + 3 * (-0.1)[br] 0.9[sup]3[/sup] ≥ 1 - 0.3[br] 0.729 ≥ 0.7[br][br] Even with a negative value of 'x', the inequality remains true, as 0.729 is greater than 0.7.[br][br]Bernoulli's Inequality finds applications in various fields of mathematics and serves as a useful tool for proving more complex inequalities and theorems. It is an essential concept in the study of mathematical analysis and plays a role in various branches of mathematics, including calculus and number theory.[/size]