Numerical inequalities and their properties. We solve the system of inequalities - properties and methods for calculating action with inequalities Examples

The system of inequalities is customary to call a record of several inequalities under the sign of the figure bracket (at the same time the number and type of inequalities in the system can be arbitrary).

To solve the system, it is necessary to find the intersection of all inequality in it. The solution of inequalities in mathematics is called any value of the change in which this inequality is true. In other words, it is required to find many of all his solutions - it will be called the answer. As an example, let's try to learn to solve the system of inequalities by the interval method.

Properties of inequalities

To solve the task, it is important to know the basic properties inherent in inequalities that can be formulated as follows:

• To both parts of inequality, you can add the same function defined in the field of permissible values \u200b\u200b(OTZ) of this inequality;
• If f (x)\u003e g (x) and h (x) - any function defined in the odd inequality, then f (x) + h (x)\u003e g (x) + h (x);
• If both parts of inequality are multiplied by a positive function defined in the OST of this inequality (or on a positive number), then we obtain inequality, equivalent to the original;
• If both parts of inequality are multiplied by a negative function defined in the OTZ of this inequality (or on a negative number) and the sign of inequality to change to the opposite, then the resulting inequality is equivalent to this inequality;
• Inequalities of the same sense can be reached by rear, and the inequalities of the opposite sense can be subtracted at least;
• Inequalities of one sense with positive parts can be multiplied by replenishment, and inequalities formed by non-negative functions can be raised to a positive degree.

To solve the inequality system, you need to solve each inequality separately, and then compare them. As a result, a positive or negative answer will be obtained, which means whether the system has a solution or not.

Interval method

When solving a system of inequalities, mathematics often resort to the method of intervals as to one of the most efficient. It allows you to reduce the solution of inequality f (x)\u003e 0 (<, <, >) To solve the equation F (x) \u003d 0.

The essence of the method is as follows:

• Find the area of \u200b\u200bpermissible inequality values;
• Lead the inequality to the form f (x)\u003e 0 (<, <, >), that is, transfer the right side to the left and simplify;
• Solve equation f (x) \u003d 0;
• Picture on a numeric direct function. All points marked on OTZ and limiting it, divide this set for the so-called sign intervals. In each such interval, the F (X) function is determined;
• Record the answer in the form of combining individual sets, on which F (x) has an appropriate sign. OTZ points, which are boundary, turn on (or not included) in response after additional check.

The field of valid numbers has an ordering property (clause 6, p. 35): For any numbers A, B take place one and only one of the three ratios: or. In this case, the record A\u003e B means that the difference is positive, and the recording difference is negative. Unlike the field of valid numbers, the field complex numbers It is not ordered: for complex numbers, the concepts of "more" and "less" are not determined; Therefore, this chapter discusses only actual numbers.

Relations We call inequalities, the number a and b - members (or parts) of inequality, signs\u003e (more) and inequalities A\u003e B and C\u003e D are called inequalities of the same (or one and the same) meaning; inequalities A\u003e B and from the definition of inequality immediately follows that

1) any positive number is greater than zero;

2) any negative number is less than zero;

3) any positive number is greater than any negative number;

4) of two negative numbers more then, the absolute value of which is less.

All these statements allow simply geometric interpretation. Let the positive direction of the numerical axis go to the right of the starting point; Then, what would be the signs of numbers, more of them are depicted by a point that is a right point depicting a smaller number.

Inequality possess the following basic properties.

1. Asymmetry (irreversibility): if, then, and back.

Indeed, if the difference is positive, then the difference is negative. It is said that when recalculating members of inequality, it is necessary to change the meaning of inequality to the opposite.

2. Transitivity: If, then. Indeed, from the positivity of differences and positivity

In addition to the signs of inequality, the signs of inequality are also applied as follows: the record means that either therefore, for example, you can write as well. Usually inequalities recorded using signs are called strict inequalities, and unstricted inequalities recorded using signs. Accordingly, the signs themselves call signs of strict or non-strict inequality. Properties 1 and 2, considered above, are true for incredible inequalities.

Consider now actions that can be made on one or more inequalities.

3. From adding to members of the inequality of the same number, the meaning of inequality does not change.

Evidence. Let the inequality and an arbitrary number be given. By definition, the difference is positive. Add two numbers two opposite numbers From what it will not change, i.e.

This equality can be rewritten so:

From this it follows that the difference is positive, i.e. what

and this was necessary to prove.

This is based on the possibility of skewing any member of inequality from one part of it to another with the opposite sign. For example, from inequality

follows that

4. When multiplying members of inequality per one and the same positive number, the meaning of inequality does not change; When multiplying members of inequality on the same negative number, the meaning of inequality varies to the opposite.

Evidence. Let then if since the product of positive numbers is positive. Opening a bracket in the left part of the last inequality, we obtain, i.e.. Similarly, the case is considered.

Exactly the same conclusion can be made regarding the division of parts of inequality to any different number from zero, since the division by the number is equivalent to multiplication by the number and the numbers have the same signs.

5. Let the members of the inequality are positive. Then, when erending his members in the same positive degree, the meaning of inequality does not change.

Evidence. Let this case, by the property of transitivity and. Then due to monotonous increase power function with and positive we will have

In particular, if where -natural number, then we get

i.e., when removing the root from both parts of inequality with positive member, the meaning of inequality does not change.

Let members of inequalities are negative. Then it is not difficult to prove that in the construction of his members in an odd natural degree of inequality will not change, and when it is erected into an even natural degree, it will change to the opposite. Of inequalities with negative members, you can also extract the root of an odd degree.

Let, further, members of inequality have different signs. Then, when it was erected into an odd degree, the meaning of inequality will not change, and when it is erected in an even degree of the meaning of the receiving inequality, nothing specified in general is impossible to say. In fact, when the number is erected into an odd degree, the number of the number is preserved and therefore the meaning of inequality does not change. If inequality is erected into an even degree, inequality with positive member is formed, and its meaning will depend on the absolute values \u200b\u200bof the members of the initial inequality, the inequality of the same meaning may be inequality as the initial, inequality of the opposite meaning and even equality!

All of the above-described inequality is useful to check the following example.

Example 1. Early the following inequalities to the specified degree, changing the sign of inequality to the opposite or on the sign of equality.

a) 3\u003e 2 into degree 4; b) to the extent 3;

c) to the degree 3; d) to the degree 2;

e) to the degree 5; e) to the degree 4;

g) 2\u003e -3 to the degree 2; h) to the degree 2

6. From inequality it is possible to move to inequality between if members of the inequality are both positive or both are negative, then there is an inequality of the opposite meaning between their reverse values:

Evidence. If a and b are one sign, then their work is positive. We divide into inequality

i.e., what was required to get.

If members of inequality have opposite signs, the inequality between their inverse values \u200b\u200bhas the same meaning, since the signs of the reverse values \u200b\u200bare the same as the signs of the values \u200b\u200bthemselves.

Example 2. Check the last property 6 on the following inequalities:

7. Logarithming inequalities can be performed only in the case when members of inequalities are positive (negative numbers and zero logarithms do not have).

Let be . Then when will be

and with will be

The correctness of these statements is based on the monotony of the logarithmic function, which increases, if the basis and decreases

So, in the logarithming of inequality consisting of positive members, based on the basis, a greater unit, an inequality of the same meaning is formed as this, and during its logarithming on a positive basis, a smaller unit - the inequality of the opposite meaning.

8. If, if, but, then.

It immediately follows from the properties of monotony indicative function (p. 42), which increases in case of decreasing if

When the inequality of the same meaning is formed in the early addition of inequality as the same sense as the data.

Evidence. We prove this statement for two inequalities, although it is true for any number of inequalities folded. Let the inequality be given

By definition, the number will be positive; Then their amount is also positive, i.e.

Grouping otherwise components, we get

and, therefore,

and this was necessary to prove.

It is impossible to say anything defined in the general case about the meaning of inequality obtained by the addition of two or several inequalities of different meaning.

10. If from one inequality to subtract the other inequality of the opposite sense, then the inequality is formed the same meaning as the first.

Evidence. Let two inequalities of different meaning given. The second of them by the ratio of irreversibility can be rewritten as follows: D\u003e s. Moving now two inequalities of the same sense and get inequality

of the same meaning. From the latter found

and this was necessary to prove.

It is impossible to say anything defined in the general case about the meaning of inequality obtained by subtracting from one inequality of other inequality of the same meaning.

Inequalities in mathematics play a prominent role. At school basically we are dealing with numerical inequalitiesWith the definition of which we will begin this article. And then list and justify properties of numerical inequalitieswhere all the principles of working with inequalities are based.

Immediately note that many properties of numerical inequalities are similar. Therefore, we will state the material according to the same scheme: we formulate the property, give it a justification and examples, after which we turn to the following property.

Navigating page.

Numeric inequalities: definition, examples

When we introduced the concept of inequality, they noticed that inequalities often determine their appearance. So inequalities we called having a meaning of algebraic expressions that contain signs are not equal to ≠, less<, больше >, less than or equal to ≤ or more or equal ≥. Based on the determination, it is convenient to define a numerical inequality:

Meeting with numerical inequalities occurs in mathematics lessons in the first grade immediately after acquaintance with the first natural numbers from 1 to 9, and acquaintance with the comparison operation. True, there they are called simply inequalities, lowering the definition of "numeric". For clarity, it will not prevent a couple of examples of the simplest numeric inequalities from that stage of their study: 1<2 , 5+2>3 .

And then Ot natural numbers Knowledge applies to other types of numbers (whole, rational, valid numbers), the rules of their comparison are studied, and this significantly expands the species diversity of numeric inequalities: -5\u003e -72, 3\u003e -0.275 · (7-5.6) ,.

Properties of numerical inequalities

In practice, working with inequalities allows a number properties of numerical inequalities. They arise from the concept of inequality that introduced by us. In relation to the numbers, this concept is given by the following statement, which can be considered the definition of "less" ratios and "more" on the set of numbers (it is often referred to as the difference definition of inequality):

Definition.

• number a. more numbers b Then and only if the difference A-B is a positive number;
• the number A is less than the number B if and only if the difference A-B is a negative number;
• the number A is equal to the number B if and only if the difference A-B is zero.

This definition can be removed into the definition of the relationship "less than or equal" and "greater than or equal." Here is its wording:

Definition.

• number a greater than or equal to the number B then and only if A-B is a non-negative number;
• the number A is less than or equal to the number B if and only if A-B is an inability.

We will use these definitions in the proof of the properties of numerical inequalities, to which we go to the survey.

Basic properties

Review Let's start with the three main properties of inequalities. Why are they basic? Because they are a reflection of the properties of inequalities in the general sense, and not only in relation to numerical inequalities.

Numeric inequalities recorded using signs< и >Characteristic:

As for the numerical inequalities recorded using the signs of non-strict inequality ≤ and ≥, then they have the property of reflexivity (and not antireflexivity), since the inequalities A≤A and A≥A include the case of equality a \u003d a. Also, they are also characteristic of antisymmetry and transitivity.

So, numeric inequalities recorded with the signs of ≤ and ≥, possess the properties:

• reflexiveness a≥a and a≤a - faithful inequalities;
• antisymmetry, if A≤b, then b≥a, and if A≥B, then b≤a.
• transitivity, if a≤b and b≤c, then a≤c, and also, if a≥b and b≥c, then A≥C.

Their proof is very similar to those already cited, so we will not stop at them, but we turn to other important properties of numerical inequalities.

Other important properties of numerical inequalities

Supplement the main properties of numerical inequalities of another series of results that are of great practical importance. They are based on the methods of assessing the values \u200b\u200bof expressions, they are based on principles solutions of inequalities etc. Therefore, it is advisable to deal well with them.

In this item, the properties of inequalities will be formulated only for one sign of strict inequality, but it is worth it in mind that similar properties will be fair and for the opposite sign to it, as well as for signs of non-strategic inequalities. Let us explain this on the example. Below we formulate and prove such an inequality property: if a

• if a\u003e b, then A + C\u003e B + C;
• if A≤b, then A + C≤B + C;
• if A≥B, then A + C≥B + C.

For convenience, imagine the properties of numerical inequalities in the form of a list, with this we will give a corresponding statement, write it formally using letters, lead proof, after which it is possible to show examples of use. And at the end of the article we will reduce all the properties of numerical inequalities in the table. Go!