Basic properties of inequalities. Numerical inequalities and their properties

The following properties are true for any numerical expressions.

Property 1. If we add the same thing to both sides of a true numerical inequality numeric expression, then we get the correct numerical inequality, that is, it 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 previous property, we add the numerical expression to both sides of this inequality, and we obtain: .

Using the associative property of the addition operation, we have: , therefore , hence .

Consequence. Any term can be transferred from one part of a numerical inequality to another with 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 inequality sign from which we are subtracting, that is: ;

Proof. By definition of true numerical inequalities . 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 .

The property is proved in a similar way.

Property 5. If both sides of a valid numerical inequality are multiplied by the same numerical expression, which takes positive value, without changing the sign of the inequality, we obtain the correct numerical inequality, that is:

Proof. From what . We have: Then . Using the distributive nature of the multiplication operation relative to subtraction, we have: .

Then by definition the relation is “greater than”.

The property is proved in a similar way.

Property 6. If both sides of a valid numerical inequality are multiplied by the same numerical expression, which takes negative meaning, changing the inequality sign to the opposite one, we get the 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: . By property 5, we get: . Using the associativity of the multiplication operation, we have: hence .

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 a correct numerical inequality, that is:

;

We omit the proof of this property.

Example 1. Are inequalities And equivalent?

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.

Lesson and presentation on the topic: "Basic properties of numerical inequalities and methods for solving them."

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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, we can use inequalities to 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 $a>b$ to 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 a negative number.

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.

This property can be shown more clearly 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 $a bc$.
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 $a, b ,c, d$ - 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 Inequalities of the form $a>b$ and $c d$) are called inequalities of opposite meaning.

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 they are raised to the same natural power, then an inequality with the same meaning will be obtained.
Note: if $n$ – odd number, then for any 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 a) $3a$.
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 $-4.5<3a<6.3$.

B) Let's use property 3. Multiply by a negative number, which means the sign of the inequality changes.
$-2*3.1>-2*b>-2*5.3$.
$-10.3
c) Adding inequalities of the same meaning, we obtain an inequality of the same meaning.
$-1.5+3.1 $1.6

D) Multiply all parts of the inequality $3.1 $-5.3<-b<-3.1$.
Now let's perform the addition operation.
$-1.5-5.3 $-6.8

D) All parts of the inequality are positive, squaring them, we obtain an inequality of the same meaning.
${3.1}^2 $9.61

E) The degree of inequality is odd, then you can safely raise it to a power and not change the sign.
${(-1.5)}^3 $-3.375

G) Let's use property 7.
$\frac(1)(5.3)<\frac{1}{b}<\frac{1}{3.1}$.
$\frac(10)(53)<\frac{1}{b}<\frac{10}{31}$.

Example 2.
Compare the numbers:
a) $\sqrt(5)+\sqrt(7)$ and $2+\sqrt(8)$.
b) $π+\sqrt(8)$ and $4+\sqrt(10)$.

Solution.
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 Find estimates of numbers.
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 $2+\sqrt(3)$.

The field of real numbers has the property of ordering (Section 6, p. 35): for any numbers a, b, one and only one of three relations holds: or . In this case, the entry a > b means that the difference is positive, and the entry difference is negative. Unlike the field of real numbers, the field of complex numbers is not ordered: for complex numbers the concepts of “more” and “less” are not defined; Therefore, this chapter deals only with real numbers.

We call the relations inequalities, the numbers a and b are terms (or parts) of the inequality, the signs > (greater than) and Inequalities a > b and c > d are called inequalities of the same (or one and the same) meaning; inequalities a > b and c From the definition of inequality it immediately follows that

1) any positive number 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, the one whose absolute value is smaller is greater.

All these statements admit of a simple geometric interpretation. Let the positive direction of the number axis go to the right of the starting point; then, whatever the signs of the numbers, the larger of them is represented by a point lying to the right of the point representing the smaller number.

Inequalities have the following basic properties.

1. Asymmetry (irreversibility): if , then , and vice versa.

Indeed, if the difference is positive, then the difference is negative. They say that when rearranging the terms of an inequality, the meaning of the inequality must be changed to the opposite.

2. Transitivity: if , then . Indeed, from the positivity of the differences it follows that

In addition to inequality signs, inequality signs and are also used. They are defined as follows: the entry means that either or Therefore, for example, you can write, and also. Typically, inequalities written using signs are called strict inequalities, and those written using signs are called non-strict inequalities. Accordingly, the signs themselves are called signs of strict or non-strict inequality. Properties 1 and 2 discussed above are also true for non-strict inequalities.

Let us now consider the actions that can be performed on one or more inequalities.

3. Adding the same number to the terms of an inequality does not change the meaning of the inequality.

Proof. Let an inequality and an arbitrary number be given. By definition, the difference is positive. Let's add two opposite numbers to this number, which will not change it, i.e.

This equality can be rewritten as follows:

It follows from this that the difference is positive, i.e. that

and this was what had to be proven.

This is the basis for the possibility of any member of the inequality being skewed from one part to another with the opposite sign. For example, from the inequality

follows that

4. When multiplying the terms of an inequality by the same positive number, the meaning of the inequality does not change; When the terms of an inequality are multiplied by the same negative number, the meaning of the inequality changes to the opposite.

Proof. Let then If then since the product of positive numbers is positive. Opening the parentheses on the left side of the last inequality, we obtain , i.e. . The case is considered in a similar way.

Exactly the same conclusion can be drawn regarding the division of the parts of the inequality by any number other than zero, since division by a number is equivalent to multiplication by a number and the numbers have the same signs.

5. Let the terms of the inequality be positive. Then, when its terms are raised to the same positive power, the meaning of the inequality does not change.

Proof. Let in this case, by the transitivity property, and . Then, due to the monotonic increase of the power function for and positive, we will have

In particular, if where is a natural number, then we get

that is, when extracting the root from both sides of an inequality with positive terms, the meaning of the inequality does not change.

Let the terms of the inequality be negative. Then it is not difficult to prove that when its terms are raised to an odd natural power, the meaning of the inequality does not change, but when raised to an even natural power, it changes to the opposite. From inequalities with negative terms one can also extract the root of odd degree.

Let, further, the terms of the inequality have different signs. Then, when raising it to an odd power, the meaning of the inequality does not change, but when raising it to an even power, in the general case, nothing definite can be said about the meaning of the resulting inequality. In fact, when a number is raised to an odd power, the sign of the number is preserved and therefore the meaning of the inequality does not change. When an inequality is raised to an even power, an inequality with positive terms is formed, and its meaning will depend on the absolute values of the terms of the original inequality; an inequality with the same meaning as the original one, an inequality of the opposite meaning, and even equality can be obtained!

It is useful to check everything that has been said about raising inequalities to powers using the following example.

Example 1. Raise the following inequalities to the indicated power, changing the inequality sign to the opposite or equal sign, if necessary.

a) 3 > 2 to the power of 4; b) to the degree 3;

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

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

g) 2 > -3 to the power of 2; h) to the power of 2,

6. From an inequality we can move on to an inequality between if the terms of the inequality are both positive or both negative, then between their reciprocals there is an inequality of the opposite meaning:

Proof. If a and b are of the same sign, then their product is positive. Divide by inequality

i.e., what was required to be obtained.

If the terms of an inequality have opposite signs, then the inequality between their reciprocals has the same meaning, since the signs of the reciprocals are the same as the signs of the quantities themselves.

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

7. Logarithm of inequalities can be done only in the case when the terms of the inequalities are positive (negative numbers and zero logarithms do not have).

Let . Then there will be

and when there will be

The correctness of these statements is based on the monotonicity of the logarithmic function, which increases if the base and decreases with

So, when taking the logarithm of an inequality consisting of positive terms to a base greater than one, an inequality of the same meaning as the given one is formed, and when taking the logarithm to a positive base less than one, an inequality of the opposite meaning is formed.

8. If, then if, but, then.

This immediately follows from the monotonicity properties of the exponential function (Section 42), which increases in the case and decreases if

When adding termwise inequalities of the same meaning, an inequality of the same meaning as the data is formed.

Proof. Let us prove this statement for two inequalities, although it is true for any number of added inequalities. Let the inequalities be given

By definition, the numbers will be positive; then their sum also turns out to be positive, i.e.

Grouping the terms differently, we get

and therefore

and this was what had to be proven.

It is impossible to say anything definite in the general case about the meaning of an inequality obtained by adding two or more inequalities of different meanings.

10. If from one inequality we subtract term by term another inequality of the opposite meaning, then an inequality of the same meaning as the first is formed.

Proof. Let two inequalities with different meanings be given. The second of them, according to the property of irreversibility, can be rewritten as follows: d > c. Let us now add two inequalities of the same meaning and obtain the inequality

the same meaning. From the latter we find

and this was what had to be proven.

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

Numerical inequalities and their properties

The presentation details the content of the topics NUMERICAL INEQUALITIES and PROPERTIES OF NUMERICAL INEQUALITIES, and provides examples of proving numerical inequalities. (Algebra 8th grade, author Makarychev Yu.N.)

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“Numerical inequalities and their properties”

Numerical inequalities

and their properties

mathematics teacher at municipal educational institution "Upshinskaya secondary school"

Orsha district of the Republic of Mari El

(To the textbook by Yu.A. Makarychev Algebra 8

Numerical inequalities

The result of comparing two or more numbers is written in the form of inequalities using the signs  ,  , =

We compare numbers using various rules (methods). It is convenient to have a generalized a method of comparison that covers all cases.

Definition:

Number A is greater than b if the difference ( a – b) is a positive number.

Number A is less than b if the difference ( a – b) is a negative number.

Number A equals the number b if the difference ( a – b) – equal to zero

A general way to compare numbers

Example 1.

Application of a generalized method of comparing numbers to prove inequalities

Example 2. Prove that the arithmetic mean of two positive numbers is not less than the geometric mean of these numbers.

If both sides of a true inequality are multiplied or divided by the same positive number, you get a true inequality.

If both sides of a true inequality are multiplied or divided by the same negative number and the sign of the inequality is reversed, you get a true inequality.

P = 3a

Multiply by 3 both sides of each of the inequalities

54.2 ∙ 3 a ∙ 3

162,6

Applying the Properties of Numerical Inequalities

We learned about inequalities at school, where we use numerical inequalities. In this article we will consider the properties of numerical inequalities, from which the principles of working with them are built.

The properties of inequalities are similar to the properties of numerical inequalities. The properties, its justification will be considered, and examples will be given.

Yandex.RTB R-A-339285-1

Numerical inequalities: definition, examples

When introducing the concept of inequalities, we have that their definition is made by the type of record. There are algebraic expressions that have signs ≠,< , >, ≤ , ≥ . Let's give a definition.

Definition 1

Numerical inequality called an inequality in which both sides have numbers and numerical expressions.

We consider numerical inequalities in school after studying natural numbers. Such comparison operations are studied step by step. The initial ones look like 1< 5 , 5 + 7 >3. After which the rules are supplemented, and the inequalities become more complicated, then we obtain inequalities of the form 5 2 3 > 5, 1 (2), ln 0. 73 - 17 2< 0 .

Properties of numerical inequalities

To work with inequalities correctly, you must use the properties of numerical inequalities. They come from the concept of inequality. This concept is defined using a statement, which is designated as “more” or “less.”

Definition 2

the number a is greater than b when the difference a - b is a positive number;
the number a is less than b when the difference a - b is a negative number;
the number a is equal to b when the difference a - b is zero.

The definition is used when solving inequalities with the relations “less than or equal to,” “greater than or equal to.” We get that

Definition 3

a is greater than or equal to b when a - b is a non-negative number;
a is less than or equal to b when a - b is a non-positive number.

The definitions will be used to prove the properties of numerical inequalities.

Basic properties

Let's look at 3 main inequalities. Use of signs< и >characteristic of the following properties:

Definition 4

anti-reflexivity, which says that any number a from the inequalities a< a и a >a is considered incorrect. It is known that for any a the equality a − a = 0 holds, hence we obtain that a = a. So a< a и a >a is incorrect. For example, 3< 3 и - 4 14 15 >- 4 14 15 are incorrect.
asymmetry. When the numbers a and b are such that aa, and if a > b, then b< a . Используя определение отношений «больше», «меньше» обоснуем его. Так как в первой части имеем, что a a. The second part of it is proved in a similar way.

Example 1

For example, given the inequality 5< 11 имеем, что 11 >5, which means its numerical inequality − 0, 27 > − 1, 3 will be rewritten as − 1, 3< − 0 , 27 .

Before moving on to the next property, note that with the help of asymmetry you can read the inequality from right to left and vice versa. In this way, numerical inequalities can be modified and swapped.

Definition 5

transitivity. When the numbers a, b, c meet the condition ab and b > c , then a > c .

Evidence 1

The first statement can be proven. Condition a< b и b < c означает, что a − b и b − c являются отрицательными, а разность а - с представляется в виде (a − b) + (b − c) , что является отрицательным числом, потому как имеем сумму двух отрицательных a − b и b − c . Отсюда получаем, что а - с является отрицательным числом, а значит, что a < c . Что и требовалось доказать.

The second part with the transitivity property is proved in a similar way.

Example 2

We consider the analyzed property using the example of inequalities − 1< 5 и 5 < 8 . Отсюда имеем, что − 1 < 8 . Аналогичным образом из неравенств 1 2 >1 8 and 1 8 > 1 32 it follows that 1 2 > 1 32.

Numerical inequalities, which are written using weak inequality signs, have the property of reflexivity, because a ≤ a and a ≥ a can have the case of equality a = a. They are characterized by asymmetry and transitivity.

Definition 6

Inequalities that have the signs ≤ and ≥ in their writing have the following properties:

reflexivity a ≥ a and a ≤ a are considered true inequalities;
antisymmetry, when a ≤ b, then b ≥ a, and if a ≥ b, then b ≤ a.
transitivity, when a ≤ b and b ≤ c, then a ≤ c, and also, if a ≥ b and b ≥ c, then a ≥ c.

The proof is carried out in a similar way.

Other important properties of numerical inequalities

To supplement the basic properties of inequalities, results that are of practical importance are used. The principle of the method is used to estimate the values of expressions, on which the principles of solving inequalities are based.

This paragraph reveals the properties of inequalities for one sign of strict inequality. The same is done for non-strict ones. Let's look at an example, formulating the inequality if a< b и c являются любыми числами, то a + c < b + c . Справедливыми окажутся свойства:

if a > b, then a + c > b + c;
if a ≤ b, then a + c ≤ b + c;
if a ≥ b, then a + c ≥ b + c.

For a convenient presentation, we give the corresponding statement, which is written down and evidence is given, examples of use are shown.

Definition 7

Adding or calculating a number to both sides. In other words, when a and b correspond to the inequality a< b , тогда для любого такого числа имеет смысл неравенство вида a + c < b + c .

Evidence 2

To prove this, the equation must satisfy the condition a< b . Тогда (a + c) − (b + c) = a + c − b − c = a − b . Из условия a < b получим, что a − b < 0 . Значит, (a + c) − (b + c) < 0 , откуда a + c < b + c . Множество действительных числе могут быть изменены с помощью прибавления противоположного числа – с.

Example 3

For example, if we increase both sides of the inequality 7 > 3 by 15, then we get that 7 + 15 > 3 + 15. This is equal to 22 > 18.

Definition 8

When both sides of the inequality are multiplied or divided by the same number c, we obtain a true inequality. If you take a negative number, the sign will change to the opposite. Otherwise it looks like this: for a and b the inequality holds when ab·c.

Evidence 3

When there is a case c > 0, it is necessary to construct the difference between the left and right sides of the inequality. Then we get that a · c − b · c = (a − b) · c . From condition a0, then the product (a − b) · c will be negative. It follows that a · c − b · c< 0 , где a · c < b · c . Другая часть доказывается аналогичным образом.

When proving, division by an integer can be replaced by multiplication by the inverse of the given one, that is, 1 c. Let's look at an example of a property on certain numbers.

Example 4

Both sides of inequality 4 are allowed< 6 умножаем на положительное 0 , 5 , тогда получим неравенство вида − 4 · 0 , 5 < 6 · 0 , 5 , где − 2 < 3 . Когда обе части делим на - 4 , то необходимо изменить знак неравенства на противоположный. отсюда имеем, что неравенство примет вид − 8: (− 4) ≥ 12: (− 4) , где 2 ≥ − 3 .

Now let us formulate the following two results, which are used in solving inequalities:

Corollary 1. When changing the signs of parts of a numerical inequality, the sign of the inequality itself changes to the opposite, as a− b . This follows the rule of multiplying both sides by - 1. It is applicable for transition. For example, − 6< − 2 , то 6 > 2 .
Corollary 2. When replacing reciprocal numbers parts of a numerical inequality to the opposite, its sign also changes, and the inequality remains true. Hence we have that a and b are positive numbers, a1 b .

When dividing both sides of inequality a 3 2 is true, we have that 1 5< 2 3 . При отрицательных a и b c условием, что a 1 b may be incorrect.

Example 5

For example, − 2< 3 , однако, - 1 2 >1 3 are an incorrect equation.

All points are united by the fact that actions on parts of the inequality give the correct inequality at the output. Let's consider properties where initially there are several numerical inequalities, and its result is obtained by adding or multiplying its parts.

Definition 9

When numbers a, b, c, d are valid for inequalities a< b и c < d , тогда верным считается a + c < b + d . Свойство можно формировать таким образом: почленно складывать числа частей неравенства.

Proof 4

Let's prove that (a + c) − (b + d) is a negative number, then we get that a + c< b + d . Из условия имеем, что a < b и c < d . Выше доказанное свойство позволяет прибавлять к обеим частям одинаковое число. Тогда увеличим неравенство a < b на число b , при c < d , получим неравенства вида a + c < b + c и b + c < b + d . Полученное неравенство говорит о том, что ему присуще свойство транзитивности.

The property is used for term-by-term addition of three, four or more numerical inequalities. The numbers a 1 , a 2 , … , a n and b 1 , b 2 , … , b n satisfy the inequalities a 1< b 1 , a 2 < b 2 , … , a n < b n , можно доказать метод математической индукции, получив a 1 + a 2 + … + a n < b 1 + b 2 + … + b n .

Example 6

For example, given three numerical inequalities of the same sign − 5< − 2 , − 1 < 12 и 3 < 4 . Свойство позволяет определять то, что − 5 + (− 1) + 3 < − 2 + 12 + 4 является верным.

Definition 10

Termwise multiplication of both sides results in a positive number. When a< b и c < d , где a , b , c и d являются положительными числами, тогда неравенство вида a · c < b · d считается справедливым.

Evidence 5

To prove this, we need both sides of the inequality a< b умножить на число с, а обе части c < d на b . В итоге получим, что неравенства a · c < b · c и b · c < b · d верные, откуда получим свойство транизитивности a · c < b · d .

This property is considered valid for the number of numbers by which both sides of the inequality must be multiplied. Then a 1 , a 2 , … , a n And b 1, b 2, …, b n are positive numbers, where a 1< b 1 , a 2 < b 2 , … , a n < b n , то a 1 · a 2 · … · a n< b 1 · b 2 · … · b n .

Note that when writing inequalities there are non-positive numbers, then their term-by-term multiplication leads to incorrect inequalities.

Example 7

For example, inequality 1< 3 и − 5 < − 4 являются верными, а почленное их умножение даст результат в виде 1 · (− 5) < 3 · (− 4) , считается, что − 5 < − 12 это является неверным неравенством.

Consequence: Termwise multiplication of inequalities a< b с положительными с a и b , причем получается a n < b n .

Properties of numerical inequalities

Let us consider the following properties of numerical inequalities.

a< a , a >a - incorrect inequalities,
a ≤ a, a ≥ a are true inequalities.
If aa - antisymmetry.
If a< b и b < c то a < c - транзитивность.
If a< b и c - любоое число, то a + b < b + c .
If ab·c.

Corollary 1: if a-b.

Corollary 2: if a and b are positive numbers and a1 b .

If a 1< b 1 , a 2 < b 2 , . . . , a n < b n , то a 1 + a 2 + . . . + a n < b 1 + b 2 + . . . + b n .
If a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n are positive numbers and a 1< b 1 , a 2 < b 2 , . . . , a n < b n , то a 1 · a 2 · . . . · a n < b 1 · b 2 · . . . b n .

Corollary 1: If a< b , a And b are positive numbers, then a n< b n .

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