Linear inequalities. Inequalities

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 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 a negative number 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 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 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 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.

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.

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Numerical inequalities: definition, examples

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<, 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 is extended 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), .

Properties of numerical inequalities

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.

number a more number b if and only if the difference a−b is a positive number;
number a less number b if and only if the difference a−b is a negative number;
the number a is equal to the number b if and only if the difference a−b is zero.

This definition can be reworked into the definition of the relations “less than or equal to” and “greater than or equal to.” Here is his wording:

Definition.

number a is greater than or equal to b if and only if a−b is a non-negative number;
a is less than or equal to b if and only if a−b is a non-positive number.

We will use these definitions when proving the properties of numerical inequalities, to a review of which we proceed.

Basic properties

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:

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

Their proof is very similar to those already given, so we will not dwell on them, but move on to other important properties of numerical inequalities.

Other important properties of numerical inequalities

Let us supplement the basic properties of numerical inequalities with a series of results that have a large practical significance. Methods for estimating the values of expressions are based on them; principles are based on them solutions to inequalities and so on. 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

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 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. Go!

Adding (or subtracting) any number to both sides of a true numerical inequality gives the 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
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:

Bibliography.

Moro M.I.. Mathematics. Textbook for 1 class. beginning school In 2 hours. Part 1. (First half of the year) / M. I. Moro, S. I. Volkova, S. V. Stepanova. - 6th ed. - M.: Education, 2006. - 112 p.: ill.+Add. (2 separate l. ill.). - ISBN 5-09-014951-8.

Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.

Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.

Mordkovich A. G. Algebra. 8th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.

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.
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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, 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 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 parts of a numerical inequality with the opposite numbers, 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 a3 2 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 aa - incorrect inequalities,
a ≤ a, a ≥ a are true inequalities.

If aa - antisymmetry.

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 .

If you notice an error in the text, please highlight it and press Ctrl+Enter

Inequalities are called linear the left and right sides of which are linear functions with respect to the unknown quantity. These include, for example, inequalities:

2x-1-x+3; 7x0;

5 >4 - 6x 9- x< x + 5 .

1) Strict inequalities: ax +b>0 or ax+b<0

2) Non-strict inequalities: ax +b≤0 or ax+b≫ 0

Let's analyze this task. One of the sides of the parallelogram is 7 cm. What must be the length of the other side so that the perimeter of the parallelogram is greater than 44 cm?

Let the required side be X cm. In this case, the perimeter of the parallelogram will be represented by (14 + 2x) cm. The inequality 14 + 2x > 44 is a mathematical model of the problem of the perimeter of a parallelogram. If we replace the variable in this inequality X on, for example, the number 16, then we obtain the correct numerical inequality 14 + 32 > 44. In this case, they say that the number 16 is a solution to the inequality 14 + 2x > 44.

Solving the inequality name the value of a variable that turns it into a true numerical inequality.

Therefore, each of the numbers is 15.1; 20;73 act as a solution to the inequality 14 + 2x > 44, but the number 10, for example, is not its solution.

Solve inequality means to establish all its solutions or to prove that there are no solutions.

The formulation of the solution to the inequality is similar to the formulation of the root of the equation. And yet it is not customary to designate the “root of inequality.”

The properties of numerical equalities helped us solve equations. Similarly, the properties of numerical inequalities will help solve inequalities.

When solving an equation, we replace it with another, simpler equation, but equivalent to the given one. The answer to inequalities is found in a similar way. When changing an equation to an equivalent equation, they use the theorem about transferring terms from one side of the equation to the opposite and about multiplying both sides of the equation by the same non-zero number. When solving an inequality, there is a significant difference between it and an equation, which lies in the fact that any solution to an equation can be verified simply by substitution into the original equation. In inequalities, this method is absent, since it is not possible to substitute countless solutions into the original inequality. Therefore, there is an important concept, these arrows<=>is a sign of equivalent, or equivalent, transformations. The transformation is called equivalent, or equivalent, if they do not change the set of solutions.

Similar rules for solving inequalities.

If we move any term from one part of the inequality to another, replacing its sign with the opposite one, we obtain an inequality equivalent to this one.

If both sides of the inequality are multiplied (divided) by the same positive number, we obtain an inequality equivalent to this one.

If both sides of the inequality are multiplied (divided) by the same negative number, replacing the inequality sign with the opposite one, we obtain an inequality equivalent to the given one.

Using these rules Let us calculate the following inequalities.

1) Let's analyze the inequality 2x - 5 > 9.

This linear inequality, we will find its solution and discuss the basic concepts.

2x - 5 > 9<=>2x>14(5 was moved to the left side with the opposite sign), then we divided everything by 2 and we have x > 7. Let us plot the set of solutions on the axis x

We have obtained a positively directed beam. We note the set of solutions either in the form of inequality x > 7, or in the form of the interval x(7; ∞). What is a particular solution to this inequality? For example, x = 10 is a particular solution to this inequality, x = 12- this is also a particular solution to this inequality.

There are many partial solutions, but our task is to find all the solutions. And there are usually countless solutions.

Let's sort it out example 2:

2) Solve inequality 4a - 11 > a + 13.

Let's solve it: A move it to one side 11 move it to the other side, we get 3a< 24, и в результате после деления обеих частей на 3 the inequality has the form a<8 .

4a - 11 > a + 13<=>3a< 24 <=>a< 8 .

Let's also display the set a< 8 , but already on the axis A.

We either write the answer in the form of inequality a< 8, либо A(-∞;8), 8 does not turn on.

Theory:

When solving inequalities, the following rules are used:

1. Any term of the inequality can be transferred from one part
inequality into another with the opposite sign, but the sign of the inequality does not change.

2. Both sides of the inequality can be multiplied or divided by one
and the same positive number without changing the inequality sign.

3. Both sides of the inequality can be multiplied or divided by one
and the same negative number, changing the inequality sign to
opposite.

Solve inequality − 8 x + 11< − 3 x − 4
Solution.

1. Let's move the penis − 3 x to the left side of the inequality, and the term 11 - to the right side of the inequality, while changing the signs to the opposite ones − 3 x and at 11 .
Then we get

− 8 x + 3 x< − 4 − 11

− 5 x< − 15

2. Let's divide both sides of the inequality − 5 x< − 15 to a negative number − 5 , and the inequality sign < , will change to > , i.e. we move on to an inequality of the opposite meaning.
We get:

− 5 x< − 15 | : (− 5 )

x > − 15 : (− 5 )

x > 3

x > 3— solution of a given inequality.

Pay attention!

There are two options for writing a solution: x > 3 or as a number interval.

Let us mark the set of solutions to the inequality on the number line and write the answer in the form of a numerical interval.

x ∈ (3 ; + ∞ )

Answer: x > 3 or x ∈ (3 ; + ∞ )

Algebraic inequalities.

Quadratic inequalities. Rational inequalities of higher degrees.

Methods for solving inequalities depend mainly on what class the functions that make up the inequality belong to.

I. Quadratic inequalities, that is, inequalities of the form

ax 2 + bx + c > 0 (< 0), a ≠ 0.

To solve the inequality you can:

Factor the square trinomial, that is, write the inequality in the form

a (x - x 1) (x - x 2) > 0 (< 0).

Plot the roots of the polynomial on the number line. The roots divide the set of real numbers into intervals, in each of which there is a corresponding quadratic function will be of constant sign.

Determine the sign of a (x - x 1) (x - x 2) in each interval and write down the answer.

If a square trinomial has no roots, then for D<0 и a>0 square trinomial is positive for any x.

Solve inequality. x 2 + x - 6 > 0.

Factor the quadratic trinomial (x + 3) (x - 2) > 0

Answer: x (-∞; -3) (2; +∞).

2) (x - 6) 2 > 0

This inequality is true for any x except x = 6.

Answer: (-∞; 6) (6; +∞).

3) x² + 4x + 15< 0.

Here D< 0, a = 1 >0. The square trinomial is positive for all x.

Answer: x Î Ø.

Solve inequalities:

1 + x - 2x²< 0. Ответ:

3x² - 12x + 12 ≤ 0. Answer:

3x² - 7x + 5 ≤ 0. Answer:

2x² - 12x + 18 > 0. Answer:

For what values of a does the inequality

x² - ax > holds for any x? Answer:

II. Rational inequalities of higher degrees, that is, inequalities of the form

a n x n + a n-1 x n-1 + … + a 1 x + a 0 > 0 (<0), n>2.

Polynomial highest degree should be factorized, that is, the inequality should be written in the form

a n (x - x 1) (x - x 2) ·…· (x - x n) > 0 (<0).

Mark the points on the number line where the polynomial vanishes.

Determine the signs of the polynomial on each interval.

1) Solve the inequality x 4 - 6x 3 + 11x 2 - 6x< 0.

x 4 - 6x 3 + 11x 2 - 6x = x (x 3 - 6x 2 + 11x -6) = x (x 3 - x 2 - 5x 2 + 5x +6x - 6) =x (x - 1)(x 2 -5x + 6) =

x (x - 1) (x - 2) (x - 3). So x (x - 1) (x - 2) (x - 3)<0

Answer: (0; 1) (2; 3).

2) Solve the inequality (x -1) 5 (x + 2) (x - ½) 7 (2x + 1) 4<0.

Let us mark the points on the number axis at which the polynomial vanishes. These are x = 1, x = -2, x = ½, x = - ½.

At the point x = - ½ there is no change of sign because the binomial (2x + 1) is raised to an even power, that is, the expression (2x + 1) 4 does not change sign when passing through the point x = - ½.

Answer: (-∞; -2) (½; 1).

3) Solve the inequality: x 2 (x + 2) (x - 3) ≥ 0.

This inequality is equivalent to the following set

The solution to (1) is x (-∞; -2) (3; +∞). The solution to (2) is x = 0, x = -2, x = 3. Combining the solutions obtained, we obtain x О (-∞; -2] (0) (0) )