The concept of inequality, related definitions. The simplest inequalities Open number ray

Infinity and minus infinity in any inequality are always written with a parenthesis.

If both parentheses in an entry are parentheses, the number gap is called open. The ends of the open gap are not a solution to the inequality and are not included in the answer.

The end of the span with square bracket is included in the response.

The interval is always recorded from left to right, from smallest to largest.

The solution of the simplest linear inequalities can be schematically represented as a diagram:

Consider examples of solving the simplest linear inequalities.

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They read: "x is more than twelve."

Decision :

The inequality is not strict, on the number line 12 is represented by a punctured point.

We mentally add an arrow to the inequality sign: -\u003e. The arrow indicates that hatching goes from 12 to the right, to plus infinity:

Since the inequality is strict and the point x=12 is punctured, we write 12 in response with a parenthesis.

They read: "x belongs to the open interval from twelve to infinity."

They read: “x is greater than minus three point seven tenths”

Decision :

The inequality is not strict, so -3.7 on the number line is depicted as a filled dot. Mentally add an arrow to the inequality sign: -≥. The arrow is pointing to the right, so the hatching from -3.7 goes to the right, to infinity:

Since the inequality is not strict and the point x= -3.7 is filled in, we write -3.7 in response with a square bracket.

They read: “X belongs to the interval from minus three point seven tenths to infinity, including minus three point seven tenths.”

They read: “x is less than zero point two tenths” (or “x is less than zero point two tenths”).

Decision :

The inequality is strict, 0.2 on the number line is represented by a punctured point. We mentally add an arrow to the inequality sign:<—. Стрелочка подсказывает, что от 0,2 штриховка уходит влево, к минус бесконечности:

The inequality is strict, the dot is punctured, 0.2 — with a parenthesis.

They read: “X belongs to the open interval from minus infinity to zero point two.”

They read: "X is less than or equal to five."

Decision :

The inequality is not strict, on the number line we represent 5 as a filled dot. We mentally add an arrow to the inequality sign: ≤-. The hatching direction is to the left, towards minus infinity:

The inequality is not strict, the dot is filled, 5 is in square brackets.

They read: "x belongs to the interval from minus infinity to five, including five."

We met with inequalities at school, where we use numerical inequalities. In this article, we consider the properties of numerical inequalities, some of which are built principles for working with them.

The properties of inequalities are similar to the properties of numerical inequalities. Properties, its justifications will be considered, we will give examples.

Numerical inequalities: definition, examples

When introducing the concept of inequality, we have that their definition is made according to 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.

Numerical inequalities are considered at school after studying natural numbers. Such comparison operations are studied step by step. Initial look like 1< 5 , 5 + 7 >3 . After that, 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. Such a concept is specified using a statement, which is denoted as "greater than" or "less than".

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 equal to zero.

The definition is used when solving inequalities with relations "less than or equal", "greater than or equal". 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 in proving the properties of numerical inequalities.

Basic properties

Consider 3 main inequalities. Use of signs< и >characteristic with properties:

Definition 4

• anti-reflexivity, which says that any number a from the inequalities a< a и a >a is considered invalid. It is known that for any a the equality a − a = 0 holds, hence we get 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 a< b , то b >a , and if a > b , then b< a . Используя определение отношений «больше», «меньше» обоснуем его. Так как в первой части имеем, что a < b , тогда a − b является отрицательным числом. А b − a = − (a − b) положительное число, потому как число противоположно отрицательному числу a − b . Отсюда следует, что b >a. The second part is proved in a similar way.

Example 1

For example, given the inequality 5< 11 имеем, что 11 >5 , then its numerical inequality − 0 , 27 > − 1 , 3 will be rewritten in the form − 1 , 3< − 0 , 27 .

Before moving on to the next property, we note that with the help of asymmetry, one can read the inequality from right to left and vice versa. Thus, numerical inequality can be changed and interchanged.

Definition 5

• transitivity. When the numbers a , b , c meet the condition a< b и b < c , тогда a < c , и если a >b and b > c , then a > c .

Proof 1

The first assertion 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

The analyzed property is considered on 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 non-strict 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 the notation 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 of evaluation of the values ​​of expressions is applied, on which the principles of solving inequalities are based.

This section reveals the properties of inequalities for one sign of strict inequality. The same is done for nonstrict ones. Consider 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 proofs are 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 .

Proof 2

To prove this, it is necessary that the equation satisfies 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 both parts of the inequality 7 > 3 are increased by 15 , then we get that 7 + 15 > 3 + 15 . This is equal to 22 > 18 .

Definition 8

When both parts of the inequality are multiplied or divided by the same number c, we get the correct inequality. If we take the number c negative, then the sign will change to the opposite. Otherwise, it looks like this: for a and b, the inequality holds when a< b и c являются положительными числами, то a· c < b · c , а если v является отрицательным числом, тогда a · c >bc.

Proof 3

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

In the proof, division by an integer can be replaced by multiplication by the inverse of the given one, that is, 1 c . Consider an example of a property on certain numbers.

Example 4

Both parts of the inequality are allowed 4< 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 we formulate the following two results that are used in solving inequalities:

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

When dividing both parts of the inequality a< b разрешается на число a · b . Данное свойство используется при верном неравенстве 5 >3 2 we have that 1 5< 2 3 . При отрицательных a и b c условием, что a < b , неравенство 1 a >1 b may be incorrect.

Example 5

For example, − 2< 3 , однако, - 1 2 >1 3 are an invalid equality.

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

Definition 9

When the numbers a , b , c , d are valid for the 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

We 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 are subject to 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 parts results in a positive number. For a< b и c < d , где a , b , c и d являются положительными числами, тогда неравенство вида a · c < b · d считается справедливым.

Proof 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 to be 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

Consider below the properties of numerical inequalities.

1. a< a , a >a - false inequalities,
   a ≤ a , a ≥ a are valid inequalities.
2. If a< b , то b >a - antisymmetry.
3. If a< b и b < c то a < c - транзитивность.
4. If a< b и c - любоое число, то a + с < b + c .
5. If a< b и c - положительное число, то a · c < b · c ,
   If a< b и c - отрицательное число, то a · c >bc.

Corollary 1: if a< b , то - a >-b.

Corollary 2: if a and b are positive numbers and a< b , то 1 a >1b.

1. 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 .
2. 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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Definitions and Properties

We will call inequality two numeric or literal expressions connected by signs >,<, ≥, ≤ или ≠.

Example: 5 > 3

This inequality says that the number 5 is greater than the number 3. The acute angle of the inequality sign should be directed towards the smaller number. This inequality is true because 5 is greater than 3.

If a watermelon weighing 5 kg is placed on the left pan of the scale, and a watermelon weighing 3 kg is placed on the right pan, then the left pan will outweigh the right one, and the scale screen will show that the left pan is heavier than the right one:

If 5 > 3 then 3< 5 . То есть левую и правую часть неравенства можно поменять местами, изменив знак неравенства на противоположный. В ситуации с весами: большой арбуз можно положить на правую чашу, а маленький арбуз на левую. Тогда правая чаша перевесит левую, и экран покажет знак <

If in the inequality 5 > 3 , without touching the left and right parts, change the sign to< , то получится неравенство 5 < 3 . Это неравенство не является верным, поскольку число 3 не может быть больше числа 5.

The numbers that are located on the left and right sides of the inequality will be called members this inequality. For example, in the inequality 5 > 3, the members are the numbers 5 and 3.

Consider some important properties for the inequality 5 > 3 .
In the future, these properties will work for other inequalities as well.

Property 1.

If the same number is added or subtracted to the left and right parts of the inequality 5 > 3, then the sign of the inequality will not change.

For example, let's add the number 4 to both parts of the inequality. Then we get:

Now let's try to subtract some number from both sides of the inequality 5 > 3, say the number 2

We see that the left side is still larger than the right one.

From this property it follows that any term of the inequality can be transferred from one part to another part by changing the sign of this term. The inequality sign will not change.

For example, in the inequality 5 > 3, let's move term 5 from the left side to the right side by changing the sign of this term. After moving term 5 to the right side, nothing will remain on the left side, so we write 0 there

0 > 3 − 5

0 > −2

We see that the left side is still larger than the right one.

Property 2.

If both parts of the inequality are multiplied or divided by the same positive number, then the sign of the inequality does not change.

For example, let's multiply both sides of the inequality 5 > 3 by some positive number, say by the number 2. Then we get:

We see that the left side is still larger than the right one.

Now let's try divide both parts of the inequality 5 > 3 by some number. Divide them by 2

We see that the left side is still larger than the right one.

Property 3.

If both sides of the inequality are multiplied or divided by the same a negative number, then the inequality sign will be reversed.

For example, let's multiply both sides of the inequality 5 > 3 by some negative number, say -2. Then we get:

Now let's try divide both parts of the inequality 5 > 3 by some negative number. Let's divide them by -1

We see that the left side has become smaller than the right. That is, the sign of inequality has changed to the opposite.

In itself, inequality can be understood as a certain condition. If the condition is met, then the inequality is true. Conversely, if the condition is not met, then the inequality is not true.

For example, to answer the question whether the inequality 7 > 3 is true, you need to check whether the condition is satisfied "is 7 more than 3" . We know that the number 7 is greater than the number 3. That is, the condition is met, and hence the inequality 7 > 3 is true.

Inequality 8< 6 не является верным, поскольку не выполняется условие "8 is less than 6".

Another way to determine whether an inequality is correct is to take the difference from the left and right sides of the given inequality. If the difference is positive, then the left side is greater than the right side. Conversely, if the difference is negative, then the left side is less than the right side. More precisely, this rule looks like this:

Number a more number b if the difference a-b positive. Number a less than number b if the difference a-b negative.

For example, we found that the inequality 7 > 3 is true because the number 7 is greater than the number 3. Let's prove this using the rule above.

Compose the difference from terms 7 and 3. Then we get 7 − 3 = 4 . According to the rule, the number 7 will be greater than the number 3 if the difference 7 − 3 is positive. We have it equal to 4, that is, the difference is positive. So the number 7 is greater than the number 3.

Let us check with the help of the difference whether the inequality 3< 4 . Составим разность, получим 3 − 4 = −1 . Согласно правилу, число 3 будет меньше числа 4, если разность 3 − 4 окажется отрицательной. У нас она равна −1, то есть разность отрицательна. А значит число 3 меньше числа 4.

Let's check whether the inequality 5 > 8 is true. Compose the difference, we get 5 − 8 = −3. According to the rule, the number 5 will be greater than the number 8 if the difference 5 − 8 is positive. Our difference is −3, that is, it is not positive. So the number 5 not more the number 3. In other words, the inequality 5 > 8 is not true.

Strict and non-strict inequalities

Inequalities containing signs >,< называют strict. And inequalities containing signs ≥, ≤ are called non-strict.

We considered examples of strict inequalities earlier. These are the inequalities 5 > 3 , 7< 9 .

Non-strict, for example, is the inequality 2 ≤ 5 . This inequality is read as follows: "2 is less than or equal to 5" .

The entry 2 ≤ 5 is incomplete. The full record of this inequality is as follows:

2 < 5 or 2 = 5

Then it becomes obvious that the inequality 2 ≤ 5 consists of two conditions: "two less than five" and "two equals five" .

A nonstrict inequality is true if at least one of its conditions is satisfied. In our example, the condition is true "2 is less than 5". This means that the inequality 2 ≤ 5 is also true.

Example 2. The inequality 2 ≤ 2 is true because one of its conditions is satisfied, namely 2 = 2.

Example 3. The inequality 5 ≤ 2 is not true because none of its conditions is satisfied: neither 5< 2 ни 5 = 2 .

double inequality

Number 3 is greater than number 2 and less than number 4 . In the form of an inequality, this statement can be written as follows: 2< 3 < 4 . Такое неравенство называют двойным.

A double inequality may contain signs of non-strict inequalities. For example, if number 5 is greater than or equal to number 2 and less than or equal to number 7 , then we can write that 2 ≤ 5 ≤ 7

To correctly write a double inequality, first write the term in the middle, then the term on the left, then the term on the right.

For example, let's write that the number 6 is greater than the number 4 and less than the number 9.

First write down 6

On the left, we write that this number is greater than the number 4

On the right, we write that the number 6 is less than the number 9

Variable Inequality

Inequality, like equality, can contain a variable.

For example, inequality x> 2 contains a variable x. Usually such an inequality needs to be solved, that is, to find out for what values x this inequality becomes true.

To solve an inequality means to find such values ​​of a variable x, under which this inequality becomes true.

The value of the variable at which the inequality becomes true is called solving the inequality.

Inequality x> 2 becomes true when x=3, x=4, x=5, x=6 and so on ad infinitum. We see that this inequality has not one solution, but many solutions.

In other words, by solving the inequality x> 2 is the set of all numbers greater than 2. For these numbers, the inequality will be true. Examples:

3 > 2

4 > 2

5 > 2

The number 2, located on the right side of the inequality x> 2 , we will call border this inequality. Depending on the sign of the inequality, the boundary may or may not belong to the set of solutions to the inequality.

In our example, the inequality boundary does not belong to the set of solutions, since when substituting the number 2 into the inequality x> 2 turns out not correct inequality 2 > 2 . The number 2 cannot be greater than itself, since it is equal to itself (2 = 2) .

Inequality x> 2 is strict. It can be read like this: x is strictly greater than 2″ . That is, all values ​​accepted by the variable x must be strictly greater than 2. Otherwise, the inequality will not be true.

If we were given a non-strict inequality x≥ 2 , then the solutions of this inequality would be all numbers that are greater than 2, including the number 2 itself. In this inequality, the boundary 2 belongs to the set of solutions to the inequality, since when substituting the number 2 into the inequality x≥ 2 we obtain the correct inequality 2 ≥ 2 . It was said earlier that a nonstrict inequality is true if at least one of its conditions is satisfied. The inequality 2 ≥ 2 satisfies the condition 2 = 2 , so the inequality 2 ≥ 2 is also true.

How to solve inequalities

The process of solving inequalities is in many ways similar to the process of solving equations. When solving inequalities, we will apply the properties that we studied at the beginning of this lesson, such as: transferring terms from one part of the inequality to another part, changing the sign; multiplying (or dividing) both sides of the inequality by the same number.

These properties allow us to obtain an inequality that is equivalent to the original one. Equivalent inequalities are called inequalities whose solutions are the same.

When solving equations, we performed identical transformations until a variable remained on the left side of the equation, and the value of this variable remained on the right side (for example: x=2, x=5). In other words, the original equation was replaced by an equivalent equation until an equation of the form x = a, where a variable value x. Depending on the equation, there could be one, two, an infinite number of roots, or not at all.

And when solving inequalities, we will replace the original inequality with an inequality equivalent to it until the variable of this inequality remains on the left side, and its boundary on the right side.

Example 1. Solve inequality 2 x> 6

So, you need to find such values x , when substituting them into 2 x> 6 we get the correct inequality.

At the beginning of this lesson, it was said that if both parts of the inequality are divided by some positive number, then the sign of the inequality will not change. If we apply this property to an inequality containing a variable, then we get an inequality equivalent to the original one.

In our case, if we separate both parts of the inequality 2 x> 6 by some positive number, then we get an inequality that is equivalent to the original inequality 2 x> 6.

So let's divide both sides of the inequality by 2.

On the left side there is a variable x, and the right side became equal to 3. We got an equivalent inequality x> 3. This completes the solution, since the variable remains on the left side, and the inequality boundary on the right side.

Now we can conclude that the solutions of the inequality x> 3 are all numbers that are greater than 3. These are the numbers 4, 5, 6, 7 and so on ad infinitum. For these values, the inequality x> 3 would be correct.

4 > 3

5 > 3

6 > 3

7 > 3

Note that the inequality x> 3 is strict. " The variable x is strictly greater than three."

And because the inequality x> 3 is equivalent to the original inequality 2 x> 6 , then their solutions will coincide. In other words, the values ​​that fit the inequality x> 3 will also fit inequality 2 x> 6. Let's show it.

Take, for example, the number 5 and substitute it first into the equivalent inequality we have obtained x> 3 , and then to the original 2 x> 6 .

We see that in both cases the correct inequality is obtained.

After the inequality is solved, the answer must be written in the form of the so-called number span in the following way:

This expression says that the values ​​taken by the variable x, belong to the numerical interval from three to plus infinity.

In other words, all numbers from three to plus infinity are solutions to the inequality x> 3 . Sign ∞ in mathematics means infinity.

Considering that the concept of a numerical interval is very important, let us dwell on it in more detail.

Numeric spans

Numeric gap call the set of numbers on the coordinate line, which can be described using an inequality.

Suppose we want to draw a set of numbers from 2 to 8 on the coordinate line. To do this, first mark points with coordinates 2 and 8 on the coordinate line, and then select with strokes the area that is located between coordinates 2 and 8. These strokes will play the role of numbers , located between the numbers 2 and 8

Let's call the numbers 2 and 8 borders number gap. When drawing a numerical interval, the points for its boundaries are depicted not as points as such, but as circles that can be seen.

The boundaries may or may not belong to the numerical range.

If the boundaries do not belong numerical interval, then they are depicted on the coordinate line in the form empty circles.

If the boundaries belong numerical interval, then the circles must paint over.

In our drawing, the circles were left empty. This meant that the boundaries 2 and 8 do not belong to the numerical gap. This means that our numerical range will include all numbers from 2 to 8, except for the numbers 2 and 8.

If we want to include borders 2 and 8 in the numerical range, then the circles need to be filled in:

In this case, the number range will include all numbers from 2 to 8, including the numbers 2 and 8.

In writing, a numerical interval is indicated by indicating its boundaries using round or square brackets.

If the boundaries do not belong parentheses.

If the boundaries belong numerical gap, then the borders are framed square brackets.

The figure shows two numerical intervals from 2 to 8 with the corresponding designations:

In the first figure, the numerical gap is indicated by parentheses, since boundaries 2 and 8 do not belong this number interval.

In the second figure, the numerical gap is indicated by square brackets, since boundaries 2 and 8 belong this number interval.

Using numerical intervals, you can write answers to inequalities. For example, the answer to the double inequality 2 ≤ x≤ 8 is written like this:

x ∈ [ 2 ; 8 ]

That is, first the variable included in the inequality is written down, then, using the membership sign ∈, they indicate to which numerical interval the values ​​of this variable belong. In this case, the expression x∈ [ 2 ; 8 ] indicates that the variable x, included in the inequality 2 ≤ x≤ 8, takes all values ​​between 2 and 8 inclusive. For these values, the inequality will be true.

Pay attention to the fact that the answer is written using square brackets, since the boundaries of the inequality 2 ≤ x≤ 8 , namely, the numbers 2 and 8 belong to the set of solutions to this inequality.

The set of solutions to the inequality 2 ≤ x≤ 8 can also be represented using a coordinate line:

Here the boundaries of the numerical interval 2 and 8 correspond to the boundaries of the inequality 2 ≤ x x 2 ≤ x≤ 8 .

In some sources, boundaries that do not belong to the numerical gap are called open .

They are called open because the numerical interval remains open due to the fact that its boundaries do not belong to this numerical interval. The empty circle on the coordinate line of mathematics is called punched out point . To poke a point means to exclude it from the numerical interval or from the set of solutions to an inequality.

And in the case when the boundaries belong to the numerical interval, they are called closed(or closed), since such boundaries close (close) a numerical gap. The filled circle on the coordinate line also indicates that the borders are closed.

There are varieties of numerical intervals. Let's consider each of them.

number beam

number beam x ≥ a, where a x- solving the inequality.

Let be a= 3 . Then the inequality x ≥ a will take the form x≥ 3 . The solutions of this inequality are all numbers that are greater than 3, including the number 3 itself.

Draw a number ray given by the inequality x≥ 3, on the coordinate line. To do this, mark on it a point with coordinate 3, and the rest area to its right highlight with dashes. It is the right side that stands out, since the solutions of the inequality x≥ 3 are numbers greater than 3. And larger numbers on the coordinate line are located to the right

x≥ 3 , and the area marked with strokes corresponds to the set of values x, which are solutions of the inequality x≥ 3 .

Point 3, which is the boundary of the number ray, is shown as a filled circle, since the boundary of the inequality x≥ 3 belongs to the set of its solutions.

In writing, the number line given by the inequality x ≥ a,

[ a; +∞)

It can be seen that on one side the border is framed by a square bracket, and on the other by a round bracket. This is due to the fact that one boundary of the numerical ray belongs to it, and the other does not, since infinity itself has no boundaries and it is understood that on the other side there is no number that closes this numerical ray.

Considering that one of the boundaries of the number line is closed, this gap is often called closed number beam.

Let's write the answer to the inequality x≥ 3 using the number ray notation. We have a variable a is 3

x ∈ [ 3 ; +∞)

This expression says that the variable x included in the inequality x≥ 3, takes all values ​​from 3 to plus infinity.

In other words, all numbers from 3 to plus infinity are solutions to the inequality x≥ 3 . Boundary 3 belongs to the solution set because the inequality x≥ 3 is nonstrict.

A closed number ray is also called a number interval, which is given by the inequality x ≤ a . Inequality solutions x ≤ a a , including the number itself a.

For example, if a x≤ 2 . On the coordinate line, boundary 2 will be depicted as a filled circle, and the entire area located left, will be highlighted with dashes. This time, the left side is highlighted, since the solutions to the inequality x≤ 2 are numbers less than 2. And smaller numbers on the coordinate line are located to the left

x≤ 2 , and the dashed area corresponds to the set of values x, which are solutions of the inequality x≤ 2 .

Point 2, which is the boundary of the number ray, is shown as a filled circle, since the boundary of the inequality x≤ 2 belongs to the set of its solutions.

Let's write the answer to the inequality x≤ 2 using number ray notation:

x ∈ (−∞ ; 2 ]

x≤ 2. Boundary 2 belongs to the set of solutions, since the inequality x≤ 2 is nonstrict.

Open number beam

Open number beam is called the numerical interval, which is given by the inequality x > a, where a is the boundary of this inequality, x- solution of inequality.

An open number line is similar in many ways to a closed number line. The difference is that the border a does not belong to the interval, as well as the boundary of the inequality x > a does not belong to the set of its solutions.

Let be a= 3 . Then the inequality takes the form x> 3 . The solutions of this inequality are all numbers that are greater than 3, except for the number 3

On the coordinate line, the boundary of the open number ray given by the inequality x> 3 will be displayed as an empty circle. The entire area to the right will be highlighted with strokes:

Here point 3 corresponds to the inequality boundary x > 3 , and the area highlighted with strokes corresponds to the set of values x, which are solutions of the inequality x > 3 . Point 3, which is the boundary of the open numerical ray, is shown as an empty circle, since the boundary of the inequality x > 3 does not belong to the set of its solutions.

x > a , denoted as follows:

(a; +∞)

Parentheses indicate that the boundaries of the open number ray do not belong to it.

Let's write the answer to the inequality x> 3 using the notation of an open numerical beam:

x ∈ (3 ; +∞)

This expression says that all numbers from 3 to plus infinity are solutions to the inequality x> 3 . Boundary 3 does not belong to the solution set because the inequality x> 3 is strict.

An open number ray is also called a number interval, which is given by the inequality x< a , where a is the boundary of this inequality, x— solution of inequality . Inequality solutions x< a are all numbers less than a , excluding number a.

For example, if a= 2 , then the inequality takes the form x< 2. On the coordinate line, boundary 2 will be shown as an empty circle, and the entire area to the left will be highlighted with strokes:

Here point 2 corresponds to the inequality boundary x< 2 , and the area marked with strokes corresponds to the set of values x, which are solutions of the inequality x< 2. Point 2, which is the boundary of the open numerical ray, is shown as an empty circle, since the boundary of the inequality x< 2 does not belong to the set of its solutions.

In writing, the open number beam given by the inequality x< a , denoted as follows:

(−∞ ; a)

Let's write the answer to the inequality x< 2 using the notation of an open numerical beam:

x ∈ (−∞ ; 2)

This expression says that all numbers from minus infinity to 2 are solutions to the inequality x< 2. Boundary 2 does not belong to the set of solutions because the inequality x< 2 is strict.

Line segment

segment a ≤ x ≤ b, where a and b x- solution of inequality.

Let be a = 2 , b= 8 . Then the inequality a ≤ x ≤ b takes the form 2 ≤ x≤ 8 . Solutions to the inequality 2 ≤ x≤ 8 are all numbers that are greater than 2 and less than 8. Moreover, the boundaries of the inequality 2 and 8 belong to the set of its solutions, since the inequality 2 ≤ x≤ 8 is nonstrict.

Draw the segment given by the double inequality 2 ≤ x≤ 8 on the coordinate line. To do this, mark the points on it with coordinates 2 and 8, and mark the area between them with strokes:

≤ x≤ 8 , and the dashed area corresponds to the set of values x x≤ 8 . Points 2 and 8, which are the boundaries of the segment, are shown as filled circles, since the boundaries of the inequality 2 ≤ x≤ 8 belong to the set of its solutions.

On the letter, the segment given by the inequality a ≤ x ≤ b denoted as follows:

[ a; b ]

Square brackets on both sides indicate that the boundaries of the segment belong him. Let us write the answer to the inequality 2 ≤ x

x ∈ [ 2 ; 8 ]

This expression says that all numbers from 2 to 8 inclusive are solutions to the inequality 2 ≤ x≤ 8 .

Interval

interval is called the numerical interval, which is given by the double inequality a< x < b , where a and b are the boundaries of this inequality, x- solution of inequality.

Let be a = 2, b = 8. Then the inequality a< x < b will take the form 2< x< 8 . Решениями этого двойного неравенства являются все числа, которые больше 2 и меньше 8, исключая числа 2 и 8.

Let's depict the interval on the coordinate line:

Here points 2 and 8 correspond to the boundaries of inequality 2< x< 8 , а выделенная штрихами область соответствует множеству значений x < x< 8 . Точки 2 и 8, являющиеся границами интервала, изображены в виде пустых кружков, поскольку границы неравенства 2 < x< 8 не принадлежат множеству его решений.

In writing, the interval given by the inequality a< x < b, denoted as follows:

(a; b)

Parentheses on both sides indicate that the interval boundaries do not belong him. Let us write down the answer to inequality 2< x< 8 с помощью этого обозначения:

x ∈ (2 ; 8)

This expression says that all numbers from 2 to 8, excluding numbers 2 and 8, are solutions to inequality 2< x< 8 .

Half interval

Half interval is called the numerical interval, which is given by the inequality a ≤ x< b , where a and b are the boundaries of this inequality, x- solution of inequality.

A half-interval is also called a numerical interval, which is given by the inequality a< x ≤ b .

One of the boundaries of the half-interval belongs to it. Hence the name of this numerical interval.

In the situation with a half-interval a ≤ x< b it (the half-interval) belongs to the left boundary.

And in the situation with a half-interval a< x ≤ b it owns the right border.

Let be a= 2 , b= 8 . Then the inequality a ≤ x< b takes the form 2 ≤ x < 8 . Решениями этого двойного неравенства являются все числа, которые больше 2 и меньше 8, включая число 2, но исключая число 8.

Draw the interval 2 ≤ x < 8 на координатной прямой:

x < 8 , а выделенная штрихами область соответствует множеству значений x, which are solutions to the inequality 2 ≤ x < 8 .

Point 2, which is left border half-interval, is shown as a filled circle, since the left boundary of the inequality 2 ≤ x < 8 belongs many of his solutions.

And point 8, which is right border half-interval is shown as an empty circle, since the right boundary of the inequality 2 ≤ x < 8 not belongs many of his solutions.

a ≤ x< b, denoted as follows:

[ a; b)

It can be seen that on one side the border is framed by a square bracket, and on the other by a round bracket. This is due to the fact that one boundary of the half-interval belongs to it, while the other does not. Let us write the answer to the inequality 2 ≤ x < 8 с помощью этого обозначения:

x ∈ [ 2 ; 8)

This expression says that all numbers from 2 to 8, including the number 2 but excluding the number 8, are solutions to the inequality 2 ≤ x < 8 .

Similarly, on the coordinate line, one can depict the half-interval given by the inequality a< x ≤ b . Let be a= 2 , b= 8 . Then the inequality a< x ≤ b will take the form 2< x≤ 8 . The solutions to this double inequality are all numbers that are greater than 2 and less than 8, excluding the number 2, but including the number 8.

Draw half-interval 2< x≤ 8 on the coordinate line:

Here points 2 and 8 correspond to the boundaries of inequality 2< x≤ 8 , and the dashed area corresponds to the set of values x, which are solutions of inequality 2< x≤ 8 .

Point 2, which is left border half-interval, is shown as an empty circle, since the left boundary of the inequality 2< x≤ 8 not belong many of his solutions.

And point 8, which is right border half-interval, is shown as a filled circle, since the right boundary of the inequality 2< x≤ 8 belongs many of his solutions.

In writing, the half-interval given by the inequality a< x ≤ b, denoted like this: a; b] . Let us write down the answer to inequality 2< x≤ 8 using this notation:

x ∈ (2 ; 8 ]

This expression says that all numbers from 2 to 8, excluding number 2, but including number 8, are solutions to inequality 2< x≤ 8 .

Image of numerical intervals on the coordinate line

A numeric span can be specified using an inequality, or using a notation (parentheses or square brackets). In both cases, one must be able to represent this numerical interval on the coordinate line. Let's look at a few examples.

Example 1. Draw the numerical interval given by the inequality x> 5

We recall that an inequality of the form x> a an open numerical ray is specified. In this case, the variable a equals 5. Inequality x> 5 is strict, so the border 5 will be displayed as an empty circle. We are interested in all values x, which are greater than 5, so the entire area on the right will be highlighted with strokes:

Example 2. Draw the number interval (5; +∞) on the coordinate line

This is the same number span that we depicted in the previous example. But this time it is set not with the help of inequality, but with the help of the notation of the numerical interval.

Boundary 5 is surrounded by a parenthesis, which means it does not belong to the gap. Accordingly, the circle remains empty.

The +∞ symbol indicates that we are interested in all numbers that are greater than 5. Accordingly, the entire area to the right of the 5 border is highlighted with strokes:

Example 3. Draw the number interval (−5; 1) on the coordinate line.

Round brackets on both sides denote intervals. The boundaries of the interval do not belong to it, so the boundaries of −5 and 1 will be displayed on the coordinate line as empty circles. The entire area between them will be highlighted with strokes:

Example 4. Draw the numerical interval given by the inequality −5< x< 1

This is the same number span that we depicted in the previous example. But this time it is specified not with the help of the interval notation, but with the help of a double inequality.

An inequality of the form a< x < b , the interval is set. In this case, the variable a equals −5 , and the variable b is equal to one. Inequality −5< x< 1 is strict, so the boundaries of −5 and 1 will be drawn as empty circles. We are interested in all values x, which are greater than −5 but less than one, so the entire area between the points −5 and 1 will be highlighted with strokes:

Example 5. Draw numerical intervals [-1; 2] and

This time we will draw two gaps on the coordinate line at once.

Square brackets on both sides denote segments. The boundaries of the segment belong to it, so the boundaries of the segments [-1; 2] and will be depicted on the coordinate line as filled circles. The entire area between them will be highlighted with strokes.

To clearly see the gaps [−1; 2] and , the first can be depicted on the upper area, and the second on the bottom. So let's do it:

Example 6. Draw numerical intervals [-1; 2) and (2; 5]

Square brackets on one side and round brackets on the other denote half-intervals. One of the boundaries of the half-interval belongs to it, and the other does not.

In the case of the half-interval [-1; 2) the left border will belong to him, but the right one will not. This means that the left border will be displayed as a filled circle. The right border will be displayed as an empty circle.

And in the case of a half-interval (2; 5] only the right border will belong to it, but the left one will not. This means that the left border will be displayed as a filled circle. The right border will be displayed as an empty circle.

Draw the interval [-1; 2) on the upper region of the coordinate line, and the interval (2; 5] — on the lower one:

Examples of solving inequalities

An inequality that, by identical transformations, can be reduced to the form ax > b(or to the view ax< b ), we will call linear inequality with one variable.

In a linear inequality ax > b , x is the variable whose values ​​are to be found, a is the coefficient of this variable, b is the boundary of inequality, which, depending on the sign of inequality, may either belong to the set of its solutions or not belong to it.

For example, inequality 2 x> 4 is an inequality of the form ax > b. In it, the role of the variable a plays the number 2, the role of a variable b(boundary inequality) plays the number 4.

Inequality 2 x> 4 can be made even simpler. If we divide both its parts by 2, then we get the inequality x> 2

The resulting inequality x> 2 is also an inequality of the form ax > b, that is, a linear inequality with one variable. In this inequality, the role of the variable a unit plays. Earlier we said that coefficient 1 is not recorded. The role of the variable b plays number 2.

Based on this information, let's try to solve some simple inequalities. During the solution, we will perform elementary identity transformations in order to obtain an inequality of the form ax > b

Example 1. Solve the inequality x− 7 < 0

Add to both sides of the inequality the number 7

x− 7 + 7 < 0 + 7

On the left side will remain x, and the right side becomes equal to 7

x< 7

By elementary transformations, we have reduced the inequality x− 7 < 0 к равносильному неравенству x< 7 . Решениями неравенства x< 7 являются все числа, которые меньше 7. Граница 7 не принадлежит множеству решений, поскольку неравенство строгое.

When the inequality is brought to the form x< a (or x > a), it can be considered already solved. Our inequality x− 7 < 0 тоже приведено к такому виду, а именно к виду x< 7 . Но в большинстве школ требуют, чтобы ответ был записан с помощью числового промежутка и проиллюстрирован на координатной прямой.

Let's write the answer using a numerical interval. In this case, the answer will be an open number ray (recall that the number ray is given by the inequality x< a and is denoted as (−∞ ; a)

x ∈ (−∞ ; 7)

On the coordinate line, boundary 7 will be displayed as an empty circle, and the entire area to the left of the boundary will be highlighted with strokes:

To check, we take any number from the interval (−∞ ; 7) and substitute it into the inequality x< 7 вместо переменной x. Take, for example, the number 2

2 < 7

It turned out the correct numerical inequality, which means that the solution is correct. Let's take some other number, for example, the number 4

4 < 7

It turned out the correct numerical inequality. So the decision is correct.

And because the inequality x< 7 равносильно исходному неравенству x - 7 < 0 , то решения неравенства x< 7 будут совпадать с решениями неравенства x - 7 < 0 . Подставим те же тестовые значения 2 и 4 в неравенство x - 7 < 0

2 − 7 < 0

−5 < 0 — Верное неравенство

4 − 7 < 0

−3 < 0 Верное неравенство

Example 2. Solve inequality −4 x < −16

Divide both sides of the inequality by −4. Do not forget that when dividing both parts of the inequality to a negative number, inequality sign changes to the opposite:

We have reduced the inequality −4 x < −16 к равносильному неравенству x> 4 . Inequality solutions x> 4 will be all numbers that are greater than 4. Boundary 4 does not belong to the set of solutions, since the inequality is strict.

x> 4 on the coordinate line and write the answer as a numerical interval:

Example 3. Solve the inequality 3y + 1 > 1 + 6y

Reschedule 6 y from the right side to the left side by changing the sign. And we will transfer 1 from the left side to the right side, again changing the sign:

3y− 6y> 1 − 1

Here are similar terms:

−3y > 0

Divide both sides by −3. Do not forget that when dividing both parts of the inequality by a negative number, the inequality sign is reversed:

Inequality solutions y< 0 являются все числа, меньшие нуля. Изобразим множество решений неравенства y< 0 на координатной прямой и запишем ответ в виде числового промежутка:

Example 4. Solve the inequality 5(x− 1) + 7 ≤ 1 − 3(x+ 2)

Let's expand the brackets in both parts of the inequality:

Move -3 x from the right side to the left side by changing the sign. We will transfer the terms −5 and 7 from the left side to the right side, again changing the signs:

Here are similar terms:

Divide both sides of the resulting inequality by 8

Solutions to the inequality are all numbers that are less than . The boundary belongs to the solution set since the inequality is not strict.

Example 5. Solve the inequality

Multiply both sides of the inequality by 2. This will get rid of the fraction on the left side:

Now we move 5 from the left side to the right side by changing the sign:

After reducing similar terms, we obtain the inequality 6 x> 1 . Divide both parts of this inequality by 6. Then we get:

Solutions to the inequality are all numbers greater than . The boundary does not belong to the solution set because the inequality is strict.

Draw the set of solutions to the inequality on the coordinate line and write the answer as a numerical interval:

Example 6. Solve the inequality

Multiply both sides by 6

After reducing similar terms, we obtain the inequality 5 x< 30 . Разделим обе части этого неравенства на 5

Inequality solutions x< 6 являются все числа, которые меньше 6. Граница 6 не принадлежит множеству решений, поскольку неравенство является x< 6 строгим.

Draw the set of solutions to the inequality x< 6 на координатной прямой и запишем ответ в виде числового промежутка:

Example 7. Solve the inequality

Multiply both sides of the inequality by 10

In the resulting inequality, open the brackets on the left side:

Transfer members without x to the right side

We present similar terms in both parts:

Divide both parts of the resulting inequality by 10

Inequality solutions x≤ 3.5 are all numbers that are less than 3.5. The 3.5 boundary belongs to the solution set since the inequality is x≤ 3.5 non-strict.

Draw the set of solutions to the inequality x≤ 3.5 on the coordinate line and write the answer as a numerical interval:

Example 8. Solve inequality 4< 4x< 20

To solve such an inequality, we need a variable x free from the coefficient 4. Then we can say in what interval is the solution of this inequality.

To release a variable x from the coefficient, you can divide the term 4 x by 4. But the rule in inequalities is that if we divide a member of the inequality by some number, then the same must be done with the rest of the terms included in this inequality. In our case, we need to divide by 4 all three terms of the inequality 4< 4x< 20

Solutions to inequality 1< x< 5 являются все числа, которые больше 1 и меньше 5. Границы 1 и 5 не принадлежат множеству решений, поскольку неравенство 1 < x< 5 является строгим.

Draw the set of solutions to inequality 1< x< 5 на координатной прямой и запишем ответ в виде числового промежутка:

Example 9. Solve the inequality −1 ≤ −2 x≤ 0

Divide all terms of the inequality by −2

We got the inequality 0.5 ≥ x≥ 0 . It is desirable to write a double inequality so that the smaller term is located on the left and the larger one is on the right. Therefore, we rewrite our inequality as follows:

0 ≤ x≤ 0,5

Solutions to the inequality 0 ≤ x≤ 0.5 are all numbers that are greater than 0 and less than 0.5. The bounds 0 and 0.5 belong to the set of solutions, since the inequality 0 ≤ x≤ 0.5 is non-strict.

Draw the set of solutions to the inequality 0 ≤ x≤ 0.5 on the coordinate line and write the answer as a numerical interval:

Example 10. Solve the inequality

Multiply both inequalities by 12

Let's open the brackets in the resulting inequality and present like terms:

Divide both sides of the resulting inequality by 2

Inequality solutions x≤ −0.5 are all numbers that are less than −0.5. The boundary −0.5 belongs to the set of solutions because the inequality x≤ −0.5 is nonstrict.

Draw the set of solutions to the inequality x≤ −0.5 on the coordinate line and write the answer as a numerical interval:

Example 11. Solve the inequality

Multiply all parts of the inequality by 3

Now subtract 6 from each part of the resulting inequality

We divide each part of the resulting inequality by −1. Do not forget that when dividing all parts of the inequality by a negative number, the inequality sign is reversed:

Solutions to the inequality 3 ≤ a≤ 9 are all numbers that are greater than 3 and less than 9. Boundaries 3 and 9 belong to the set of solutions, since the inequality 3 ≤ a≤ 9 is non-strict.

Draw the set of solutions to the inequality 3 ≤ a≤ 9 on the coordinate line and write the answer as a numerical interval:

When there are no solutions

There are inequalities that have no solutions. Such, for example, is the inequality 6 x> 2(3x+ 1). In the process of solving this inequality, we will come to the fact that the inequality sign > does not justify its location. Let's see what it looks like.

Expanding the brackets on the right side of this inequality, we get 6 x> 6x+ 2 . Reschedule 6 x from the right side to the left side, changing the sign, we get 6 x− 6x> 2 . We give similar terms and obtain the inequality 0 > 2, which is not true.

For better understanding, we rewrite the reduction of like terms on the left side as follows:

We got the inequality 0 x> 2 . On the left side is the product, which will be equal to zero for any x. And zero cannot be greater than the number 2. Hence the inequality 0 x> 2 has no solutions.

x> 2 , then it has no solutions and the original inequality 6 x> 2(3x+ 1) .

Example 2. Solve the inequality

Multiply both sides of the inequality by 3

In the resulting inequality, we transfer the term 12 x from the right side to the left side by changing the sign. Then we give like terms:

The right side of the resulting inequality for any x will be equal to zero. And zero is not less than -8. Hence the inequality 0 x< −8 не имеет решений.

And if the reduced equivalent inequality 0 x< −8 , то не имеет решений и исходное неравенство .

Answer: no solutions.

When there are infinite solutions

There are inequalities that have an infinite number of solutions. Such inequalities become true for any x .

Example 1. Solve the inequality 5(3x− 9) < 15x

Let's expand the brackets on the right side of the inequality:

Reschedule 15 x from the right side to the left side, changing the sign:

Here are the similar terms on the left side:

We got the inequality 0 x< 45 . On the left side is the product, which will be equal to zero for any x. And zero is less than 45. So the solution of the inequality 0 x< 45 is any number.

x< 45 has an infinite number of solutions, then the original inequality 5(3x− 9) < 15x has the same solutions.

The answer can be written as a numerical interval:

x ∈ (−∞; +∞)

This expression says that the solutions of the inequality 5(3x− 9) < 15x are all numbers from minus infinity to plus infinity.

Example 2. Solve the inequality: 31(2x+ 1) − 12x> 50x

Let's expand the brackets on the left side of the inequality:

Let's reschedule 50 x from the right side to the left side by changing the sign. And we will transfer term 31 from the left side to the right side, again changing the sign:

Here are similar terms:

We got the inequality 0 x >-31 . On the left side is the product, which will be equal to zero for any x. And zero is greater than −31 . So the solution of the inequality 0 x< −31 is any number.

And if the reduced equivalent inequality 0 x >−31 has an infinite number of solutions, then the original inequality 31(2x+ 1) − 12x> 50x has the same solutions.

Let's write the answer as a numerical interval:

x ∈ (−∞; +∞)

Tasks for independent solution

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