..... Click the link for more information. , but it does contain information about the expressions involved. The symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) are used in place of the equals sign in expressions of inequalities. As in the case of equations, inequalities can be transformed in various ways. The direction of the inequality remains unchanged if some number is added to both sides or subtracted from both sides or if both sides are multiplied or divided by some positive number; e.g., subtracting 10 from both sides of the inequality x < 8 gives x − 10 < −2, and multiplying the inequality by 2 gives 2x < 16. Multiplication or division by a negative number reverses the sign of the inequality; e.g., if −2x < 8, then dividing both sides by −2 results in the inequality x > −4.
inequality
In mathematics, a statement of an order relationship—greater than, greater than or equal to, less than, or less than or equal to—between two numbers or algebraic expressions. Inequalities can be posed either as questions, much like equations, and solved by similar techniques, or as statements of fact in the form of theorems. For example, the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the remaining side. Mathematical analysis relies on many such inequalities (e.g., the Cauchy-Schwarz inequality) in the proofs of its most important theorems.
inequality
inequality [‚in·i′kwäl·əd·ē]
Inequality
(mathematics), a relation between two numbers or quantities indicating which of them is greater or smaller. An inequality is denoted by the symbol <, which is drawn pointed toward the smaller number. Thus, 2 > 1 and 1 < 2 state the same thing, namely, 2 is greater than 1 and 1 is less than 2. Sometimes we make use of multiple inequalities as in a < b < c. In order to express the fact that of two numbers a and b, the first is greater than or equal to the second, we write a ≥ b (or b ≤ a) and read “a is greater than or equal to b” (or “a is less than or equal to b“ or simply “a is not less than b“ (or “b is not greater than a”). The notation α ≠ b signifies that the numbers a and b are unequal but does not indicate which is larger. All these relations are termed inequalities.
Inequalities have many properties in common with equalities. Thus, an inequality remains valid if the same number is added to or subtracted from both sides. We can similarly multiply both sides of an inequality by a positive number. However, if both sides are multiplied by a negative number, then the sense of the inequality is reversed, that is, the symbol > is replaced by <, and < by >. From the inequalities A < B and C < D follow the inequalities A + C < B + D and A —D < B —C; in other words, inequalities of the same sense (A < B and C < D) may be added term by term, and inequalities of opposite sense (A < B and D > C) may be subtracted term by term. If the numbers A, B. C, and D are positive, then the inequalities A < B and C < D also imply AC < BD and A/D < B/C; in other words, inequalities of the same sense (between positive numbers) may be multiplied term by term, and those of opposite sense may be divided term by term.
Inequalities containing quantities that can assume different numerical values may be true for some values of these quantities and false for others. Thus, the inequality x2 —4x + 3 > 0 is true for x = 4 and false for x = 2. To solve such inequalities is to determine the limits within which the quantities entering into the inequalities must be taken in order for the inequalities to hold. Thus, by rewriting the inequality x2 —4x + 3 > 0 in the form (x —1) (x —3) > 0, we see that the latter holds for all x satisfying either one of the inequalities x < 1, x > 3, and these yield the solution of the original inequality.
We now give a few examples of important inequalities.
(1) “Triangle” inequality. For any real or complex numbers a1, a2, . . . ,an,
ǀa1 + a2 + . . . anǀ ≤ ǀa1ǀ + ǀa2ǀ + . . . + ǀanǀ
(2) Inequality for means. The most famous inequality relates the harmonic mean, the geometric mean, the arithmetic mean, and the root-mean-square:
The numbers a1, a2, . . . , an are assumed to be positive.
(3) Linear inequalities. Consider the system of inequalities
ai1x1 + ai2x2 + . . . + ain xn ≥ bi
(i = 1, 2, . .. ,m)
The totality of solutions of this system is a convex polyhedron in n-dimensional space (x1,x2, . . . xn). The task of the theory of linear inequalities consists of studying the properties of this polyhedron. Certain problems in the theory of linear inequalities are closely related to the theory of best approximations, which was created by P. L. Chebyshev.
(See also, , , , and .)
Inequalities are very important in many branches of mathematics. Diophantine approximations, an entire branch of number theory, are completely based on inequalities. Analytic number theory often operates with inequalities. The axiomatic development of inequalities is given in algebra. Linear inequalities play a large role in the theory of linear programming. In geometry, inequalities are constantly encountered in the theory of convex bodies and in isoperimetric problems. In probability theory, many laws are formulated in terms of inequalities (for example, the Chebyshev inequality). Differential inequalities are used in the theory of differential equations (the Chaplygin method). In the theory of functions, various inequalities are constantly used for derivatives of polynomials and trigonometric polynomials. In functional analysis, the definition of norm in a function space requires that it satisfy the triangle inequality
ǀǀx + y ǀǀ ≤ ǀǀxǀǀ + ǀǀyǀǀ
Many classical inequalities virtually define or estimate the norm of a linear functional or a linear operator in some space.
REFERENCES
Korovkin, P. P. Neravenstva, 3rd ed. Moscow, 1966. Hardy, G. H. , S. E. Littlewood, and G. Pólya. Neravenstva. Moscow, 1948. (Translated from English.)Want to thank TFD for its existence? Tell a friend about us, add a link to this page, add the site to iGoogle, or visit the webmaster's page for free fun content.
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