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There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.
The simplest is the slope-intercept form: = +, from which one can immediately see the slope a and the initial value () =, which is the y-intercept of the graph = (). Given a slope a and one known value () =, we write the point-slope form:
The feasible regions of linear programming are defined by a set of inequalities. In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size.
Two-dimensional linear inequalities are expressions in two variables of the form: + < +, where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane. [2]
Bernoulli's inequality; Bernstein's inequality (mathematical analysis) Bessel's inequality; Bihari–LaSalle inequality; Bohnenblust–Hille inequality; Borell–Brascamp–Lieb inequality; Brezis–Gallouet inequality; Carleman's inequality; Chebyshev–Markov–Stieltjes inequalities; Chebyshev's sum inequality; Clarkson's inequalities ...
A non-vertical line can be defined by its slope m, and its y-intercept y 0 (the y coordinate of its intersection with the y-axis). In this case, its linear equation can be written = +. If, moreover, the line is not horizontal, it can be defined by its slope and its x-intercept x 0. In this case, its equation can be written
Using this form, vertical lines correspond to equations with b = 0. One can further suppose either c = 1 or c = 0, by dividing everything by c if it is not zero. There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the standard form.
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.