Inequalities are the expressions indicating comparison of numbers. They are strict (more, it is less) and mild (more or equally, it is less or equally). To solve inequality – means to find all those values of variables at which substitution the right numerical record turns out.
Used the concept "inequality" of Ancient Greece. So, in the 3rd century BC Archimedes, calculating circle length, established that the perimeter of a circle is equal "to the trebled diameter there is a lot of which is less than seventh part of diameter, but more than ten seventy first". In other words, he established borders for number π: 3 10/71 <π<3 1/7. Mathematical expression a> b means that number a is more than number b. If an is written down <b, it means that an is less than b. For mild inequalities: a≥b means that number a is more or equally to number b, a≤b – number a less or equally to number b. In mild inequalities of number can coincide. The simplest inequalities can be linear, containing modules, rational, irrational. More difficult inequalities – indicative, logarithmic, trigonometrical, mixed. A special type of tasks – inequalities with parameters. Graphically the solution of inequality is represented by a half-space which can be limited and unlimited. To find a solution, it is useful to replace the sign of inequality with an equal-sign, to solve the received equation and to build the schedule. To solve irrational inequality, it is necessary to transfer all fractions to the left part, to reduce to a common denominator, to factorize numerator and a denominator, to apply a method of intervals. For the solution of exponential equations it is necessary to use properties of degrees, logarithmic – properties of logarithms. Eventually, all difficult inequalities are solved their data to protozoa. At the decision all transitions have to be equivalent. Begin the solution of all inequalities with finding of ODZ, area of permissible values. You watch equivalence of transformations. That is, each your step should not narrow or expand ODZ. Starting the solution of logarithmic inequalities, learn definition of a logarithm, property of logarithms, transformation formulas. Get the hand in the solution of the logarithmic equations. Consider that properties of logarithms differ depending on the basis: when it more than unit, and when from zero to some.