Inequation

The inequations, unlike the equations, are inequalities between monomials, or polynomials, for which we seek the solution of one or more literal variables, called unknowns (as for the equations). Some examples of inequations are:

  • \(a<b\)
  • \(x+y+z\leq 1\)
  • \(n>1\)
  • \(x\neq 0\)

Intervals

In mathematics, an interval is defined as the set of all the elements of an ordered set which are preceded by an element (far left of the interval) and precede another element (far right of the interval).

The interval is said to be closed if the values of the extremes are part of it; a closed interval can be closed to the right (or left) if only the right (or left) end is part of it. The interval, on the other hand, is said to be open if the values of the extremes are not part of it.

Classification of intervals

  • Proper and bounded:
    • \((a,b)=\{x|a<x<b\}\) open interval
    • \([a,b]=\{x|a\le x \le b\}\) closed interval
    • \([a,b)=\{x|a\le x<b\}\) left-closed, right-open
    • \((a,b]=\{x|a<x\le b\}\) left-open, right-closed
  • Left-bounded and right-unbounded
    • \((a,+\infty)=\{x|x>a\}\) left-open
    • \([a,+\infty)=\{x|x\ge a\}\) left-closed
  • Left-unbounded and right-bounded:
    • \((-\infty,b)=\{x|x<b\}\) right-open
    • \((-\infty,b]=\{x|x\le b\}\) right-closed
  • \((-\infty,+\infty)=\mathbb{R}\) unbounded at both ends
  • \([a,a]=\{a\}\) a point (degenerate)
  • \([b,a]=(b,a)=[b,a)=(b,a]=(a,a)=[a,a)=(a,a]={}=\emptyset\) empty set