Any school math program includes material on inequalities. They surround the student everywhere: in formulas, algebraic axioms and problems. What are inequalities and what does the solution of inequalities look like?
Inequality in its condition implies a distinction between the two parts of an expression. There are two types in total: strict and non-strict. Non-stringent inequalities allow a variant 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, inequalities have a whole range of values in the answer, including both integers and many fractional ones. To give a complete and only correct answer, they do not record the exact values, but their intervals. The solution of inequalities occurs most often by the method of intervals, where it is checked in which part of the coordinate interval all the conditions are fulfilled, which make it possible to make the correct inequality. The answer is written in the form "the unknown belongs to the coordinate interval with these boundaries." An example of recording an answer is x Є (7; 10], where the parenthesis denotes a strict inequality, and the square one is not strict (that is, 10 is one of the possible answers, and 7 is not). If the interval of possible solutions to the inequality goes to infinity, then the infinity sign in the answer is always highlighted with a parenthesis.
There are many kinds of inequalities, but the most difficult questions arise in two cases: this is a solution to irrational and fractional inequalities.
What is irrational inequality? This is an inequality, one part of which is the root of the function. Such an inequality looks rather complicated both for an inexperienced student and for many students of mathematical departments. However, the solution to irrational inequalities is quite simple: you just need to raise all the inequality to the extent at the root of which is one of its parts. It is worth observing only one rule: if one of the functions is negative, raising to an even degree will distort inequality and make it different from the original by its very essence. Therefore, the solution of irrational inequalities is one of those moments at which the lion's share of the schoolchildren and students being examined is mistaken.
The solution of fractional inequalities is also quite simple. Fractional inequality is one in which one of the parts is a fraction. What to do to make the right solution to 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 effortlessly calculate the correct range of solutions to inequality.
There are a huge number of types of inequalities, and the solutions to many of them vary. It is necessary to know and present the correct method for solving each of them in order to competently be able to make a condition, write down the answer and get high scores for the work. How similar are the solutions to irrational and fractional inequalities? First of all, the fact that they are solved by simplification by eliminating an uncomfortable factor (in one case, the root, in the second, the denominator of the function). Therefore, every pupil and student must remember: barely noticing the root or denominator in the inequality, he must react and either raise both sides 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, incidentally, are extremely rare). Therefore, we can confidently say that the solution of inequalities proposed above will be true in almost a hundred percent of cases. Success in your studies!