The other side of equality is inequality. In this article we will introduce the concept of inequalities, and give some basic information about them in the context of mathematics.
First, let's look at what inequality is and introduce the concepts of not equal, greater than, less. Next we’ll talk about writing inequalities using the signs not equal, less than, greater than, less than or equal to, greater than or equal to. After this, we will touch on the main types of inequalities, give definitions of strict and non-strict, true and false inequalities. Next, let us briefly list the main properties of inequalities. Finally, let's look at doubles, triples, etc. inequalities, and let’s look at the meaning they carry.
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What is inequality?
Concept of inequality, like , is associated with the comparison of two objects. And if equality is characterized by the word “identical,” then inequality, on the contrary, speaks of the difference between the objects being compared. For example, the objects and are the same; we can say about them that they are equal. But the two objects are different, that is, they not equal or unequal.
The inequality of compared objects is recognized along with the meaning of words such as higher, lower (inequality in height), thicker, thinner (inequality in thickness), further, closer (inequality in distance from something), longer, shorter (inequality in length) , heavier, lighter (weight inequality), brighter, dimmer (brightness inequality), warmer, colder, etc.
As we already noted when getting acquainted with equalities, we can talk both about the equality of two objects as a whole, and about the equality of some of their characteristics. The same applies to inequalities. As an example, we give two objects and . Obviously, they are not the same, that is, in general they are unequal. They are not equal in size, nor are they equal in color, however, we can talk about the equality of their shapes - they are both circles.
In mathematics, the general meaning of inequality remains the same. But in its context we are talking about the inequality of mathematical objects: numbers, values of expressions, values of any quantities (lengths, weights, areas, temperatures, etc.), figures, vectors, etc.
Not equal, greater, less
Sometimes it is the very fact that two objects are unequal that is of value. And when the values of any quantities are compared, then, having found out their inequality, they usually go further and find out what quantity more, and which one – less.
We learn the meaning of the words “more” and “less” almost from the first days of our lives. On an intuitive level, we perceive the concept of more and less in terms of size, quantity, etc. And then we gradually begin to realize that in fact we are talking about comparing numbers, corresponding to the number of certain objects or the values of certain quantities. That is, in these cases we find out which number is greater and which is less.
Let's give an example. Consider two segments AB and CD, and compare their lengths 
. Obviously, they are not equal, and it is also obvious that the segment AB is longer than the segment CD. Thus, according to the meaning of the word “longer”, the length of the segment AB is greater than the length of the segment CD, and at the same time the length of the segment CD is less than the length of the segment AB.
Another example. In the morning the air temperature was recorded at 11 degrees Celsius, and in the afternoon – 24 degrees. According to 11 is less than 24, therefore, the temperature value in the morning was less than its value at lunchtime (the temperature at lunchtime became higher than the temperature in the morning).
Writing inequalities using signs
The letter has several symbols for recording inequalities. The first one is not equal sign, it represents a crossed out equal sign: ≠. The unequal sign is placed between unequal objects. For example, the entry |AB|≠|CD|
means that the length of the segment AB is not equal to the length of the segment CD. Likewise, 3≠5 – three does not equal five.
The greater than sign > and the less than sign ≤ are used similarly. The greater sign is written between larger and smaller objects, and the less sign is written between smaller and larger objects. Let us give examples of the use of these signs. The entry 7>1 is read as seven over one, and you can write that the area of triangle ABC is less than the area of triangle DEF using the ≤ sign as SABC≤SDEF.
Also widely used is the greater than or equal to sign of the form ≥, as well as the less than or equal to ≤ sign. We'll talk more about their meaning and purpose in the next paragraph.
Let us also note that algebraic notations with the signs not equal to, less than, greater than, less than or equal to, greater than or equal to, similar to those discussed above, are called inequalities. Moreover, there is a definition of inequalities in the sense of the way they are written:
Definition. Inequalities<, >, ≤, ≥.
are meaningful algebraic expressions composed using the signs ≠,
Let us also note that algebraic notations with the signs not equal to, less than, greater than, less than or equal to, greater than or equal to, similar to those discussed above, are called inequalities. Moreover, there is a definition of inequalities in the sense of the way they are written:
Strict and non-strict inequalities signs of strict inequalities, and the inequalities written with their help are strict inequalities.
In its turn
Let us also note that algebraic notations with the signs not equal to, less than, greater than, less than or equal to, greater than or equal to, similar to those discussed above, are called inequalities. Moreover, there is a definition of inequalities in the sense of the way they are written:
The signs less than or equal to ≤ and greater than or equal to ≥ are called signs of weak inequalities, and the inequalities compiled using them are non-strict inequalities.
The scope of application of strict inequalities is clear from the information above. Why are weak inequalities needed? In practice, with their help it is convenient to model situations that can be described by the phrases “no more” and “no less.” The phrase “no more” essentially means less or the same; it is answered by a less than or equal sign of the form ≤. Likewise, “not less” means the same or more, and is associated with the greater than or equal sign ≥.
From here it becomes clear why the signs< и >are called signs of strict inequalities, and ≤ and ≥ - non-strict. The former exclude the possibility of equality of objects, while the latter allow it.
To conclude this section, we will show a couple of examples of using non-strict inequalities. For example, using the greater than or equal sign, you can write the fact that a is a non-negative number as |a|≥0. Another example: it is known that the geometric mean of two positive numbers a and b is less than or equal to their arithmetic mean, that is, 
.
True and false inequalities
Inequalities can be true or false.
Let us also note that algebraic notations with the signs not equal to, less than, greater than, less than or equal to, greater than or equal to, similar to those discussed above, are called inequalities. Moreover, there is a definition of inequalities in the sense of the way they are written:
Inequality is faithful, if it corresponds to the meaning of the inequality introduced above, otherwise it is unfaithful.
Let us give examples of true and false inequalities. For example, 3≠3 is an incorrect inequality, since the numbers 3 and 3 are equal. Another example: let S be the area of some figure, then S<−7 – неверное неравенство, так как известно, что площадь фигуры по определению выражается неотрицательным числом. И еще пример неверного неравенства: |AB|>|AB| . But the inequalities are −3<12 , |AB|≤|AC|+|BC| и |−4|≥0 – верные. Первое из них отвечает , второе – выражает triangle inequality, and the third is consistent with the definition of the modulus of a number.
Note that along with the phrase “true inequality” the following phrases are used: “fair inequality”, “there is inequality”, etc., meaning the same thing.
Properties of inequalities
According to the way we introduced the concept of inequality, we can describe the main properties of inequalities. It is clear that an object cannot be equal to itself. This is the first property of inequalities. The second property is no less obvious: if the first object is not equal to the second, then the second is not equal to the first.
The concepts “less” and “more” introduced on a certain set define the so-called “less” and “more” relations on the original set. The same applies to the relations “less than or equal to” and “greater than or equal to.” They also have characteristic properties.
Let's start with the properties of the relations to which the signs correspond< и >. Let us list them, after which we will give the necessary comments for clarification:
- anti-reflexivity;
- antisymmetry;
- transitivity.
The anti-reflexivity property can be written using letters as follows: for any object a the inequalities a>a and a b , then b a. Finally, the transitivity property is that from a b and b>c it follows that a>c . This property is also perceived quite naturally: if the first object is smaller (larger) than the second, and the second is smaller (larger) than the third, then it is clear that the first object is even smaller (larger) than the third.
In turn, the relations “less than or equal to” and “greater than or equal to” have the following properties:
- reflexivity: the inequalities a≤a and a≥a hold (since they include the case a=a);
- antisymmetry: if a≤b, then b≥a, and if a≥b, then b≤a;
- transitivity: from a≤b and b≤c it follows that a≤c, and from a≥b and b≥c it follows that a≥c.
Double, triple inequalities, etc.
The property of transitivity, which we touched upon in the previous paragraph, allows us to compose so-called double, triple, etc. inequalities that are chains of inequalities. As an example, let us give the double inequality a Now let's look at how to understand such records. They should be interpreted in accordance with the meaning of the signs they contain. For example, double inequality a In conclusion, we note that sometimes it is convenient to use notations in the form of chains containing both equal and not equal signs, as well as strict and non-strict inequalities. For example, x=2 Bibliography. Today we will learn how to use the interval method to solve weak inequalities. In many textbooks, non-strict inequalities are defined as follows: A non-strict inequality is an inequality of the form f (x) ≥ 0 or f (x) ≤ 0, which is equivalent to the combination of a strict inequality and the equation: Translated into Russian, this means that the non-strict inequality f (x) ≥ 0 is the union of the classical equation f (x) = 0 and the strict inequality f (x) > 0. In other words, now we are interested not only in positive and negative regions on a straight line, but also points where the function is zero. Before solving loose inequalities, let's remember how an interval differs from a segment: In order not to confuse intervals with segments, special notations have been developed for them: an interval is always indicated by punctured dots, and a segment by filled dots. For example: In this figure the segment and interval (9; 11) are marked. Please note: the ends of the segment are marked with filled dots, and the segment itself is indicated by square brackets. With the interval, everything is different: its ends are gouged out, and the brackets are round. Interval method for non-strict inequalities What was all this lyrics about segments and intervals? It’s very simple: to solve non-strict inequalities, all intervals are replaced by segments - and you get the answer. Essentially, we simply add to the answer obtained by the interval method the boundaries of these same intervals. Compare the two inequalities: Task. Solve the strict inequality: (x − 5)(x + 3) > 0 We solve using the interval method. We equate the left side of the inequality to zero: (x − 5)(x + 3) = 0; There is a plus sign on the right. You can easily verify this by substituting billion into the function: f (x) = (x − 5)(x + 3) All that remains is to write out the answer. Since we are interested in positive intervals, we have: x ∈ (−∞; −3) ∪ (5; +∞) Task. Solve the weak inequality: (x − 5)(x + 3) ≥ 0 The beginning is the same as for strict inequalities: the interval method works. We equate the left side of the inequality to zero: (x − 5)(x + 3) = 0; We mark the resulting roots on the coordinate axis: In the previous problem, we already found out that there is a plus sign on the right. Let me remind you that you can easily verify this by substituting a billion into the function: f (x) = (x − 5)(x + 3) All that remains is to write down the answer. Since the inequality is not strict, and we are interested in positive values, we have: x ∈ (−∞; −3] ∪ ∪ ∪ , and (−∞; −3] ∪ Task. Solve the inequality: x (12 − 2x )(3x + 9) ≥ 0 x (12 − 2x )(3x + 9) = 0; x ≥ 6 ⇒ f (x ) = x (12 − 2x )(3x + 9) → (+) (−) (+) = (−)< 0; In this lesson we will begin to study inequalities and their properties. We will consider the simplest inequalities - linear and methods for solving systems and sets of inequalities. We often compare certain objects by their numerical characteristics: goods by their prices, people by their height or age, smartphones by their diagonal, or the results of teams by the number of goals scored in a match. Relationships of the form or are called inequalities. After all, it is written in them that the numbers are not equal, but greater or less than each other. To compare natural numbers in decimal notation, we order the digits: But this method is not always convenient. The easiest way is to compare positive numbers, because they denote quantities. Indeed, if a number can be equivalently represented as the sum of a number with some other number, then greater than: . Equivalent entry: . This definition can be extended not only to positive numbers, but also to any two numbers: Numbermore number
(written as or ) if the number is positive .
Accordingly, if the number is negative, then . For example, let's compare two fractions: and . You can’t tell right away which one is bigger. Therefore, let's turn to the definition and consider the difference: Got a negative number, Means, . On the number axis, the larger number will always be located to the right, the smaller number to the left (Fig. 1). Rice. 1. On the number axis, the larger number is located to the right, the smaller number is to the left Why are such formal definitions needed? Our understanding is one thing, and technology is another. If you formulate a strict algorithm for comparing numbers, then you can entrust it to a computer. There is a plus in this - this approach saves us from performing routine operations. But there is also a minus - the computer exactly follows the given algorithm. If the computer is given the task: the train must leave the station at, then even if you find yourself on the platform at, you will not be on time for this train. Therefore, the algorithms that we assign to the computer to perform various calculations or solve problems must be very accurate and as formalized as possible. As in the case of equalities, you can perform certain operations on inequalities and obtain equivalent inequalities. Let's look at some of them. 1.
If, Thatfor any number.
Those. you can add or subtract the same number to both sides of the inequality. We already have a good image - scales. If one of the scales is overweight, then no matter how much we add (or take away) to both scales, this situation will not change (Fig. 2). Rice. 2. If the scales are not balanced, then after adding (subtracting) the same number of weights to them they will remain in the same unbalanced position This action can be formulated differently: you can transfer terms from one part of the inequality to another, changing their sign to the opposite: . 2.
If, ThatAnd To understand this property, we can again use the analogy with scales: if, for example, the left bowl was outweighed, then if we take two left bowls and two right ones, the advantage will definitely remain. The same situation for , bowls, etc. Even if we take half of each of the bowls, the situation will not change either (Fig. 3). Rice. 3. If the scales are not balanced, then after taking away half of each of them, they will remain in the same unbalanced position If you multiply or divide both sides of the inequality by a negative number, then the sign of the inequality will change to the opposite. The analogy for this operation is a little more complicated - there are no negative quantities. The fact that for negative numbers the opposite is true will help here (the larger the absolute value of the number, the smaller the number itself): For numbers of different signs it is even easier: As for multiplying by a negative number, you can perform an equivalent two-part operation: first multiply by the opposite positive number - as we already know, the inequality sign will not change: . In the first property we wrote: , but at the same time we said that you can not only add, but also subtract. Why? Because subtracting a number is the same as adding its opposite number: Similarly with the second property: division is multiplication by the reciprocal number: . Therefore, in the second property we are talking not only about multiplication by a number, but also about division. 3. For positive numbersAnd, If, That.
We know this property well: if we divide the cake among people, then the more , the less everyone gets. For example: , therefore (indeed, the fourth part of the cake is clearly smaller than the third part of the same cake) (Fig. 4). Rice. 4. A fourth of a cake is smaller than a third of the same cake. 4.
IfAnd, That.
Continuing the analogy with scales: if on some scales the left pan outweighs the right one and on others the situation is the same, then by pouring the contents of the left bowls separately and separately the contents of the right bowls, we again obtain that the left bowl outweighs (Fig. 5). Rice. 5. If the left pans of two scales outweigh the right ones, then by pouring separately the contents of the left and separately the contents of the right bowls, it turns out that the left pan outweighs 5. For the positive, IfAnd, That.
Here the analogy is a little more complex, but also clear: if the left bowl is heavier than the right and we take more left bowls than right ones, then we will definitely get a more massive bowl (Fig. 6). Rice. 6. If the left bowl is heavier than the right, then if you take more left bowls than right bowls, you will get a more massive bowl The last two properties are intuitive: when we add or multiply larger numbers, we end up with a larger number. Most of these properties can be rigorously proven using various algebraic axioms and definitions, but we will not do this. For us, the process of proof is not as interesting as the directly obtained result, which we will use in practice. So far, we have talked about inequalities as a way of writing the result of comparing two numbers: or. But inequalities can also be used to record various information about restrictions for a particular object. In life, we often use such restrictions to describe, for example: Russia is millions of people from Kaliningrad to Vladivostok; You can carry no more than kg in an elevator, and you can put no more than kg in a bag. Constraints can also be used to classify objects. For example, depending on age, different categories of the population are distinguished - children, adolescents, youth, etc. In all the examples considered, a common idea can be identified: a certain quantity is limited from above or below (or from both sides at once). If is the lifting capacity of the elevator, and is the permissible mass of goods that can be placed in the package, then the information described above can be written as follows: , etc. In the examples we looked at, we were a little inaccurate. The wording “no more” implies that exactly kg can be transported in an elevator, and exactly kg can be put in a bag. Therefore, it would be more correct to write it this way: or . Naturally, it’s inconvenient to write this way, so they came up with a special sign: , which reads “less than or equal to.” Such inequalities are called not strict(respectively, inequalities with signs - strict). They are used when a variable can not only be strictly greater or less, but can also be equal to the boundary value. Solving the inequality All such values of a variable are called, upon substitution of which the resulting numerical inequality will be true. Consider, for example, the inequality: . Numbers are solutions to this inequality, because the inequalities are true. But numbers are not solutions, since numerical inequalities are not true. Solve inequality, which means finding all the values of the variables for which the inequality is true. Let's return to inequality. Its solutions can be equivalently described as: all real numbers that are greater than . It is clear that there are an infinite number of such numbers, so how can we write down the answer in this case? Let's turn to the number axis: all numbers greater than , are located to the right of . Let's shade this area, thereby showing that this will be the answer to our inequality. To show that a number is not a solution, it is enclosed in an empty circle, or, in other words, a dot is poked out (Fig. 7). Rice. 7. The number line shows that the number is not a solution (punctured point) If the inequality is not strict and the chosen point is a solution, then it is enclosed in a filled circle. Rice. 8. The number line shows that the number is a solution (shaded dot) It is convenient to write the final answer using gaps. The interval is written according to the following rules: The sign denotes infinity, i.e. shows that the number can take on an arbitrarily large () or arbitrarily small value (). We can write the answer to the inequality as follows: or simply: . This means that the unknown belongs to the specified interval, i.e. can take any value from this range. If both brackets of the gap are round, as in our example, then such a gap is also called interval. Usually the solution to the inequality is an interval, but other options are possible, for example, the solution can be a set consisting of one or more numbers. For example, an inequality has only one solution. Indeed, for any other values, the expression will be positive, which means that the corresponding numerical inequality will not be satisfied. Inequalities may not have solutions. In this case, the answer is written as (“The variable belongs to the empty set”). There is nothing unusual in the fact that the solution to an inequality can be the empty set. Indeed, in real life, restrictions can also lead to the fact that there is not a single element that satisfies the requirements. For example, there are definitely no people taller than meters and weighing up to kg. The set of such people does not contain a single element, or, as they say, it is an empty set. Inequalities can be used not only to record known information, but also, as mathematical models, to solve various problems. Let you have rubles. How many ruble ice creams can you buy with this money? Another example. We have rubles and we need to buy ice cream for our friends. At what price can we choose ice cream to buy? In life, each of us knows how to solve such simple problems in our heads, but the task of mathematics is to develop a convenient tool with which you can solve not one specific problem, but a whole class of different problems, regardless of what we are talking about - the number of servings of ice cream, cars for transporting goods or rolls of wallpaper for a room. Let's rewrite the condition of the first problem about ice cream in mathematical language: one serving costs rubles, the number of servings that we can buy is unknown to us, let's denote it as . Then the total cost of our purchase: rubles. And, according to the condition, this amount should not exceed rubles. Getting rid of names, we get a mathematical model: . Similarly for the second problem (where is the cost of a serving of ice cream): . Constructions are the simplest examples of inequalities with a variable, or linear inequalities. Inequalities are called linear type There is nothing new for us in this definition: the difference between linear inequalities and linear equations is only in replacing the equal sign with an inequality sign. The name is also associated with the linear function, which appears on the left side of the inequality (Fig. 9). Rice. 9. Graph of a Linear Function Accordingly, the algorithm for solving linear inequalities is almost the same as the algorithm for solving linear equations: Let's look at a few examples. Example 1. Solve linear inequality: . Solution Let's move the term with the unknown from the right side of the inequality to the left: . We divide both sides by a negative number, the inequality sign changes to the opposite: . Let's make a drawing on the axis (Fig. 10). Rice. 10. Illustration for example 1 There is no left edge of the gap, so we write . The left edge of the interval is a strict inequality, so we write it with a parenthesis. We get the interval: . Example 2. Solve linear inequality: Solution Let's open the brackets on the left and right sides of the inequality: . Let us present similar terms: . Let's make a drawing on the axis (Fig. 11). Rice. 11. Illustration for example 2 We get the interval: . Example 1. Solve linear inequality: Solution Let's expand the brackets: Let’s move all terms with a variable to the left side, and without a variable to the right side: Let's look at similar terms: We get: . There is no unknown, what to do? Actually nothing new again. Remember what we did in such cases for linear equations: if the equality is true, then the solution is any real number; if the equality is incorrect, then the equation has no solutions. We do the same here. If the resulting numerical inequality is true, then the unknown can take any value: ( - the set of all real numbers). But this can be depicted on the numerical axis as follows (Fig. 1): Rice. 1. The unknown can take any value And using the interval write it like this: . If the numerical inequality turns out to be incorrect, then the original inequality has no solutions: . In our case, the inequality is not true, so the answer is: . In various tasks we may encounter not one, but several conditions or restrictions at once. For example, to solve a transport problem, you need to take into account the number of cars, travel time, carrying capacity, etc. Each of the conditions will be described in mathematical language by its own inequality. In this case, two options are possible: 1. All conditions are met simultaneously. Such a case is described system of inequalities. When writing, they are combined with a curly brace (you can read it as a conjunction AND): . 2. At least one of the conditions must be met. This is described set of inequalities(you can read it as a conjunction OR): . Systems and sets of inequalities can contain several variables; their number and complexity can be any. But we will study in detail the simplest case: systems and sets of inequalities with one variable. How to solve them? We need to solve each of the inequalities separately, and then everything depends on whether we are dealing with a system or a collection. If it's a system, all conditions must be met. If Sherlock Holmes determined that the criminal was blond and had the size of his feet, then only blondes with the size of his feet should remain among the suspects. Those. We will only use those values that correspond to one, the second, and, if any, the third and other conditions. They are at the intersection of all the resulting sets. If you use a number axis, then - at the intersection of all shaded parts of the axis (Fig. 12). Rice. 12. Solution of the system - the intersection of all shaded parts of the axis If it's a collection, then all values that are solutions to at least one inequality are suitable for us. If Sherlock Holmes determined that the criminal could be either a blond man or a person with a foot size, then among the suspects there should be both all blondes (regardless of shoe size) and all people with a foot size (regardless of hair color). Those. the solution to a set of inequalities will be the union of the sets of their solutions. If you use a number axis, then it is the union of all shaded parts of the axis (Fig. 13). Rice. 13. Solution of the ensemble - union of all shaded parts of the axis You can learn more about intersection and union below. The terms "intersection" and "union" refer to the concept of set. A bunch of- a set of elements that meet certain criteria. You can come up with as many examples of sets as you like: many classmates, many football players of the Russian national team, many cars in the neighboring yard, etc. You are already familiar with numerical sets: the set of natural numbers, integers, rational numbers, real numbers. There are also empty sets, they do not contain elements. Solutions to inequalities are also sets of numbers. The intersection of two setsAnd is called a set that contains all the elements that belong simultaneously to both the set and the set (Fig. 1). Rice. 1. Intersection of sets and For example, the intersection of the set of all women and the set of presidents of all countries will be all women presidents. Union of two setsAnd is called a set that contains all elements that belong to at least one of the sets or (Fig. 2). Rice. 2. Union of sets and For example, the union of many Zenit football players in the Russian national team and Spartak football players in the Russian national team will be all the Zenit and Spartak football players who play for the national team. By the way, the intersection of these sets will be the empty set (a player cannot play for two clubs at the same time). You have already encountered the union and intersection of numerical sets when you were looking for the LCM and GCD of two numbers. If and are sets consisting of prime factors obtained by decomposing numbers, then the gcd is obtained from the intersection of these sets, and the gcd is obtained from the union. Example: Example 3. Solve the system of inequalities: Solution Let's solve the inequalities separately. In the first inequality, we move the term without a variable to the right side with the opposite sign: . Let us present similar terms: . Let us divide both sides of the inequality by a positive number, the sign of the inequality does not change: In the second inequality, we move the term with the variable to the left side, and without the variable to the right side: Let us divide both sides of the inequality by a positive number, the sign of the inequality does not change: Let us depict the solutions of individual inequalities on the number axis. By condition, we have a system of inequalities, so we are looking for the intersection of solutions (Fig. 14). Rice. 14. Illustration for example 3 In essence, the first part of solving systems and sets of inequalities with one variable comes down to solving individual linear inequalities. You can practice this yourself (for example, using our tests and simulators), and we will dwell in more detail on finding unions and intersections of solution sets. Example 4. Let the following solution of individual equations of the system be obtained: Solution Let's shade the area on the axis corresponding to the solution of the first equation (Fig. 15); the solution to the second equation is an empty set; there is nothing corresponding to it on the axis. Rice. 15. Illustration for example 4 This is a system, so you need to look for the intersection of solutions. But there are none. This means that the answer to the system will also be an empty set: . Example 5. Another example: . Solution The difference is that this is already a set of inequalities. Therefore, you need to select a region on the axis that corresponds to the solution of at least one of the equations. We get the answer: . For example, the inequality is the expression \(x>5\). If \(a\) and \(b\) are numbers or , then the inequality is called numerical. It's actually just comparing two numbers. Such inequalities are divided into faithful And unfaithful. For example: \(17+3\geq 115\) is an incorrect numerical inequality, since \(17+3=20\), and \(20\) is less than \(115\) (and not greater than or equal to). If \(a\) and \(b\) are expressions containing a variable, then we have inequality with variable. Such inequalities are divided into types depending on the content: \(2x+1\geq4(5-x)\) Variable only to the first power \(3x^2-x+5>0\) There is a variable in the second power (square), but there are no higher powers (third, fourth, etc.) \(\log_(4)((x+1))<3\) \(2^(x)\leq8^(5x-2)\) If you substitute a number instead of a variable into an inequality, it will turn into a numeric one. For example, if we substitute the number \(7\) into the linear inequality \(x+6>10\), we get the correct numerical inequality: \(13>10\). And if we substitute \(2\), there will be an incorrect numerical inequality \(8>10\). That is, \(7\) is a solution to the original inequality, but \(2\) is not. However, the inequality \(x+6>10\) has other solutions. Indeed, we will get the correct numerical inequalities when substituting \(5\), and \(12\), and \(138\)... And how can we find all possible solutions? For this they use For our case we have: \(x+6>10\) \(|-6\) That is, any number greater than four will suit us. Now you need to write down the answer. Solutions to inequalities are usually written numerically, additionally marking them on the number axis with shading. For our case we have: Answer:
\(x\in(4;+\infty)\) There is one big trap in inequalities that students really “love” to fall into: Why is this happening? To understand this, let's look at the transformations of the numerical inequality \(3>1\). It is correct, three is indeed greater than one. First, let's try to multiply it by any positive number, for example, two: \(3>1\) \(|\cdot2\) As we can see, after multiplication the inequality remains true. And no matter what positive number we multiply by, we will always get the correct inequality. Now let’s try to multiply by a negative number, for example, minus three: \(3>1\) \(|\cdot(-3)\) The result is an incorrect inequality, because minus nine is less than minus three! That is, in order for the inequality to become true (and therefore, the transformation of multiplication by negative was “legal”), you need to reverse the comparison sign, like this: \(−9<− 3\). The rule written above applies to all types of inequalities, not just numerical ones. \(2x+2-1<7+8x\) Let's move \(8x\) to the left, and \(2\) and \(-1\) to the right, not forgetting to change the signs \(2x-8x<7-2+1\) \(-6x<6\) \(|:(-6)\) Let's divide both sides of the inequality by \(-6\), not forgetting to change from “less” to “more” Let's mark a numerical interval on the axis. Inequality, therefore we “prick out” the value \(-1\) itself and do not take it as an answer Let's write the answer as an interval Answer:
\(x\in(-1;\infty)\) Inequalities, just like equations, can have restrictions on , that is, on the values of x. Accordingly, those values that are unacceptable according to the DZ should be excluded from the range of solutions. Example:
Solve the inequality \(\sqrt(x+1)<3\) Solution:
It is clear that in order for the left side to be less than \(3\), the radical expression must be less than \(9\) (after all, from \(9\) just \(3\)). We get: \(x+1<9\) \(|-1\) All? Any value of x smaller than \(8\) will suit us? No! Because if we take, for example, the value \(-5\) that seems to fit the requirement, it will not be a solution to the original inequality, since it will lead us to calculating the root of a negative number. \(\sqrt(-5+1)<3\) Therefore, we must also take into account the restrictions on the value of X - it cannot be such that there is a negative number under the root. Thus, we have the second requirement for x: \(x+1\geq0\) And for x to be the final solution, it must satisfy both requirements at once: it must be less than \(8\) (to be a solution) and greater than \(-1\) (to be admissible in principle). Plotting it on the number line, we have the final answer: Answer:
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Segments and intervals: what's the difference?
x − 5 = 0 ⇒ x = 5;
x + 3 = 0 ⇒ x = −3;
x − 5 = 0 ⇒ x = 5;
x + 3 = 0 ⇒ x = −3;
x = 0;
12 − 2x = 0 ⇒ 2x = 12 ⇒ x = 6;
3x + 9 = 0 ⇒ 3x = −9 ⇒ x = −3.
x ∈ (−∞ −3] ∪ .
, and then most often used the advantages of decimal notation: they began to compare the digits of numbers from the leftmost digits until the first discrepancy.
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for any positive.
Those. Both sides of the inequality can be multiplied or divided by a positive number and its sign will not change.
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. That is, when multiplying by , we must change the sign of inequality to the opposite.Learn more about addition and multiplication
. That is why we talk not only about addition, but also about subtraction.
, as well as those that can be brought to this form by equivalent transformations. For example: ; ; 
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What to do if, after reducing similar terms, the unknown
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.Intersection and union of sets
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. Let us present similar terms: .
Types of inequalities:
\(-5<2\) - верное числовое неравенство, ведь \(-5\) действительно меньше \(2\);
... and so on.
What is the solution to an inequality?
If a given value for x turns the original inequality into a true numerical one, then it is called solution to inequality. If not, then this value is not a solution. And to solve inequality– you need to find all its solutions (or show that there are none).
\(x>4\)When does the sign of an inequality change?
When multiplying (or dividing) an inequality by a negative number, it is reversed (“more” by “less”, “more or equal” by “less than or equal”, and so on)
\(6>2\)
\(-9>-3\)
With division it will work out the same way, you can check it yourself.
Solution:
Inequalities and disability
\(x<8\)
\(\sqrt(-4)<3\)
\(x\geq-1\)
Collection and use of personal information
Disclosure of information to third parties
Protection of personal information
Respecting your privacy at the company level