Solution of inequalities

Any school program in mathematics includes material on inequalities. They surround the schoolboy everywhere: in formulas, algebraic axioms and problems. What is inequality and how does the solution of inequalities look like?

Inequality assumes in its condition a distinction between the two parts of the expression. In total, there are two types: strict and non-strict. Non-simple inequalities allow an option in which their parts are equal (in this case, the signs "greater than or equal to" and "less than or equal to" are used). Strict inequalities do not allow the use of answers in which their parts become equal. In this case, the solution of inequalities includes the signs "more", "less" and "not equal".

Most often, the inequalities have a whole range of values in the answer, including both integers and many fractional ones. To give a complete and uniquely correct answer, write down not exact values, but their intervals. The solution of inequalities occurs most often by the method of intervals, where it is checked which part of the coordinate segment satisfies all the conditions that make it possible to form the correct inequality. The answer is written in the form "the unknown belongs to a segment of coordinates with given boundaries". An example of a response record is x Є (7; 10), where the parenthesis indicates a strict inequality, and the square bracket indicates a non-strict (that is, 10 is one of the possible answers, and 7 is not.) If the range of possible solutions to the inequality goes to infinity The sign of infinity in the answer is always allocated with a parenthesis.

Inequalities are many kinds, but the most complex questions arise in two cases: this is a solution of irrational and fractional inequalities.

What is irrational inequality? This inequality, one of whose parts is the root of the function. It looks like this inequality is quite difficult for an inexperienced student, and for many students of mathematical departments. However, the solution of irrational inequalities is quite simple: it is necessary simply to raise all the inequalities to a power at the root of which is one of its parts. It is necessary to observe only one rule: if one of the functions is negative, raising to an even degree distorts the inequality and makes it different from the original by its very essence. Therefore, the solution of irrational inequalities is one of those moments on which the lion's share of the schoolchildren and students being mistaken is mistaken.

The solution of fractional inequalities is also quite simple. A fractional inequality is one in which one of the parts is a fraction. What can be done to make the right decision of fractional inequalities? Simply multiply both sides of the inequality by the denominator of one of the functions. This will bring the function into a simpler form, which allows you to quickly and without much effort calculate the correct range of solutions to the inequality.

There are a huge number of types of inequalities, and the solutions of many of them differ from each other. It is necessary to know and present the correct method of solving each of them in order to be able to make a condition competently, to write down the answer and get high scores for the work. What is the solution of irrational and fractional inequalities? First of all, the fact that simplification is used to solve them by eliminating an inconvenient factor (in one case - the root, in the second - the denominator of the function). Therefore, every student and student is obliged to remember that hardly having noticed in the inequality the root or denominator, he must react and either raise both parts of the inequality to the desired degree, or multiply both sides of the inequality by the denominator. This solution method works in most cases, except for tasks of exceptional complexity (which, by the way, are extremely rare). Therefore, it can be said with certainty that the solution to the inequalities proposed above will be true in almost one hundred percent of cases. Good luck in your studies!