A system of irregularities it is customary to name whether there are two or more irregularities, to avenge an unknown magnitude.

At the same time, the formulary is illustrative, for example, so systems of irregularities:

Unravel the system of irritability - means to signify all the meanings of the unknown change, in which the skin nervnist of the system is realized, which is to say that there are no such .

Mean, for skin cream system irregularities counting the unknown change. Given the meaning that you have chosen, choose only those, which are correct and for the first and for the other unevenness. Henceforth, under the hour of setting the opposite meaning, the offenses of the system's inconsistencies become correct.

Let's sort out a number of inconsistencies:

Let's split one pіd іnshoy pair of number lines; put a value on the top x, for which the first unevenness ( x> 1) stavati vіrnim, but in the lower — value X, like solutions to other nervities ( X> 4).

Having entered the data on number lines, it is significant that the solution for both irregularities will X> 4. Vidpovid, X> 4.

butt 2.

Calculating first unevenness acceptable -3 X< -6, или x> 2, other - X> -8 or X < 8. Затем делаем по аналогии с предыдущим примером. На верхнюю числовую прямую наносим все те значения X, for which are realized first system irregularity, and on the lower number line, all values X, which have a different system's nervousness.

Having given the data, we take it that it is insulting unevenness realizable at all values X, Placement from 2 to 8. Anonymous meaning X meaningful underwhelming nervousness 2 < X< 8.

Example 3. We know

Resurrection of inconsistencies online

Before him, how to overcome nervousness, it is necessary to acquire goodness, how jealousy is.

It doesn’t matter what kind of unevenness is strict () or not strict (≤, ≥), we should proceed to the end of the equalization, replacing the sign of unevenness with jealousy (=).

Let us explain what it means to virishity nerіvnіst?

After vivchennya equals, the following picture emerges at the head of a schoolboy: it is necessary to assign such values ​​of change, for some insults, parts of equals acquire the same values. In other words, to know all the points from which equanimity wins. Everything is correct!

If we talk about nervousness, we can think about the importance of intervals (vіdrіzkіv), on which nervousness is victorious. If there are two changes in the unevenness, then the solutions will not be intervals, but like areas on the area. Guess yourself, what will be the unraveling of the nervousness among the three zminnykh?

How to relieve nervousness?

In a universal way to eliminate irregularities, the interval method is used (in the same way, the interval method), which is used for all defined intervals, and unevenness is specified at the boundaries of which there are intervals.

Without going into the type of nervousness, at the same time the essence, it is necessary to virishity in the same way and the same root with the upcoming signs of these decisions on the numerical axis.

How to correctly write down the breakdown of nervousness?

If you have designated the intervals for the rozv'azkіv nerіvnostі, it is necessary to correctly write down the solution itself. An important nuance - what is included between the intervals before the cherry?

Everything is simple here. As a result, the resolution of the equalization is satisfied with the ODZ and the unevenness is suvorim, between the intervals, enter the boundary of the nervousness. Otherwise - no.

Looking at the leather interval, unraveling the unevenness can be the interval itself, or napіvіnval (if one of the intervals is satisfied with the unevenness), or the other is the interval at once from the yoga between.

important moment

Don't think that the unraveling of the nervousness can be more than intervals, napіvіntermali that vodrazki. Ні, the solution can include and okremi points.

For example, the unevenness |x|≤0 has more than one solution - the whole point is 0.

And in the nervousness | x |

Need a bump calculator?

The calculator of irregularities is of the correct type. Whenever possible, an illustration of a numerical axis or a plane is drawn. It can be seen that chi enter between intervals at the rozvyazannya chi ni - the points seem to be filled with chi pierced.

You can interpret the online calculator of irregularities, who correctly knew the roots of equalization, put them on the numerical axis, and corrected them on intervals (and boundaries) to understand the unevenness?

If your opinion differs from the calculator's, then it is definitely necessary to re-verify your decision again and show a pardon.

If you use less “iksi” and less than all abscissa, then “greeks” are added at once and the field of activity expands to the entire coordinate plane. Dali behind the text the phrase "linear unevenness" is comprehensible to the two-world meaning, which will become clear in a few seconds.

Crim of analytical geometry, relevant material for the low level of mathematical analysis, economic and mathematical modeling, I recommend that you study the lecture with seriousness.

Linear irregularities

There are two types of linear irregularities:

1) Suvori unevenness: .

2) Not strict unevenness: .

What geometrical replacement of these irregularities? If linear alignment is given directly, then linear unevenness is assigned napіvploschina.

For the understanding of the lower information, it is necessary to know the various straight lines on the plane and to be straight. To blame the problems in this part, read the conclusion Graphs and power functions- Paragraph about the linear function.

Let's get over the simplest linear irregularities. Blakitna dream of any two-man - coordinate plane, on which there is nothing:

As you can see, the entire abscissa is assigned to equals - “iplayer” is always (if any value “iks”) equals zero

Let's look at the inconsistency. How to understand yoga informally? "Іgrek" is always (whenever "iks" is significant) is positive. Obviously, scho nerіvnіst signifies the upper pіvploshchina - even there and there are all speckles with positive "igreeks".

That slope, like the unevenness of a non-suvor, to the upper flat dodatkovo add up all by itself.

Similarly: unevenness satisfies all points of the lower napіvploschina, nesvoorі nerіvnostі vіdpovidaє lower navіvshina + vos.

From the top of the ordinates, that very prosaic story:

- nerіvnіst put the right to the right;
– nerіvnіst put the right half-plane, including all ordinates;
– nerіvnіst to put the left half-plane;
– inconsistency to set the left half-plane, including all ordinates.

On the other hand, we can see the unevenness, for some people, one of the most important.

Daily "iplayer":

Abo day "iks":

Such inconsistencies can be dealt with in two ways, be kind, look at the offense come. Pretentiously zgadaemo-zakrіpimo shkіlnі dії z nerіvnosti Area of ​​assigned function.

butt 1

Virishity of linear inconsistencies:

What does rozvyazati linear unevenness mean?

Virishiti linear unevenness - tse means to know the flat area, points that satisfy this unevenness (plus the straight line itself, as unevenness is not bad). Solution, as a rule, more graphic.

The best viconate of the chair, and then comment everything:

a) We eliminate the nervousness

First way

A way to guess history with coordinate axes, as we have seen more. The idea of ​​​​believing in the converted nervousness - sob in the left part of the omission of one change without the same constants, in the other - change "iks".

rule: In the case of unevenness, the additions are transferred from part to part with a change of sign, with which the sign of NAIREVNESS do not change(for example, as if the sign is "smaller", then so i get rid of "smaller").

We transfer the “p'yatirka” to the right of the part with a change of sign:

rule POSITIVE do not change.

Now damn straight (dashed blue line). The straight line is drawn by a dotted line for those reasons why the unevenness Suvore, and the points that lie on this straight line are not included until the decision.

What sense of nervousness? "Іks" zavzhdi (whenever the meaning of "gravets") is less, lower. It is obvious that the whiskers of the left side of the plane are satisfied with what solidity. Qiu on the surface, in principle, can be shaded, but I will circle with small blue arrows, so as not to transform the armchair into an artistic palette.

Other way

Tse є universal way. READ IMPORTANTLY!

The back of the head is straight. For clarity, before speech, equal dotally at the sight.

Now we choose to be a point of the plane, do not lie straight. Most of all, the lasa point itself, splendidly. Let's imagine the coordinates of the points in the unevenness:

Otrimano unevenness is wrong(To put it simply, you can’t be like this), it means that the point is not satisfied with the unevenness.

The key rule of our task:
not enough unevenness, then USE points given on the surface not satisfied tsіy nerіvnostі.
- Like a point on a flat surface (so you can’t lie straight) satisfied unevenness, then USE points given on the surface satisfy tsіy nerіvnostі.

You can protest: be it a right-handed point in a straight line, you are not satisfied with the nervousness.

What kind of vysnovok іz conducted dosvіdu z dot? Go nowhere, the unevenness satisfies all the points on the other side - the left side of the plane (you can distort it).

b) We eliminate the nervousness

First way

Let's fix the inconsistency:

rule: Offenses of part of the nervousness can be multiplied (divided) by NEGATIVE number, has its own sign of unevenness CHANGE to the protilezhny (for example, as if the sign “more or less one”, then become “smaller or one”).

We multiply the insults of part of the nervousness on:

Let's call it straight (red color), moreover, call it a strong line, because we have unevenness nasty, and it’s obvious that a decision should be made.

Having analyzed the unevenness, we come to the visnovka, which is the lower nape plane (+ the straight line itself).

The visible half-plane is hatched or marked with arrows.

Other way

Let's go straight. Let's choose a flat point of the plane (not to lay straight lines), for example, and let's put її coordinates into our unevenness:

Otrimano virna nerіvnіst, Otzhe, the point is satisfied with the unevenness, and vzagali - ALL points of the lower half-plane are satisfied with this unevenness.

Here, at the last point, we “spent” the water surface.

The solution of the task is marked with a red straight line and red arrows.

The first way of presentation is especially suitable for me, the other way is formal.

butt 2

Virishity of linear inconsistencies:

This is an example of an independent solution. Try to solve the task in two ways (before speech, it is a good way to re-verify the decision). At the beginning of the lesson, there will be less than a bag chair.

I think, after all the victors at the butts, you will happen to be friends with them, it doesn’t matter, but the simplest nervousness at kshtalt and so on.

Let's move on to the third, zagalny vpadka, if the nervousness has the presence of resentment changes:

As a variant, the optional term "ce" can be null.

butt 3

Know the napіvploshchini, scho podpovіdat advancing nerіvnosti:

Solution: Here the universal method of solving with the substitution of a point is victorious.

a) Let's make a straight line, draw a dotted line at your line, shards of unevenness and not directly see the solution.

We choose the last point of the plane, so as not to overlap with this straight line, for example, and let's put these coordinates into our unevenness:

Otrimano unevenness is wrong, Otzhe, the point and ALL points of the given surface are not satisfied with unevenness. The solution to the unevenness will be another flat surface, adored by blue sparkles:

b) We eliminate the nervousness. The back of the head is straight forward. It’s awkward, maybe canonical direct proportion. The line is drawn by a suciliar, to him that the nervousness is not bad.

We choose a full point of the plane, so as not to lie straight. I would like to renew the cob of coordinates, but, unfortunately, it won't work at once. Tom happened to work with another friend. Nayvigidnishe take a point with small values ​​of coordinates, for example, . Let's put these coordinates into our unevenness:

Otrimano virna nerіvnіst, Otzhe, the point and all points of the given surface are satisfied with the unevenness. Shukana napіvploshchina is marked with red arrows. In addition, the decision to enter itself is straight.

butt 4

Know the napіvploshchini, scho podpovіdat nerіvnosti:

This is an example of an independent solution. Outwardly, a decision, a piece of fine design, and an example of a lesson.

Rozbero zvorotne zavdannya:

butt 5

a) Dana is straight. Significance napіvploshchina, in which there is a point, at which it is directly guilty to enter until the decision.

b) Dana is straight. Significance napіvploshchina, in which there is a point. She herself should not enter directly at the decision.

Solution: there is nothing to use in the armchair, and the solution will be analytical Nothing folding:

a) Foldable additional rich term and let's calculate its value in points:
. In this rank, shukana nerivnist will be the sign of "smaller". For the mind, it’s straight to enter at the decision, to that the unevenness will be unbearable:

b) We add up the polynomial and calculate its value in points:
. In this order, nerіvnіst will be the sign of "greater". Behind the mind, do not directly enter at the decision, then, we will judge the nervousness:.

Vidpovid:

A creative butt for an independent wedding:

butt 6

Given points and lines. The middle of the listing points to know tі, yakі time from the cob of coordinates to lie on one side of the given straight line.

A small hint: it is necessary to fold the unevenness on the cob, which indicates the flat area, in which the cob of coordinates is known. Analytical solution and insight into the lesson.

Systems of linear irregularities

The system of linear irregularities - tse, as you know, the system is composed of decal irregularities. Lol, having seen that one =) Zhachok - tse zhachok, knife - tse knife. Adzhe pravda - it was simple and affordable! But, seriously, I don’t want to induce some kind of butt at the insolent looking, then we’ll immediately move on to impudent food:

What does it mean to untie a system of linear irregularities?

Resolve the system of linear irregularities- tse means know the impersonal point of the plane, yakі satisfy skin nervousness of the system.

In the simplest way, apply a system of irregularities that define the coordinates of a rectangular coordinate system (“little twins” are found at the very beginning of the lesson):

The system of irregularities defines the first coordinate quarter (upper right). Coordinates, be it the points of the first quarter, for example, and etc. satisfy skin unevenness of the system.

Similarly:
- set the system of irregularities to a friend's coordinate quarter (upper left);
– the system of irregularities defines the third coordinate quarter (bottom left);
- set the system of irregularities to the fourth coordinate quarter (lower right).

The system of linear irregularities can solve, tobto buti crazy. I know the simplest example: . It is quite obvious that "ix" cannot be more than three and less than two at the same time.

Solutions to the system of irregularities can be direct, for example: . Swan, crayfish, without pike, to pull from two different sides. That vіz i ninі there - the decisions of the system are straight.

And the widest slope, if the solutions of the system are equal area area. Solution area maybe booty not fringed(for example, coordinates) or obmezhenoyu. The area is fenced, the decision is called system solution.

butt 7

Resolve the system of linear irregularities

In practice, in the majority of vipadkiv, the mother is brought to the right with non-strict irregularities, that part of the lesson that is lost, the round dances will lead the very stench.

Solution: those who are full of irregularities are not guilty How many errors can the system have? Those skils are welcome. Golovnya, to reach a rational algorithm, inspire a solution:

1) We know the simplest inconsistencies on the back. Irregularities signify the first coordinate quarter, including the cordon of coordinate axes. It’s already much easier, because the area sounded significant for a joke. On the armchair, it is indicated by arrows in the middle of the plane (red and blue arrows)

2) Nerіvnіst friend for simplicity - here we see “graves”. In the first place, I will be very direct, but, in a different way, after the transformation of the nervousness to the mind, I suddenly realized that all the “ixi” are smaller, lower 6. It is indicated by green arrows in the opposite direction. Well, for a joke, the region has become even smaller - such a straight cut is not surrounded by the beast.

3) At the rest of the stage, the nervousness “with the new ammunition” is fading: . Algorithm solutions were reportedly reviewed at the front paragraph. In short: the back will be straight, then for the help of the last point, we will need a flat surface.

Stand up, children, stand in the colo:


The area of ​​\u200b\u200bthe solution of the system is a bagatokutnik, on the armchair veins of circling with a raspberry line and shading. Overdone the troch =) In the area of ​​​​the solution, it’s enough to either shade it, or circle it more boldly with a simple olive.

Whether the point of this bagatokudnika is satisfied with the SKIN unevenness of the system (you can reverse it for interest).

Vidpovid: system solutions є bagatokutnik

When drawn up on a clean copy, it was not good to reportly write, for which points you would be straight (divinary lesson Graphs and power functions), and how they signified the napіvploschini (div. the first paragraph of the lesson). However, in practice, the best thing for you will be protected and just the right chair. The roses themselves can be carried out on black or navit orally.

Krіm bagatokutnika solution of the system, practical, let it go and more, zustrichaєtsya open the area. Try to pick up the offensive butt on your own. For the sake of accuracy, there are no tortoises here - the algorithm is the same, it's just that the region is not surrounded.

butt 8

Check the system

The solution is to follow the example of the lesson. You, better for everything, will have other letters for the heights of the conquered area. It’s not important, smut, to know the peaks correctly and to correctly inspire the area.

Not rare, if the tasker needs to prompt the area to solve the system, but to know the coordinates of the area's vertices. In two front butts, the coordinates of these points were obvious, but in practice everything is far from ice:

butt 9

Check the system and know the coordinates of the vertices of the taken area

Solution: depicted on the armchair area of ​​\u200b\u200bthe solution of the system of the system Unevenness to put the left side of the plane of the ordinates, and there are no more freebies here. After rozrahunkiv on the clean / black or deep rozumovyh processes, we will eventually come to the decision area:

At this level, we can continue to look at the rational inequalities of both systems, the same: a system of linear and square inequalities. On the other hand, we guess that such a system of two linear irregularities is from one change. Let's take a look at the system of square irregularities and the technique of their implementation on the example of specific tasks. Let's take a closer look at the names of the Dahu method. Let's look at the typical solutions of systems and, for example, the lesson, let's look at the solution of the system with linear and square unevenness.

2. Electronic teaching and learning complex for preparing grades 10-11 to introductory studies in computer science, mathematics, Russian language.

3. Center for education "Technology of education" ().

4. Distributed College.ru from mathematics ().

1. Mordkovich A.G. Ta іn Algebra Grade 9: A task book for learning about sacred and illuminating mortgages / A. G. Mordkovich, T. N. Mishustina ta іn - 4th type. - M.: Mnemozina, 2002.-143 p.: il. No. 58 (a, c); 62; 63.

The article explores the topic of inconsistencies, discusses the meaning of systems and yogo virishennya. Will look at the examples of rozvyazannya systems of equals at school in algebra.

Designation of the system of irregularities

The systems of irregularities are assigned to the appointments of the equal systems, therefore, special respect is given to the records of that equal equal.

Appointment 1

A system of irregularities name the record equals, united by a curly bow with an impersonal solution at once to all the irregularities that enter the system.

Lower applied butts of irregularities. Given two irregularities 2 · x − 3 > 0 and 5 − x ≥ 4 · x − 11 . It is necessary to write down one equal number of others, after which it is combined for an additional curly bow:

2 x - 3 > 0, 5 - x ≥ 4 x - 11

So the very nature of the systems of irregularities is presented by the school assistants as a choice of one change, and two.

The main types of systems of irregularities

There is a place for the folding of an inexhaustible multiplicity of systems of irregularities. They classify them after the groups, which are reviewed for the singing signs. Irregularities are subdivided according to the criteria:

• number of system irregularities;
• number of changed records;
• kind of unevenness.

The number of irregularities to enter may be present in two or more. At the front point, the butt of the system was ripped apart by irregularities.

2 x - 3 > 0, 5 - x ≥ 4 x - 11

Let's look at the breakdown of the system from chotirma by inconsistencies.

x ≥ - 2 , y ≤ 5 , x + y + z ≥ 3 , z ≤ 1 - x 2 - 4 y 2

It's okay not to talk about the perfection of the system in vain. To complete the system, it is necessary to take into account all apparent irregularities.

Such systems of irregularities can have one, two, three or more. In the rest of the system image, it is clearly visible, there may be three changes: x, y, z. Rivnyannya can take revenge on one change, like a butt, or a sprat. Vykhodyachi z prikladіv, nerіvosti x + 0 · y + 0 · z ≥ − 2 and 0 · x + y + 0 · z ≤ 5 are not considered equal. Shkіlnym programs attach importance to the solution of inconsistencies with one change.

When recording the system, you can set the number of different views and different numbers of changes. Most often, the numbers of nervousness lower steps. Under the hour of preparation before drinking, systems can be used with irrational, logarithmic, ostentatious equals:

544 - 4 - x 32 - 2 - x ≥ 17 , log x 2 16 x + 20 16 ≤ 1

Such a system includes a display of logarithmic equalities.

Virishennya system of irregularities

Let's look at the butt of the rozv'yazannya systems equal from one change.

x > 7 , 2 - 3 x ≤ 0

If the value x = 8, then the decoupling of the system is more obvious, so that 8 > 7 and 2 − 3 · 8 ≤ 0 are victorious. When x = 1, the system does not fail, so the first numerical unevenness can be 1>7. So the system with two and more changes is violating itself.

Appointment 3

Virishennya system of irregularities from two and more change naming the meanings, yakі є razvyazannym vsіh nerіvnіst pri urnennі dermal at the correct numerical nerіvnіst.

How x = 1 and y = 2 will be the difference between x + y< 7 x - y < 0 , потому как выражения 1 + 2 < 7 и 1 − 2 < 0 верны. Если подставить числовую пару (3 , 5 , 3) , тогда система не даст значения переменных и неравенство будет неверным 3 , 5 − 3 < 0 .

When the system of inconsistencies is broken, they can give a lot of inconsistencies, or they can be inconsistent. Maєtsya on uvazі impersonal solution of such a system. For the present, it is a decision to speak to someone who may be empty of an impersonal decision. Even though the decision may be the same number, the same impersonal decision may be the final number of elements. Like a rich solution, even an impersonal solution to avenge impersonal numbers.

Deyakі podruchniki give a clear solution to the system of irregularities, as if to understand how to take the solution. And the deepest decisions of the system of irregularities are taken into account by all the other private decisions. Such a designation of victories is rarely done, it seems to this "a violation of the system of irregularities."

The data on the designation of systems of irregularities and the distribution are looked at as a cross-section of the multiplicity of solutions to the system's initial irregularities. I would like to give special respect to the varto to the division, dedicated to equally strong nervousness.

How did you remember the pardon in the text, be kind, see it and press Ctrl + Enter