§ 1 Universal way of comparing numbers
Let's get acquainted with the basic properties of numerical inequalities, and also consider a universal way of comparing numbers.
The result of comparing numbers can be written using equality or inequality. Inequality can be strict or lax. For example, a> 3 is a strict inequality; a≥3 is a loose inequality. The way numbers are compared depends on the kind of numbers being compared. For example, if you need to compare decimal fractions, then we compare them bit by bit; if it is necessary to compare ordinary fractions with different denominators, then it is necessary to bring them to a common denominator and compare the numerators. But there is a universal way to compare numbers. It consists in the following: find the difference between the numbers a and b; if a - b> 0, that is, a positive number, then a> b; if a - b< 0, то есть отрицательное число, то a < b; если a - b = 0, то a = b. Этот способ удобно использовать для доказательства неравенств. Например, доказать неравенство:
2b2 - 6b + 1> 2b (b- 3)
Let's use a universal comparison method. Find the difference between the expressions 2b2 - 6b + 1 and 2b (b - 3);
2b2 - 6b + 1 - 2b (b-3) = 2b2 - 6b + 1 - 2b2 + 6b; we give similar terms and get 1. Since 1 is greater than zero, a positive number, then 2b2 - 6b + 1> 2b (b-3).
§ 2 Properties of numerical inequalities
Property 1. If a> b, b> c, then a> c.
Proof. If a> b, then the difference a - b> 0, that is, a positive number. If b> c, then the difference b - c> 0 is a positive number. Add the positive numbers a - b and b - c, open the brackets and give similar terms, we get (a - b) + (b - c) = a- b + b - c = a - c. Since the sum of positive numbers is a positive number, it means that a - c is a positive number. Therefore, a> c, as required.
Property 2. If a< b, c- любое число, то a + с < b+ с. Это свойство можно трактовать так: «К обеим частям верного неравенства можно прибавить одно и то же число, при этом знак неравенства не изменится».
Proof. Find the difference between the expressions a + c and b + c, open the brackets and give similar terms, we get (a + c) - (b + c) = a + c - b - c = a - b. By condition a< b, тогда разность a - b- отрицательное число. Значит, и разность (a + с) -(b+ с) отрицательна. Следовательно, a + с < b+ с, что и требовалось доказать.
Property 3. If a< b, c - положительное число, то aс < bс.
If a< b, c- отрицательное число, то aс >bc.
Proof. Let us find the difference between the expressions ac and bc, put it outside the brackets with, then we have ac-bc = c (a-b). But since a If we multiply a negative number a-b by a positive number c, then the product c (a-b) is negative, therefore, the difference ac-bc is negative, which means that ac If the negative number a-b is multiplied by the negative number c, then the product c (a-b) will be positive, therefore, the difference ac-bc will be positive, which means that ac> bc. Q.E.D. For example, a Since division can be replaced by multiplication by the inverse number, = n ∙, the proved property can be applied to division. Thus, the meaning of this property is as follows: “Both parts of the inequality can be multiplied or divided by the same positive number without changing the sign of the inequality. Both parts of the inequality can be multiplied or divided by a negative number, in this case it is necessary to change the sign of the inequality to the opposite sign. " Consider a corollary to property 3. Consequence. If a Proof. We divide both sides of the inequality a
reduce the fractions and get The statement is proven. Indeed, for example, 2< 3, но Property 4. If a> b and c> d, then a + c> b + d. Proof. Since a> b and c> d, the differences a-b and c-d are positive numbers. Then the sum of these numbers is also a positive number (a-b) + (c-d). Expand the brackets and group (a-b) + (c-d) = a-b + c-d = (a + c) - (b + d). In view of this equality, the resulting expression (a + c) - (b + d) will be a positive number. Therefore, a + c> b + d. Inequalities of the form a> b, c> d or a< b, c< d называют неравенствами одинакового смысла, а неравенства a>b, c Property 5. If a> b, c> d, then ac> bd, where a, b, c, d are positive numbers. Proof. Since a> b and c is a positive number, using property 3, we obtain ac> bc. Since c> d and b is a positive number, then bc> bd. Therefore, by the first property ac> bd. The meaning of the proven property is as follows: "If we multiply term-by-term inequalities of the same meaning, for which the left and right sides are positive numbers, then we get an inequality of the same meaning." For example 6< a < 7, 4 < b< 5 тогда, 24 < ab < 35. Property 6. If a< b, a и b - положительные числа, то an< bn, где n- натуральное число. Proof. If we multiply term by term n given inequalities a< b, то, согласно утверждению свойства 5, получим an< bn. Прочесть доказанное утверждение можно так: «Если обе части неравенства - положительные числа, то их можно возвести в одну и ту же натуральную степень, сохранив знак неравенства». § 3 Application of properties Let's consider an example on the application of the properties we have considered. Let 33< a < 34, 3 < b< 4. Оценить сумму a + b, разность a - b, произведение a ∙ b и частное a: b. 1) Estimate the sum a + b. Using property 4, we get 33 + 3< a + b < 34 + 4 или 36 < a+ b <38. 2) Let us estimate the difference a - b. Since there is no subtraction property, we replace the difference a - b with the sum a + (- b). Let's first estimate (- b). For this, using property 3, both sides of inequality 3< b< 4 умножим на -1, при этом меняем знак неравенства на противоположный знак 3 ∙ (-1) >b ∙ (-1)> 4 ∙ (-1). We get -4< -b< -3. Теперь можно сложить два неравенства одного знака 33< a < 34 и -4< -b< -3. Имеем 2 9< a - b <31. 3) Estimate the product a ∙ b. By Property 5, we multiply the inequalities of the same sign Inequalities in mathematics play a prominent role. At school, we mainly deal with numerical inequalities, with the definition of which we will begin this article. And then we will list and justify properties of numerical inequalities, on which all the principles of working with inequalities are based. We note right away that many of the properties of numerical inequalities are similar. Therefore, we will present the material according to the same scheme: we formulate a property, give its justification and examples, and then move on to the next property. Page navigation. When we introduced the concept of inequality, we noticed that inequalities are often defined by the way they are written. So inequalities we called meaningful algebraic expressions containing signs not equal to ≠, less<, больше >, less than or equal to ≤ or greater than or equal to ≥. Based on the above definition, it is convenient to give a definition of a numerical inequality: The encounter with numerical inequalities occurs in mathematics lessons in the first grade immediately after getting acquainted with the first natural numbers from 1 to 9, and getting acquainted with the comparison operation. True, there they are simply called inequalities, omitting the definition of "numerical". For clarity, it does not hurt to give a couple of examples of the simplest numerical inequalities from that stage of their study: 1<2
, 5+2>3
. And further from natural numbers, knowledge is extended to other types of numbers (integers, rational, real numbers), the rules for their comparison are studied, and this significantly expands the species diversity of numerical inequalities: −5> −72, 3> −0.275 (7−5, 6),. In practice, working with inequalities allows the series properties of numerical inequalities... They follow from the notion of inequality introduced by us. In relation to numbers, this concept is defined by the following statement, which can be considered a definition of the relationship "less" and "more" on the set of numbers (it is often called the difference definition of inequality): Definition.
This definition can be rewritten into a definition of the relationship "less than or equal to" and "greater than or equal to". Here is its wording: Definition.
We will use these definitions in proving the properties of numerical inequalities, which we will now review. We begin our survey with three main properties of inequalities. Why are they essential? Because they are a reflection of the properties of inequalities in the most general sense, and not just in relation to numerical inequalities. Numerical inequalities written using signs< и >, typically: As for the numerical inequalities written using the signs of non-strict inequalities ≤ and ≥, they have the property of reflexivity (and not anti-reflexivity), since the inequalities a≤a and a≥a include the case of equality a = a. They are also characterized by antisymmetry and transitivity. So, numerical inequalities written using the signs ≤ and ≥ have the following properties: Their proofs are very similar to those already given, so we will not dwell on them, but move on to other important properties of numerical inequalities. Let us supplement the basic properties of numerical inequalities with a series of results that are of great practical importance. Methods for evaluating the values of expressions are based on them, the principles are based on them solutions to inequalities etc. Therefore, it is advisable to deal with them well. In this subsection, the properties of inequalities will be formulated only for one sign of a strict inequality, but it should be borne in mind that similar properties will be valid for the opposite sign, as well as for the signs of non-strict inequalities. Let us explain this with an example. Below we formulate and prove the following property of inequalities: if a
For convenience, we will present the properties of numerical inequalities in the form of a list, in this case we will give the corresponding statement, write it down formally using letters, give a proof, and then show examples of use. And at the end of the article, we will summarize all the properties of numerical inequalities in a table. Go! Adding (or subtracting) any number to both sides of a valid numeric inequality produces a valid numeric inequality. In other words, if the numbers a and b are such that a
For the proof, we compose the difference between the left and right sides of the last numerical inequality, and show that it is negative under the condition a (a + c) - (b + c) = a + c − b − c = a − b... Since by condition a
We do not dwell on the proof of this property of numerical inequalities for the subtraction of the number c, since the subtraction on the set of real numbers can be replaced by the addition of −c. For example, if you add 15 to both sides of the correct numerical inequality 7> 3, you get the correct numerical inequality 7 + 15> 3 + 15, which is the same thing, 22> 18. If both sides of a true numerical inequality are multiplied (or divided) by the same positive number c, then you get the correct numerical inequality. If both sides of the inequality are multiplied (or divided) by a negative number c, and the sign of the inequality is reversed, then the correct inequality is obtained. In literal form: if for numbers a and b the inequality a b c. Proof. Let's start with the case when c> 0. Let us compose the difference between the left and right sides of the numerical inequality being proved: a c - b c = (a - b) c. Since by condition a 0, then the product (a - b) · c will be a negative number as the product of a negative number a - b and a positive number c (which follows from). Therefore, a c - b c<0
, откуда a·c We do not dwell on the proof of the considered property for dividing both sides of a true numerical inequality by the same number c, since division can always be replaced by multiplication by 1 / c. Let us show an example of applying the analyzed property to concrete numbers. For example, you can both sides of the true numerical inequality 4<6
умножить на положительное число 0,5
, что дает верное числовое неравенство −4·0,5<6·0,5
, откуда −2<3
. А если обе части верного числового неравенства −8≤12
разделить на отрицательное число −4
, и изменить знак неравенства ≤ на противоположный ≥, то получится верное числовое неравенство −8:(−4)≥12:(−4)
, откуда 2≥−3
. Two practically valuable results follow from the property just examined of multiplying both sides of a numerical equality by a number. So we will formulate them in the form of consequences. All the properties discussed above in this subsection are united by the fact that first the correct numerical inequality is given, and from it, by means of some manipulations with the parts of the inequality and the sign, another correct numerical inequality is obtained. Now we will give a block of properties in which not one, but several correct numerical inequalities are initially given, and the new result is obtained from their joint use after adding or multiplying their parts. If the numbers a, b, c, and d satisfy the inequalities a
Let us prove that (a + c) - (b + d) is a negative number, this will prove that a + c By induction, this property extends to term-by-term addition of three, four, and, in general, any finite number of numerical inequalities. So, if the numbers a 1, a 2,…, a n and b 1, b 2,…, b n satisfy the inequalities a 1 a 1 + a 2 +… + a n . For example, we are given three correct numerical inequalities of the same sign −5<−2
, −1<12
и 3<4
. Рассмотренное свойство числовых неравенств позволяет нам констатировать, что неравенство −5+(−1)+3<−2+12+4
– тоже верное. You can multiply term-by-term numerical inequalities of the same sign, both sides of which are represented by positive numbers. In particular, for two inequalities a
For the proof, we can multiply both sides of the inequality a
The indicated property is also valid for the multiplication of any finite number of true numerical inequalities with positive parts. That is, if a 1, a 2, ..., a n and b 1, b 2, ..., b n are positive numbers, and a 1 a 1 · a 2 ·… · a n . Separately, it is worth noting that if the record of numerical inequalities contains non-positive numbers, then their term-by-term multiplication can lead to incorrect numerical inequalities. For example, the numerical inequalities 1<3
и −5<−4
– верные и одного знака, почленное умножение этих неравенств дает 1·(−5)<3·(−4)
, что то же самое, −5<−12
, а это неверное неравенство. Consequence. Term-by-term multiplication of the same true inequalities of the form a
In conclusion of the article, as promised, we will collect all the studied properties in numerical inequality property table: Bibliography. LINEAR EQUATIONS AND INEQUALITIES I § 10 Basic properties of numerical inequalities 1. If a> b, then b< а
, and vice versa, if a< b
, then b> a.
Proof. Let be a> b
... By definition, this means that the number ( a - b
) is positive. If we put a minus sign in front of it, then the resulting number is ( a - b
) will obviously be negative. That's why - ( a - b
) < 0, или b - a
< 0. А это (опять же по определению) и означает, что b< a
. We invite students to prove the opposite statement on their own. The proved property of inequalities admits a simple geometric interpretation: if point A lies on the number line to the right of point B, then point B lies to the left of point A, and vice versa (see Fig. 20). 2. If a> b, a b> c, then a> c.
Geometrically, this property is as follows. Let point A (corresponding to the number a
) lies to the right of point B (corresponding to the number b
), and point B, in turn, lies to the right of point C (corresponding to the number with
). Then point A will even more so lie to the right of point C (Fig. 21). Let us give an algebraic proof of this property of inequalities. Let be a> b
, a b> c
... This means that the numbers ( a - b
) and ( b- c
) are positive. The sum of two positive numbers is obviously positive. That's why ( a - b
) + (b- c
)> 0, or a - c
> 0. But this also means that a
> with
. 3. If a> b, then for any number with a + c> b + c, a - c > b - with.
In other words, if the same number is added to both sides of the numerical inequality or the same number is subtracted from both sides, then the inequality will not be violated. Proof. Let be a> b
... It means that a - b
> 0. But a - b
= (a + c
) - (b + c
). That's why ( a + c
) - (b + c
)> 0. And by definition, this means that a + c> b + c
... It is shown similarly that a - c > b - with
. For example, if 1 1/2 is added to both sides of the inequality 5> 4, then we get Consequence. Any term in one part of the numerical inequality can be transferred to the other part of the inequality by changing the sign of this term to the opposite. Let, for example, a + b> c
... It is required to prove that a> c - b
... To prove from both sides of this inequality, it suffices to subtract the number b
. 4. Let be a> b... If c> 0, then ac> bc
. If with< 0
, then ace< bс
. In other words, if both sides of the numerical inequality are multiplied by a positive number, then the inequality is not violated; In short, this property is formulated as follows: The inequality persists when multiplied by a positive number term by term and reverses sign when multiplied by a negative number term by term. For example, multiplying the inequality 5> 1 term-by-7, we get 35> 7. The term-by-term multiplication of the same inequality by - 7 gives - 35< - 7. Proof of the 4th property. Let be a> b... This means that the number a - b positively. Product of two positive numbers a - b and with
is obviously also positive, i.e. ( a - b
) with
> 0, or The case is considered similarly when the number with
negatively. Product of a positive number a - b
by a negative number with
is obviously negative, i.e. Consequence. The inequality sign is preserved when dividing by a positive number, and reversed when dividing by a negative number. This follows from the fact that division by the number with
= / = 0 is equivalent to multiplying by the number 1 /
c
. Exercises
81. Can the inequality 2> 1 be multiplied term by a) a
2 + 1; b) | a
|; v) a
; d) 1 - 2a + a
2 so that the inequality sign is preserved? 82. Is it always 5 NS
more than 4 NS
, a - at
smaller at
? 83. What can be a number NS
if it is known that - NS
> 7? 84. Arrange in ascending order of numbers: a) a 2, 5a 2, 2a 2; b) 5 a
, 2a
; v) a
, a
2 , a
3. 85. Arrange in descending order of numbers a - b
, a
- 2b
, a
- 3b
. 86. Give a geometric interpretation to the third property of numerical inequalities. The set of all real numbers can be represented as the union of three sets: a set of positive numbers, a set of negative numbers and a set consisting of one number - the number zero. To indicate that the number a positively, use the recording a> 0, use a different notation to indicate a negative number a< 0
. The sum and product of positive numbers are also positive numbers. If the number a negative, then the number -a positively (and vice versa). For any positive number a, there is a positive rational number r, what r< а
... These facts form the basis of the theory of inequalities. By definition, the inequality a> b (or, which is the same, b< a) имеет место в том и только в том случае, если а - b >0, that is, 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, otherwise, that the number -a positively. But this is the case if and only if the number a negatively. So the inequality a< 0
means that the number but negative. The notation is also often used ab(or, equivalently, bа). (a b) [(a> b) V (a = b)] Example 1. Are the inequalities 5 0, 0 0 true? Inequality 5 0 is a complex statement consisting of two simple statements connected by a 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 disjunction, such a complex statement is true. The 00 entry is discussed in a similar way. Inequalities of the form a> b, a< b
will be called 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 the opposite meaning. Note that these two terms (inequalities of the same and opposite meaning) refer only to the form of notation of inequalities, and not to the facts themselves expressed by these inequalities. So, in relation to the inequality a< b
inequality with< d
is an inequality of the same meaning, and in the notation d> c(meaning the same thing) - inequality of the opposite meaning. Along with inequalities of the form a> b, ab so-called double inequalities are used, that is, inequalities of the form a< с < b
, ace< b
, a< cb
, a< с < b
(1) 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 the double inequality a ≤ c ≤ b can be written as follows: (a c b) [(a< c)V(a = c) & (c < b)V(c = b)] We now turn to the presentation of the basic properties and rules of action on inequalities, having agreed that in this article the letters a, b, c denote real numbers, and n means a 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 - with 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. 3) If a + b> c, then a> b - c, that is, any term can be transferred from one part of the inequality to the other 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 two inequalities of the same meaning are added together, an inequality of the same meaning is obtained. Proof. By the definition of the inequality, it suffices to show that the difference Consequence. Rules 2) and 4) imply the following rule for subtraction of inequalities: if a> b, c> d, then a - d> b - c(for the proof, it suffices to 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 both sides of the inequality are multiplied, neither the positive number of the inequality sign is preserved (that is, an inequality of the same meaning is obtained), and when multiplied by a negative number, the inequality sign changes to the opposite (that is, an inequality of the opposite meaning is obtained. Proof. If a> b, then a - b there is a positive number. Therefore, the sign of the difference ac-bc = c (a - b) matches the sign of a number with: if with is a positive number, then the difference ac - bc positive and therefore ac> bc, what 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 the term-by-term multiplication of these inequalities results in an inequality of the same meaning. 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. It is clear from the proof 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 a> b> 0, c> d, c> 0... If (when the inequalities a> b, c> d) numbers a, b, c are not all positive, then the inequality ac> bd may not be executed. For example, for 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 statement of property 6) is essential. 7) If a ≥ b> 0 and c> d> 0, then (division of inequalities). Proof. We have Comment. Note an important special case of rule 7), which is obtained for a = b = 1: if c> d> 0, then. Thus, if the terms of the inequality are positive, then in passing to reciprocal quantities we obtain an inequality of the opposite meaning. We invite the readers to check that this rule is also preserved in 7) If ab> 0 and c> d> 0, then (division of inequalities). Proof. then. We have proved above several properties of the inequalities written using the sign >
(more). However, all these properties could be formulated using the sign <
(less), since the inequality b< а
means, by definition, the same as the inequality a> b... Moreover, as is easy to verify, the properties proved above are also valid for nonstrict inequalities. For example, property 1) for nonstrict inequalities will have the following form: if ab and bc, then ace. Of course, the above does not limit the general properties of inequalities. There is still whole line general inequalities related to the consideration of power, exponential, logarithmic and trigonometric functions. The general approach for writing this kind of inequality 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). Similarly, if the function y = f (x) decreases monotonically on the segment [a, b], then at x 1> x 2 (where x 1 and NS 2 belong to this segment) we have f (x 1)< f(x 2
). Of course, what has been said does not differ from the definition of monotonicity, but this technique is very convenient for memorizing and writing inequalities. So, for example, for any natural n the function y = x n is monotonically increasing on the ray }
Numerical inequalities: definition, examples
Properties of numerical inequalities
Basic properties
Other important properties of numerical inequalities
6 1/2> 5 1/2. Subtracting 5 from both sides of this inequality, we get 0> - 1.
if both sides of the inequality are multiplied by a negative number, then the sign of the inequality will change to the opposite.
ac - bc>
0. Therefore ac> bc
.
(a - b) c<
0; therefore ac - bc<
0, whence ace< bс
.
Recording ab, by definition, means that either a> b or a = b... If we consider the entry ab as an indefinite statement, then in the notation of mathematical logic we can write
acb... By definition, a record
means that both inequalities hold:
a + c> b + c.
(a + c) - (b + c) positive. This difference can be written as follows:
(a + c) - (b + d) = (a - b) + (c - d).
Since, by condition, the numbers a - b and c - d are positive then (a + c) - (b + d) there is also a positive number.
The numerator of the fraction on the right is positive (see properties 5), 6)), the denominator is also positive. Hence,. This proves property 7).
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