Not to be confused with Inequation. “Less than”, “Greater than”, and “More than” redirect here. For the use of the “” signs as punctuation, see Bracket. For the UK insurance brand “More Th>n”, see RSA Insurance Group. In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality). The notation a ≠ b means that a is not equal to b. It does not say that one is greater than the other, or even that they can be compared in size. If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size. The notation a b means that a is greater than b. In either case, a is not equal to b. These relations are known as strict inequalities. The notation a ) and (in the case of applying a function) monotonic functions are limited to strictly monotonic functions. Transitivity The Transitive property of inequality states: For any real numbers a, b, c: If a ≥ b and b ≥ c, then a ≥ c. If a ≤ b and b ≤ c, then a ≤ c. If either of the premises is a strict inequality, then the conclusion is a strict inequality. E.g. if a ≥ b and b > c, then a > c An equality is of course a special case of a non-strict inequality. E.g. if a = b and b > c, then a > c Converse The relations ≤ and ≥ are each other’s converse: For any real numbers a and b: If a ≤ b, then b ≥ a. If a ≥ b, then b ≤ a. Addition and subtraction A common constant c may be added to or subtracted from both sides of an inequality: For any real numbers a, b, c If a ≤ b, then a + c ≤ b + c and a − c ≤ b − c. If a ≥ b, then a + c ≥ b + c and a − c ≥ b − c. i.e., the real numbers are an ordered group under addition. Multiplication and division The properties that deal with multiplication and division state: For any real numbers, a, b and non-zero c: If c is positive, then multiplying or dividing by c does not change the inequality: If a ≥ b and c > 0, then ac ≥ bc and a/c ≥ b/c. If a ≤ b and c > 0, then ac ≤ bc and a/c ≤ b/c. If c is negative, then multiplying or dividing by c inverts the inequality: If a ≥ b and c b, then 1/a > 1/b. These can also be written in chained notation as: For any non-zero real numbers a and b: If 0 0. If a ≤ b 1/a ≥ 1/b. If a a ≥ b, then 1/a ≤ 1/b 0, then 0 0 > b, then 1/a > 0 > 1/b. Applying a function to both sides Any monotonically increasing function may be applied to both sides of an inequality (provided they are in the domain of that function) and it will still hold. Applying a monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function. If the inequality is strict (a b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. The rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function. As an example, consider the application of the natural logarithm to both sides of an inequality when a and b are positive real numbers: a ≤ b ⇔ ln(a) ≤ ln(b). a on real numbers are strict total orders.
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