The set of all real numbers can be represented as the union of three sets: the set of positive numbers, the set of negative numbers and the set consisting of one number - the number zero. To indicate that the number A positive, use the recording a > 0, to indicate a negative number use another notation a< 0 .
The sum and product of positive numbers are also positive numbers. If the number A negative, then the number -A positive (and vice versa). For any positive number a there is a positive rational number r, What r< а . These facts underlie the theory of inequalities.
By definition, the inequality a > b (or, what is the same, b< a) имеет место в том и только в том случае, если а - b >0, i.e. if the number a - b is positive.
Consider, in particular, the inequality A< 0 . What does this inequality mean? According to the above definition, it means that 0 - a > 0, i.e. -a > 0 or, in other words, what is the number -A positively. But this takes place if and only if the number A negative. So inequality A< 0 means that the number but negative.
The notation is also often used ab(or, what is the same, bа).
Record ab, by definition, means that either a > b, or a = b. If we consider the record ab as an indefinite statement, then in the notation of mathematical logic we can write
(a b) [(a > b) V (a = b)]
Example 1. Are the inequalities 5 0, 0 0 true?
The inequality 5 0 is a complex statement consisting of two simple statements connected by the logical connective “or” (disjunction). Either 5 > 0 or 5 = 0. The first statement 5 > 0 is true, the second statement 5 = 0 is false. By the definition of a disjunction, such a complex statement is true.
The entry 00 is discussed similarly.
Inequalities of the form a > b, a< b we will call them strict, and inequalities of the form ab, ab- not strict.
Inequalities a > b And c > d(or A< b And With< d ) will be called inequalities of the same meaning, and inequalities a > b And c< d - inequalities of opposite meaning. Note that these two terms (inequalities of the same and opposite meaning) refer only to the form of writing the inequalities, and not to the facts themselves expressed by these inequalities. So, in relation to inequality A< b inequality With< d is an inequality of the same meaning, and in the notation d>c(meaning the same thing) - an inequality of the opposite meaning.
Along with inequalities of the form a>b, ab so-called double inequalities are used, i.e., inequalities of the form A< с < b
, ac< b
, a< cb
,
acb. By definition, a record
A< с < b
(1)
means that both inequalities hold:
A< с And With< b.
The inequalities have a similar meaning acb, ac< b, а < сb.
Double inequality (1) can be written as follows:
(a< c < b) [(a < c) & (c < b)]
and double inequality a ≤ c ≤ b can be written in the following form:
(a c b) [(a< c)V(a = c) & (c < b)V(c = b)]
Let us now proceed to the presentation of the basic properties and rules of action on inequalities, having agreed that in this article the letters a, b, c stand for real numbers, and n means natural number.
1) If a > b and b > c, then a > c (transitivity).
Proof.
Since by condition a > b And b > c, then the numbers a - b And b - c are positive, and therefore the number a - c = (a - b) + (b - c), as the sum of positive numbers, is also positive. This means, by definition, that a > c.
2) If a > b, then for any c the inequality a + c > b + c holds.
Proof.
Because a > b, then the number a - b positively. Therefore, the number (a + c) - (b + c) = a + c - b - c = a - b is also positive, i.e.
a + c > b + c.
3) If a + b > c, then a > b - c, that is, any term can be transferred from one part of the inequality to another by changing the sign of this term to the opposite.
The proof follows from property 2) it is sufficient for both sides of the inequality a + b > c add number - b.
4) If a > b and c > d, then a + c > b + d, that is, when adding two inequalities of the same meaning, an inequality of the same meaning is obtained.
Proof.
By virtue of the definition of inequality, it is sufficient to show that the difference
(a + c) - (b + c) positive. This difference can be written as follows:
(a + c) - (b + d) = (a - b) + (c - d).
Since according to the condition of the number a - b And c - d are positive, then (a + c) - (b + d) there is also a positive number.
Consequence. From rules 2) and 4) the following Rule for subtracting inequalities follows: if a > b, c > d, That a - d > b - c(for proof it is enough to apply both sides of the inequality a + c > b + d add number - c - d).
5) If a > b, then for c > 0 we have ac > bc, and for c< 0 имеем ас < bc.
In other words, when multiplying both sides of an inequality with either a positive number, the inequality sign is preserved (i.e., an inequality of the same meaning is obtained), but when multiplied by a negative number, the inequality sign changes to the opposite (i.e., an inequality of the opposite meaning is obtained.
Proof.
If a > b, That a - b is a positive number. Therefore, the sign of the difference ac-bc = c(a - b) matches the sign of the number With: If With is a positive number, then the difference ac - bc is positive and therefore ac > bс, and if With< 0 , then this difference is negative and therefore bc - ac positive, i.e. bc > ac.
6) If a > b > 0 and c > d > 0, then ac > bd, that is, if all terms of two inequalities of the same meaning are positive, then when multiplying these inequalities term by term, an inequality of the same meaning is obtained.
Proof.
We have ac - bd = ac - bc + bc - bd = c(a - b) + b(c - d). Because c > 0, b > 0, a - b > 0, c - d > 0, then ac - bd > 0, i.e. ac > bd.
Comment. From the proof it is clear that the condition d > 0 in the formulation of property 6) is unimportant: for this property to be valid, it is sufficient that the conditions be met a > b > 0, c > d, c > 0. If (if the inequalities are fulfilled a > b, c > d) numbers a, b, c will not all be positive, then the inequality ac > bd may not be fulfilled. For example, when A = 2, b =1, c= -2, d= -3 we have a > b, c > d, but inequality ac > bd(i.e. -4 > -3) failed. Thus, the requirement that the numbers a, b, c be positive in the formulation of property 6) is essential.
7) If a ≥ b > 0 and c > d > 0, then (division of inequalities).
Proof.
We have 
The numerator of the fraction on the right side is positive (see properties 5), 6)), the denominator is also positive. Hence,. This proves property 7).
Comment. Let us note an important special case of rule 7), obtained for a = b = 1: if c > d > 0, then. Thus, if the terms of the inequality are positive, then when passing to the reciprocals we obtain an inequality of the opposite meaning. We invite readers to check that this rule also holds in 7) If ab > 0 and c > d > 0, then (division of inequalities).
Proof. That.
We have proved above several properties of inequalities written using the sign > (more). However, all these properties could be formulated using the sign < (less), since inequality b< а means, by definition, the same as inequality a > b. In addition, as is easy to verify, the properties proved above are also preserved for non-strict inequalities. For example, property 1) for non-strict inequalities will have the following form: if ab and bc, That ac.
Of course, the above does not limit the general properties of inequalities. There is also a whole series of general inequalities related to the consideration of power, exponential, logarithmic and trigonometric functions. The general approach for writing this kind of inequalities is as follows. If some function y = f(x) increases monotonically on the segment [a, b], then for x 1 > x 2 (where x 1 and x 2 belong to this segment) we have f (x 1) > f(x 2). Likewise, if the function y = f(x) monotonically decreases on the interval [a, b], then when x 1 > x 2 (where x 1 And X 2 belong to this segment) we have f(x 1)< f(x 2 ). Of course, what has been said is no different from the definition of monotonicity, but this technique is very convenient for memorizing and writing inequalities.
So, for example, for any natural number n the function y = xn is monotonically increasing along the ray }
