System of inequalities.
Example 1. Find the scope of an expression
Solution. Under the sign square root must be non-negative number, which means that two inequalities must be satisfied simultaneously: In such cases, they say that the problem reduces to solving a system of inequalities

But we have not yet encountered such a mathematical model (system of inequalities). This means that we are not yet able to complete the solution to the example.

The inequalities that form a system are combined with a curly bracket (the same is true in systems of equations). For example, record

means that the inequalities 2x - 1 > 3 and 3x - 2< 11 образуют систему неравенств.

Sometimes a system of inequalities is written in the form of a double inequality. For example, a system of inequalities

can be written as a double inequality 3<2х-1<11.

In the 9th grade algebra course, we will consider only systems of two inequalities.

Consider the system of inequalities

You can select several of its particular solutions, for example x = 3, x = 4, x = 3.5. In fact, for x = 3 the first inequality takes the form 5 > 3, and the second one takes the form 7< 11. Получились два верных числовых неравенства, значит, х = 3 - решение системы неравенств. Точно так же можно убедиться в том, что х = 4, х = 3,5 - решения системы неравенств.

At the same time, the value x = 5 is not a solution to the system of inequalities. When x = 5, the first inequality takes the form 9 > 3 - a correct numerical inequality, and the second one takes the form 13< 11- неверное числовое неравенство .
To solve a system of inequalities means to find all its particular solutions. It is clear that the guessing demonstrated above is not a method for solving a system of inequalities. IN following example We will show how people usually reason when solving a system of inequalities.

Example 3. Solve the system of inequalities:

Solution.

A) Solving the first inequality of the system, we find 2x > 4, x > 2; solving the second inequality of the system, we find 3x< 13 Отметим эти промежутки на одной координатной прямой , использовав для выделения первого промежутка верхнюю штриховку, а для второго - нижнюю штриховку (рис. 22). Решением системы неравенств будет пересечение решений неравенств системы, т.е. промежуток, на котором обе штриховки совпали. В рассматриваемом примере получаем интервал
b) Solving the first inequality of the system, we find x > 2; solving the second inequality of the system, we find Let's mark these intervals on one coordinate line, using upper hatching for the first interval, and lower hatching for the second (Fig. 23). The solution to the system of inequalities will be the intersection of the solutions to the system’s inequalities, i.e. the interval where both hatches coincide. In the example under consideration we obtain a beam

V) Solving the first inequality of the system, we find x< 2; решая второе неравенство системы, находим Отметим эти промежутки на одной координатной прямой, использовав для первого промежутка верхнюю штриховку, а для второго - нижнюю штриховку (рис. 24). Решением системы неравенств будет пересечение решений неравенств системы, т.е. промежуток, на котором обе штриховки совпали. Здесь такого промежутка нет, значит, система неравенств не имеет решений.

Let us generalize the reasoning carried out in the example considered. Suppose we need to solve the system of inequalities

Let, for example, the interval (a, b) be a solution to the inequality fx 2 > g(x), and the interval (c, d) be a solution to the inequality f 2 (x) > s 2 (x). Let us mark these intervals on one coordinate line, using the upper hatch for the first interval, and the lower hatch for the second (Fig. 25). The solution to a system of inequalities is the intersection of solutions to the system’s inequalities, i.e. the interval where both hatchings coincide. In Fig. 25 is the interval (c, b).

Now we can easily solve the system of inequalities that we obtained above in example 1:

Solving the first inequality of the system, we find x > 2; solving the second inequality of the system, we find x< 8. Отметим эти промежутки (лучи) на одной координатной прямой, использовав для первого -верхнюю, а для второго - нижнюю штриховку (рис. 26). Решением системы неравенств будет пересечение решений неравенств системы, т.е. промежуток, на котором обе штриховки совпали, - отрезок . Это - область определения того выражения, о котором шла речь в примере 1.

Of course, the system of inequalities does not have to consist of linear inequalities, as it was until now; Any rational (and not only rational) inequalities can occur. Technically, working with a system of rational nonlinear inequalities is, of course, more complicated, but there is nothing fundamentally new (compared to systems of linear inequalities) here.

Example 4. Solve the system of inequalities

Solution.

1) Solve the inequality We have
Let's mark points -3 and 3 on the number line (Fig. 27). They divide the line into three intervals, and on each interval the expression p(x) = (x- 3)(x + 3) retains a constant sign - these signs are indicated in Fig. 27. We are interested in the intervals at which the inequality p(x) > 0 holds (they are shaded in Fig. 27), and the points at which the equality p(x) = 0 holds, i.e. points x = -3, x = 3 (they are marked in Fig. 2 7 with dark circles). Thus, in Fig. 27 presented geometric model solutions to the first inequality.

2) Solve the inequality We have
Let's mark points 0 and 5 on the number line (Fig. 28). They divide the line into three intervals, and on each interval the expression<7(х) = х(5 - х) сохраняет постоянный знак - эти знаки указаны на рис. 28. Нас интересуют промежутки, на которых выполняется неравенство g(х) >O (shaded in Fig. 28), and the points at which the equality g (x) - O is satisfied, i.e. points x = 0, x = 5 (they are marked in Fig. 28 with dark circles). Thus, in Fig. Figure 28 presents a geometric model for solving the second inequality of the system.

3) Let us mark the found solutions to the first and second inequalities of the system on the same coordinate line, using upper hatching for solutions to the first inequality, and lower hatching for solutions to the second (Fig. 29). The solution to the system of inequalities will be the intersection of the solutions to the system’s inequalities, i.e. the interval where both hatches coincide. Such an interval is a segment.

Example 5. Solve the system of inequalities:

Solution:

A) From the first inequality we find x >2. Let's consider the second inequality. Square trinomial x 2 + x + 2 has no real roots, and its leading coefficient (the coefficient of x 2) is positive. This means that for all x the inequality x 2 + x + 2>0 holds, and therefore the second inequality of the system has no solutions. What does this mean for the system of inequalities? This means that the system has no solutions.

b) From the first inequality we find x > 2, and the second inequality is satisfied for any values ​​of x. What does this mean for the system of inequalities? This means that its solution has the form x>2, i.e. coincides with the solution to the first inequality.

Answer:

a) no solutions; b) x >2.

This example is an illustration of the following useful

1. If in a system of several inequalities with one variable one inequality has no solutions, then the system has no solutions.

2. If in a system of two inequalities with one variable, one inequality is satisfied for any values ​​of the variable, then the solution to the system is the solution to the second inequality of the system.

Concluding this section, let us return to the problem about the intended number given at the beginning and solve it, as they say, according to all the rules.

Example 2(see p. 29). Intended natural number. It is known that if you add 13 to the square of the intended number, the sum will be more work the planned number and the number 14. If you add 45 to the square of the planned number, then the sum will be less than the product of the planned number and the number 18. What number is planned?

Solution.

First stage. Drawing up a mathematical model.
The intended number x, as we saw above, must satisfy the system of inequalities

Second stage. Working with the compiled mathematical model. Let's transform the first inequality of the system to the form
x2- 14x+ 13 > 0.

Let's find the roots of the trinomial x 2 - 14x + 13: x 2 = 1, x 2 = 13. Using the parabola y = x 2 - 14x + 13 (Fig. 30) we conclude that the inequality we are interested in is satisfied at x< 1 или x > 13.

Let us transform the second inequality of the system to the form x2 - 18 2 + 45< 0. Найдем корни трехчлена х 2 - 18x + 45: = 3, х 2 = 15.

Inequalities and systems of inequalities are one of the topics covered in high school in algebra. In terms of difficulty level, it is not the most difficult, since it has simple rules (more on them a little later). As a rule, schoolchildren learn to solve systems of inequalities quite easily. This is also due to the fact that teachers simply “train” their students on this topic. And they cannot help but do this, because it is studied in the future using other mathematical quantities, and is also tested on the OGE and the Unified State Exam. IN school textbooks The topic of inequalities and systems of inequalities is covered in great detail, so if you are going to study it, it is best to resort to them. This article only summarizes larger material and there may be some omissions.

The concept of a system of inequalities

If you turn to scientific language, then we can define the concept of “system of inequalities”. This is a mathematical model that represents several inequalities. This model, of course, requires a solution, and this will be the general answer for all the inequalities of the system proposed in the task (usually this is written in it, for example: “Solve the system of inequalities 4 x + 1 > 2 and 30 - x > 6... "). However, before moving on to the types and methods of solutions, you need to understand something else.

Systems of inequalities and systems of equations

In the process of studying new topic very often misunderstandings arise. On the one hand, everything is clear and you want to start solving tasks as soon as possible, but on the other hand, some aspects remain in the “shadow” and are not fully understood. Also, some elements of already acquired knowledge may be intertwined with new ones. As a result of this “overlapping”, errors often occur.

Therefore, before we begin to analyze our topic, we should remember the differences between equations and inequalities and their systems. To do this, we need to clarify once again what the data represents. mathematical concepts. An equation is always an equality, and it is always equal to something (in mathematics this word is denoted by the sign "="). Inequality is a model in which one quantity is either greater or less than another, or contains a statement that they are not the same. Thus, in the first case, it is appropriate to talk about equality, and in the second, no matter how obvious it may sound from the name itself, about the inequality of the initial data. Systems of equations and inequalities practically do not differ from each other and the methods for solving them are the same. The only difference is that in the first case equalities are used, and in the second inequalities are used.

Types of inequalities

There are two types of inequalities: numerical and with an unknown variable. The first type represents provided values ​​(numbers) that are unequal to each other, for example, 8 > 10. The second is inequalities containing an unknown variable (denoted by some letter Latin alphabet, most often X). This variable needs to be found. Depending on how many there are, the mathematical model distinguishes between inequalities with one (they make up a system of inequalities with one variable) or several variables (they make up a system of inequalities with several variables).

The last two types, according to the degree of their construction and the level of complexity of the solution, are divided into simple and complex. Simple ones are also called linear inequalities. They, in turn, are divided into strict and non-strict. Strict ones specifically “say” that one quantity must necessarily be either less or more, so this is pure inequality. Several examples can be given: 8 x + 9 > 2, 100 - 3 x > 5, etc. Non-strict ones also include equality. That is, one value can be greater than or equal to another value (the “≥” sign) or less than or equal to another value (the “≤” sign). Even in linear inequalities, the variable is not at the root, square, or divisible by anything, which is why they are called “simple.” Complex ones involve unknown variables that require execution to find. more mathematical operations. They are often located in a square, cube or under a root, they can be modular, logarithmic, fractional, etc. But since our task is the need to understand the solution of systems of inequalities, we will talk about a system of linear inequalities. However, before that, a few words should be said about their properties.

Properties of inequalities

The properties of inequalities include the following:

1. The inequality sign is reversed if an operation is used to change the order of the sides (for example, if t 1 ≤ t 2, then t 2 ≥ t 1).
2. Both sides of the inequality allow you to add the same number to itself (for example, if t 1 ≤ t 2, then t 1 + number ≤ t 2 + number).
3. Two or more inequalities with a sign in the same direction allow their left and right sides to be added (for example, if t 1 ≥ t 2, t 3 ≥ t 4, then t 1 + t 3 ≥ t 2 + t 4).
4. Both sides of the inequality can be multiplied or divided by the same thing positive number(for example, if t 1 ≤ t 2 and number ≤ 0, then number · t 1 ≥ number · t 2).
5. Two or more inequalities that have positive terms and a sign in the same direction allow themselves to be multiplied by each other (for example, if t 1 ≤ t 2, t 3 ≤ t 4, t 1, t 2, t 3, t 4 ≥ 0 then t 1 · t 3 ≤ t 2 · t 4).
6. Both parts of the inequality allow themselves to be multiplied or divided by the same negative number, but in this case the sign of the inequality changes (for example, if t 1 ≤ t 2 and a number ≤ 0, then the number · t 1 ≥ number · t 2).
7. All inequalities have the property of transitivity (for example, if t 1 ≤ t 2 and t 2 ≤ t 3, then t 1 ≤ t 3).

Now, after studying the basic principles of the theory related to inequalities, we can proceed directly to the consideration of the rules for solving their systems.

Solving systems of inequalities. General information. Solutions

As mentioned above, the solution is the values ​​of the variable that are suitable for all the inequalities of the given system. Solving systems of inequalities is the implementation of mathematical operations that ultimately lead to a solution to the entire system or prove that it has no solutions. In this case, they say that the variable refers to an empty numerical set (written as follows: letter denoting a variable∈ (sign “belongs”) ø (sign “empty set”), for example, x ∈ ø (read: “The variable “x” belongs to the empty set”). There are several ways to solve systems of inequalities: graphical, algebraic, substitution method. It is worth noting that they are among those mathematical models, which have several unknown variables. In the case where there is only one, the interval method is suitable.

Graphic method

Allows you to solve a system of inequalities with several unknown quantities (from two and above). Thanks to this method, a system of linear inequalities can be solved quite easily and quickly, so it is the most common method. This is explained by the fact that plotting a graph reduces the amount of writing mathematical operations. It becomes especially pleasant to take a little break from the pen, pick up a pencil with a ruler and start working. further actions with their help when a lot of work has been done and you want a little variety. However this method some people don’t like it because they have to break away from the task and switch their mental activity to drawing. However, this is a very effective method.

To solve a system of inequalities using graphic method, it is necessary to transfer all terms of each inequality to their left side. The signs will be reversed, zero should be written on the right, then each inequality needs to be written separately. As a result, functions will be obtained from inequalities. After this, you can take out a pencil and a ruler: now you need to draw a graph of each function obtained. The entire set of numbers that will be in the interval of their intersection will be a solution to the system of inequalities.

Algebraic way

Allows you to solve a system of inequalities with two unknown variables. Also, inequalities must have with the same sign inequalities (i.e. they must contain either only the “greater than” sign, or only the “less than” sign, etc.) Despite its limitations, this method is also more complex. It is applied in two stages.

The first involves actions to get rid of one of the unknown variables. First you need to select it, then check for the presence of numbers in front of this variable. If they are not there (then the variable will look like a single letter), then we do not change anything, if there are (the type of the variable will be, for example, 5y or 12y), then it is necessary to make sure that in each inequality the number in front of the selected variable is the same. To do this, you need to multiply each term of the inequalities by common multiplier, for example, if 3y is written in the first inequality, and 5y in the second, then it is necessary to multiply all terms of the first inequality by 5, and the second by 3. The result is 15y and 15y, respectively.

Second stage of solution. It is necessary to transfer the left side of each inequality to their right sides, changing the sign of each term to the opposite, and write zero on the right. Then comes the fun part: getting rid of the selected variable (otherwise known as “reduction”) while adding the inequalities. This results in an inequality with one variable that needs to be solved. After this, you should do the same thing, only with another unknown variable. The results obtained will be the solution of the system.

Substitution method

Allows you to solve a system of inequalities if it is possible to introduce a new variable. Typically, this method is used when the unknown variable in one term of the inequality is raised to the fourth power, and in the other term it is squared. Thus, this method is aimed at reducing the degree of inequalities in the system. The sample inequality x 4 - x 2 - 1 ≤ 0 is solved in this way. A new variable is introduced, for example t. They write: “Let t = x 2,” then the model is rewritten in a new form. In our case, we get t 2 - t - 1 ≤0. This inequality needs to be solved using the interval method (more on that a little later), then back to the variable X, then do the same with the other inequality. The answers received will be the solution of the system.

Interval method

This is the simplest way to solve systems of inequalities, and at the same time it is universal and widespread. It is used in secondary schools and even in higher schools. Its essence lies in the fact that the student looks for intervals of inequality on a number line, which is drawn in a notebook (this is not a graph, but just an ordinary line with numbers). Where the intervals of inequalities intersect, the solution to the system is found. To use the interval method, you need to follow these steps:

1. All terms of each inequality are transferred to the left side with the sign changing to the opposite (zero is written on the right).
2. The inequalities are written out separately, and the solution to each of them is determined.
3. The intersections of inequalities on the number line are found. All numbers located at these intersections will be a solution.

Which method should I use?

Obviously the one that seems easiest and most convenient, but there are cases when tasks require a certain method. Most often they say that you need to solve either using a graph or the interval method. Algebraic way and substitution are used extremely rarely or not used at all, since they are quite complex and confusing, and besides, they are more used for solving systems of equations rather than inequalities, so you should resort to drawing graphs and intervals. They bring clarity, which cannot but contribute to the efficient and fast execution of mathematical operations.

If something doesn't work out

While studying a particular topic in algebra, naturally, problems may arise with its understanding. And this is normal, because our brain is designed in such a way that it is not able to understand complex material at one time. Often you need to reread a paragraph, take help from a teacher, or practice solving a problem. typical tasks. In our case, they look, for example, like this: “Solve the system of inequalities 3 x + 1 ≥ 0 and 2 x - 1 > 3.” Thus, personal desire, help from outsiders and practice help in understanding any complex topic.

Solver?

A solution book is also very suitable, not for copying homework, but for self-help. In them you can find systems of inequalities with a solution, look at them (as templates), try to understand exactly how the author of the solution coped with the task, and then try to do the same on your own.

Conclusions

Algebra is one of the most difficult subjects in school. Well, what can you do? Mathematics has always been like this: for some it is easy, but for others it is difficult. But in any case, it should be remembered that general education program It is built in such a way that any student can handle it. Moreover, one must keep in mind huge amount assistants Some of them have been mentioned above.

Program for solving linear, quadratic and fractional inequalities not only gives the answer to the problem, it leads detailed solution with explanations, i.e. displays the solution process to test knowledge in mathematics and/or algebra.

Moreover, if in the process of solving one of the inequalities it is necessary to solve, for example, quadratic equation, then its detailed solution is also displayed (it contains a spoiler).

This program may be useful for high school students in preparing for tests, to parents to monitor their children’s solutions to inequalities.

This program may be useful for high school students secondary schools in preparation for tests and exams, when testing knowledge before the Unified State Exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get it done as quickly as possible? homework in mathematics or algebra? In this case, you can also use our programs with detailed solutions.

This way you can conduct your own training and/or training of yours. younger brothers or sisters, while the level of education in the field of problems being solved increases.

Rules for entering inequalities

Any Latin letter can act as a variable.
For example: \(x, y, z, a, b, c, o, p, q\), etc.

Numbers can be entered as whole or fractional numbers.
Moreover, fractional numbers can be entered not only as a decimal, but also as an ordinary fraction.

Rules for entering decimal fractions.
In decimals fractional part can be separated from the whole by either a period or a comma.
For example, you can enter decimals like this: 2.5x - 3.5x^2

Rules for entering ordinary fractions.
Only a whole number can act as the numerator, denominator and integer part of a fraction.

The denominator cannot be negative.

When entering numerical fraction The numerator is separated from the denominator by a division sign: /
Whole part separated from the fraction by an ampersand: &
Input: 3&1/3 - 5&6/5y +1/7y^2
Result: \(3\frac(1)(3) - 5\frac(6)(5) y + \frac(1)(7)y^2 \)

You can use parentheses when entering expressions. In this case, when solving inequalities, the expressions are first simplified.
For example: 5(a+1)^2+2&3/5+a > 0.6(a-2)(a+3)

Select the right sign inequalities and enter the polynomials in the boxes below.

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A little theory.

Systems of inequalities with one unknown. Numeric intervals

You became acquainted with the concept of a system in 7th grade and learned to solve systems of linear equations with two unknowns. Next we will consider systems of linear inequalities with one unknown. Sets of solutions to systems of inequalities can be written using intervals (intervals, half-intervals, segments, rays). You will also become familiar with the notation of number intervals.

If in the inequalities \(4x > 2000\) and \(5x \leq 4000\) unknown number x are the same, then these inequalities are considered together and they are said to form a system of inequalities: $$ \left\(\begin(array)(l) 4x > 2000 \\ 5x \leq 4000 \end(array)\right .$$

Brace shows that it is necessary to find such values ​​of x for which both inequalities of the system turn into correct numerical inequalities. This system- an example of a system of linear inequalities with one unknown.

The solution to a system of inequalities with one unknown is the value of the unknown at which all the inequalities of the system become true numerical inequalities. Solving a system of inequalities means finding all solutions to this system or establishing that there are none.

The inequalities \(x \geq -2 \) and \(x \leq 3 \) can be written as a double inequality: \(-2 \leq x \leq 3 \).

The solutions to systems of inequalities with one unknown are different number sets. These sets have names. Yes, on number axis the set of numbers x such that \(-2 \leq x \leq 3 \) is represented by a segment with ends at points -2 and 3.

If \(a is a segment and is denoted by [a; b]

If \(a is an interval and is denoted by (a; b)

Sets of numbers \(x\) satisfying the inequalities \(a \leq x are half-intervals and are denoted respectively [a; b) and (a; b]

Segments, intervals, half-intervals and rays are called numerical intervals.

Thus, numerical intervals can be specified in the form of inequalities.

The solution to an inequality in two unknowns is a pair of numbers (x; y) that reverses this inequality into the correct numerical inequality. Solving an inequality means finding the set of all its solutions. Thus, the solutions to the inequality x > y will be, for example, pairs of numbers (5; 3), (-1; -1), since \(5 \geq 3 \) and \(-1 \geq -1\)

Solving systems of inequalities

You have already learned how to solve linear inequalities with one unknown. Do you know what a system of inequalities and a solution to the system are? Therefore, the process of solving systems of inequalities with one unknown will not cause you any difficulties.

And yet, let us remind you: to solve a system of inequalities, you need to solve each inequality separately, and then find the intersection of these solutions.

For example, the original system of inequalities was reduced to the form:
$$ \left\(\begin(array)(l) x \geq -2 \\ x \leq 3 \end(array)\right. $$

To solve this system of inequalities, mark the solution to each inequality on the number line and find their intersection:

The intersection is the segment [-2; 3] - this is the solution to the original system of inequalities.