Main types of inequalities and their properties. Basic properties of numerical inequalities
Lesson and presentation on the topic: "Basic properties of numerical inequalities and methods for solving them."
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Combinatorics and probability theory Equations and inequalities
Introduction to Numerical Inequalities
Guys, we have already encountered inequalities, for example, when we began to get acquainted with the concept of square root. Intuitively, using inequalities you can estimate which of the given numbers is greater or less. For a mathematical description, it is enough to add a special symbol that will mean either more or less.Writing the expression $a>b$ in mathematical language means that the number $a$ more number$b$. In turn, this means that $a-b$ is a positive number.
Writing the expression $a
Like almost all mathematical objects, inequalities have certain properties. We will study these properties in this lesson.
Property 1.
If $a>b$ and $b>c$, then $a>c$.
Proof.
Obviously, $10>5$, and $5>2$, and of course $10>2$. But mathematics loves rigorous proofs for the most general case.
If $a>b$, then $a-b$ is a positive number. If $b>c$, then $b-c$ is a positive number. Let's add the two resulting positive numbers.
$a-b+b-c=a-c$.
The sum of two positive numbers is a positive number, but then $a-c$ is also a positive number. From which it follows that $a>c$. The property has been proven.
More clearly this property can be shown using a number line. If $a>b$, then the number $a$ on the number line will lie to the right of $b$. Accordingly, if $b>c$, then the number $b$ will lie to the right of the number $c$.
As can be seen from the figure, point $a$ in our case is located to the right of the point$c$, which means that $a>c$.
Property 2.
If $a>b$, then $a+c>b+c$.
In other words, if the number $a$ is greater than the number $b$, then no matter what number we add (positive or negative) to these numbers, the inequality sign will also be preserved. This property is very easy to prove. You need to do a subtraction. The variable that was added will disappear and the original inequality will be correct.
Property 3.
a) If both sides of the inequality are multiplied by a positive number, then the inequality sign is preserved.
If $a>b$ and $c>0$, then $ac>bc$.
b) If both sides of the inequality are multiplied by a negative number, then the sign of the inequality should be reversed.
If $a>b$ and $c If $a
When dividing, you should proceed in the same way (divide by a positive number - the sign remains the same, divide by a negative number - the sign changes).
Property 4.
If $a>b$ and $c>d$, then $a+c>b+d$.
Proof.
From the condition: $a-b$ is a positive number and $c-d$ is a positive number.
Then the sum $(a-b)+(c-d)$ is also a positive number.
Let's swap some terms $(a+c)-(b+d)$.
Changing the places of the terms does not change the sum.
This means $(a+c)-(b+d)$ is a positive number and $a+c>b+d$.
The property has been proven.
Property 5.
If $a, b ,c, d$ are positive numbers and $a>b$, $c>d$, then $ac>bd$.
Proof.
Since $a>b$ and $c>0$, then, using property 3, we have $ac>bc$.
Since $c>d$ and $b>0$, then, using property 3, we have $cb>bd$.
So, $ac>bc$ and $bc >bd$.
Then, using property 1, we obtain $ac>bd$. Q.E.D.
Definition.
Inequalities of the form $a>b$ and $c>d$ ($a
Then property 5 can be rephrased. When multiplying inequalities of the same meaning, whose left and right sides are positive, an inequality of the same meaning is obtained.
Property 6.
If $a>b$ ($a>0$, $b>0$), then $a^n>b^n$, where $n$ is any natural number.
If both sides of the inequality are positive numbers and raise them to the same natural degree, then we get an inequality of the same meaning.
Note: if $n$ is an odd number, then for numbers $a$ and $b$ of any sign, Property 6 is satisfied.
Property 7.
If $a>b$ ($a>0$, $b>0$), then $\frac(1)(a)
Proof.
To prove this property, it is necessary to subtract $\frac(1)(a)-\frac(1)(b)$ to obtain a negative number.
$\frac(1)(a)-\frac(1)(b)=\frac(b-a)(ab)=\frac(-(a-b))(ab)$.
We know that $a-b$ is a positive number, and the product of two positive numbers is also a positive number, i.e. $ab>0$.
Then $\frac(-(a-b))(ab)$ is a negative number. The property has been proven.
Property 8.
If $a>0$, then the following inequality holds: $a+\frac(1)(a)≥2$.
Proof.
Let's consider the difference.
$a+\frac(1)(a)-2=\frac(a^2-2a+1)(a)=\frac((a-1)^2)(a)$ is a non-negative number.
The property has been proven.
Property 9. Cauchy's inequality (the arithmetic mean is greater than or equal to the geometric mean).
If $a$ and $b$ are non-negative numbers, then the inequality holds: $\frac(a+b)(2)≥\sqrt(ab)$.
Proof.
Let's consider the difference:
$\frac(a+b)(2)-\sqrt(ab)=\frac(a-2\sqrt(ab)+b)(2)=\frac((\sqrt(a)-\sqrt(b ))^2)(2)$ is a non-negative number.
The property has been proven.
Examples of solving inequalities
Example 1.It is known that $-1.5
b) $-2b$.
c) $a+b$.
d) $a-b$.
e) $b^2$.
e) $a^3$.
g) $\frac(1)(b)$.
Solution.
a) Let’s use property 3. Multiply by a positive number, which means the inequality sign does not change.
$-1.5*3
B) Let's use property 3. Multiply by a negative number, which means the sign of the inequality changes. D) Multiply all parts of the inequality $3.1 D) All parts of the inequality are positive, squaring them, we obtain an inequality of the same meaning. E) The degree of inequality is odd, then you can safely raise it to a power and not change the sign. G) Let's use property 7. Example 2. Solution. § 1 A universal way to compare numbers Let's get acquainted with the basic properties of numerical inequalities, and also consider a universal way to compare numbers. The result of comparing numbers can be written using equality or inequality. Inequality can be strict or non-strict. For example, a>3 is a strict inequality; a≥3 is a weak inequality. The way numbers are compared depends on the type of numbers being compared. For example, if we need to compare decimal fractions, then we compare them place by digit; If you need to compare ordinary fractions with different denominators, then you need to bring them to a common denominator and compare the numerators. But there is a universal way to compare numbers. It consists of 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. Let's find the difference between the expressions 2b2 - 6b + 1 and 2b(b - 3); 2b2 - 6b + 1- 2b(b-3)= 2b2 - 6b + 1 - 2b2 + 6b; Let's add 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. Let's add the positive numbers a - b and b - c, open the brackets and add similar terms, we get (a - b) + (b - c) = a - b + b - c = a - c. Since the sum of positive numbers is a positive number, then a - c is a positive number. Therefore, a > c, which is what needed to be proved. 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. Let's find the difference between the expressions a + c and b+ c, open the brackets and add 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's find the difference between the expressions ac and bc, put c out of brackets, 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 ac If a negative number a-b is multiplied by a negative number c, then the product c(a-b) will be positive, therefore, the difference ac-bc will be positive, which means ac>bc. Q.E.D. For example, a Since division can be replaced by multiplication by the reciprocal number, = n∙, the proven property can also be applied to division. Thus, the meaning of this property is as follows: “Both sides of an inequality can be multiplied or divided by the same positive number, and the sign of the inequality does not change. Both sides of the inequality can be multiplied or divided by a negative number, but it is necessary to change the sign of the inequality to the opposite sign.” Let us consider the corollary to property 3. Consequence. If a Proof. Let us divide both sides of the inequality a
reduce the fractions and get The statement has been 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). Let's open 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 are a positive number, then, using property 3, we get 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, whose left and right sides are positive numbers, we obtain 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 n given inequalities term by term 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 of 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) Let's estimate the sum a + b. Using property 4, we get 33 + 3< a + b < 34 + 4 или 36 < a+ b <38. 2) Let's estimate the difference a - b. Since there is no subtraction property, we replace the difference a - b with the sum a + (-b). First let's estimate (- b). To do 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) Let's estimate the product a ∙ b. By property 5, we multiply inequalities of the same sign Inequalities play a prominent role in mathematics. 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 principles of working with inequalities are based. Let us immediately note that many 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, after which we 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 we called inequalities meaningful algebraic expressions containing the signs not equal to ≠, less than<, 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 meeting 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 becoming familiar with the comparison operation. True, there they are simply called inequalities, omitting the definition of “numerical”. For clarity, it wouldn’t 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 extends to other types of numbers (integer, rational, real numbers), the rules for their comparison are studied, and this significantly expands the variety of types of numerical inequalities: −5>−72, 3>−0.275 (7−5, 6) , . In practice, working with inequalities allows a number of properties of numerical inequalities. They follow from the concept of inequality we introduced. In relation to numbers, this concept is given by the following statement, which can be considered a definition of the relations “less than” and “more than” on a set of numbers (it is often called the difference definition of inequality): Definition.
This definition can be reworked into the definition of the relations “less than or equal to” and “greater than or equal to.” Here is his wording: Definition.
We will use these definitions when proving the properties of numerical inequalities, to a review of which we proceed. We begin the review with three main properties of inequalities. Why are they basic? Because they are a reflection of the properties of inequalities in the most general sense, and not only in relation to numerical inequalities. Numerical inequalities written using signs< и >, characteristic: As for numerical inequalities written using the weak inequality signs ≤ 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 proof is 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 estimating the values of expressions are based on them; principles are based on them solutions to inequalities etc. Therefore, it is advisable to understand them well. In this section, we will formulate the properties of inequalities only for one sign of strict inequality, but it is worth keeping in mind that similar properties will be valid for the opposite sign, as well as for signs of non-strict inequalities. Let's 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, while we will give the corresponding statement, write it 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. Let's go! Adding (or subtracting) any number to both sides of a true numerical inequality produces a true numerical inequality. In other words, if the numbers a and b are such that a
To prove it, let’s make up 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 subtracting a number c, since on the set of real numbers subtraction can be replaced by adding −c. For example, if you add the number 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 valid numerical inequality are multiplied (or divided) by the same positive number c, you get a valid 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 inequality will be true. In literal form: if the numbers a and b satisfy the inequality a b·c. Proof. Let's start with the case when c>0. Let's make up 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's show an example of using the analyzed property on specific numbers. For example, you can have both sides of the correct 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
. From the just discussed property of multiplying both sides of a numerical equality by a number, two practically valuable results follow. So we formulate them in the form of consequences. All the properties discussed above in this paragraph are united by the fact that first a correct numerical inequality is given, and from it, through some manipulations with the parts of the inequality and the sign, another correct numerical inequality is obtained. Now we will present a block of properties in which not one, but several correct numerical inequalities are initially given, and a 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 This property numerical inequalities are usually formulated in the following form: you can add term by term true inequalities of the same sign (by this we mean that all inequalities are written using one sign, for example,<, и под почленным сложением понимают сложение чисел, стоящих по одну сторону этого знака). 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 for the numbers a 1, a 2, …, a n and b 1, b 2, …, b n the following inequalities are true: 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 numerical inequalities of the same sign term by term, both sides of which are represented by positive numbers. In particular, for two inequalities a
To prove it, you can multiply both sides of the inequality a
This property is also true 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 notation for numerical inequalities contains non-positive numbers, then their term-by-term multiplication can lead to incorrect numerical inequalities. For example, 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. Termwise multiplication of identical true inequalities of the form a
At the end of the article, as promised, we will collect all the studied properties in table of properties of numerical inequalities: References. The following properties are true for any numerical expressions. Property 1. If we add the same numerical expression to both sides of a true numerical inequality, we obtain a true numerical inequality, that is, the following is true: ; . Proof. If . Using the commutative, associative and distributive properties of the addition operation we have: . Therefore, by definition of the relation “greater than” Property 2. If we subtract the same numerical expression from both sides of a true numerical inequality, we obtain a true numerical inequality, that is, the following is true: ; Proof. By condition Using the associative property of the addition operation, we have: , therefore Consequence. Any term can be transferred from one part of a numerical inequality to another with the opposite sign. Property 3. If we add the correct numerical inequalities term by term, we obtain the correct numerical inequality, that is, true: Proof. By property 1 we have: and, using the transitivity property of the relation “more”, we obtain: Property 4. True numerical inequalities of the opposite meaning can be subtracted term by term, preserving the sign of the inequality from which we are subtracting, that is: ; Proof. By definition of true numerical inequalities The property is proved in a similar way. Property 5. If both sides of a correct numerical inequality are multiplied by the same numerical expression that takes a positive value, without changing the sign of the inequality, then we obtain a correct numerical inequality, that is: Proof. From what Then by definition the relation is “greater than”. The property is proved in a similar way. Property 6. If both parts of a correct numerical inequality are multiplied by the same numerical expression, which takes a negative value, changing the sign of the inequality to the opposite, then we obtain a correct numerical inequality, that is: ; Property 7. If both sides of a true numerical inequality are divided by the same numerical expression that takes a positive value, without changing the sign of the inequality, then we obtain a true numerical inequality, that is: Proof. We have: The property is proved in a similar way. Property 8. If both parts of a correct numerical inequality are divided by the same numerical expression that takes a negative value, changing the sign of the inequality to the opposite, then we obtain a correct numerical inequality, that is: ; We omit the proof of this property. Property 9. If we multiply, term by term, correct numerical inequalities of the same meaning with negative parts, changing the sign of the inequality to the opposite, we obtain a correct numerical inequality, that is: We omit the proof of this property. Property 10. If we multiply, term by term, correct numerical inequalities of the same meaning with positive parts, without changing the sign of the inequality, we obtain a correct numerical inequality, that is: We omit the proof of this property. Property 11. If we divide the correct numerical inequality of the opposite meaning term by term with the positive parts, preserving the sign of the first inequality, we obtain the correct numerical inequality, that is: We omit the proof of this property. Example 1. Are inequalities Solution. The second inequality is obtained from the first inequality by adding to both its parts the same expression, which is not defined at . This means that the number cannot be a solution to the first inequality. However, it is a solution to the second inequality. So there is a solution to the second inequality that is not a solution to the first inequality. Therefore, these inequalities are not equivalent. The second inequality is a consequence of the first inequality, since any solution to the first inequality is a solution to the second. 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, ba). (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
, 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 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. 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 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 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 inequalities general view related to the consideration of power, exponential, logarithmic and trigonometric functions. 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 }
$-2*3.1>-2*b>-2*5.3$.
$-10.3
c) Adding inequalities of the same meaning, we obtain an inequality of the same meaning.
$-1.5+3.1
Now let's perform the addition operation.
$-1.5-5.3
${3.1}^2
${(-1.5)}^3
$\frac(1)(5.3)<\frac{1}{b}<\frac{1}{3.1}$.
$\frac(10)(53)<\frac{1}{b}<\frac{10}{31}$.
Compare the numbers:
a) $\sqrt(5)+\sqrt(7)$ and $2+\sqrt(8)$.
b) $π+\sqrt(8)$ and $4+\sqrt(10)$.
a) Let's square each number.
$(\sqrt(5)+\sqrt(7))^2=5+2\sqrt(35)+7=12+\sqrt(140)$.
$(2+\sqrt(8))^2=4+4\sqrt(8)+8=12+\sqrt(128)$.
Let's calculate the difference between the squares of these squares.
$(\sqrt(5)+\sqrt(7))^2-(2+\sqrt(8))^2=12+\sqrt(140)-12-\sqrt(128)=\sqrt(140) -\sqrt(128)$.
Obviously, we got a positive number, which means:
$(\sqrt(5)+\sqrt(7))^2>(2+\sqrt(8))^2$.
Since both numbers are positive, then:
$\sqrt(5)+\sqrt(7)>2+\sqrt(8)$.Problems to solve independently
1. It is known that $-2.2
a) $4a$.
b) $-3b$.
c) $a+b$.
d) $a-b$.
e) $b^4$.
e) $a^3$.
g) $\frac(1)(b)$.
2. Compare the numbers:
a) $\sqrt(6)+\sqrt(10)$ and $3+\sqrt(7)$.
b) $π+\sqrt(5)$ and $2+\sqrt(3)$. Numerical inequalities: definition, examples
Properties of numerical inequalities
Basic properties
Other important properties of numerical inequalities
.. Using the previous property, we add the numerical expression to both sides of this inequality, and we obtain: .
, hence . .
. By property 3, if . As a consequence of property 2 of this theorem, any term can be transferred from one part of the inequality to another with the opposite sign. Hence, 
. Thus, if .
. We have: 
Then 
. Using the distributive nature of the operation of multiplication relative to subtraction, we have: .
. By property 5, we get: . Using the associativity of the multiplication operation, we have: 
hence .
;
.
And 
equivalent?
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
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 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.
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).
