The video lesson “Inequalities with two variables” is intended for teaching algebra on this topic in the 9th grade of a secondary school. The video lesson contains a description of the theoretical foundations of solving inequalities, describes in detail the process of solving inequalities in a graphical way, its features, and demonstrates examples of solving tasks on the topic. The purpose of this video lesson is to facilitate the understanding of the material using a visual presentation of information, to promote the formation of skills in solving problems using the studied mathematical methods.

The main tools of the video lesson are the use of animation in the presentation of graphs and theoretical information, highlighting concepts and features important for understanding and memorizing the material in color and other graphic ways, voice explanations for the purpose of easier memorization of information and the formation of the ability to use mathematical language.

The video lesson begins by introducing the topic and an example demonstrating the concept of solving an inequality. To form an understanding of the meaning of the concept of a solution, the inequality 3x 2 -y is presented<10, в которое подставляется пара значений х=2 и у=6. Демонстрируется, как после подстановки данных значений неравенство становится верным. Понятие решения данного неравенства как пары значений (2;6) выведено на экран, подчеркивая его важность. Затем представляется определение рассмотренного понятия для запоминания его учениками или записи в тетрадь.

An important part of the ability to solve inequalities is the ability to depict the set of its solutions on a coordinate plane. The formation of such a skill in this lesson begins with a demonstration of finding a set of solutions to linear inequalities ax+by c. The peculiarities of defining the inequality are noted - x and y are variables, a, b, c are some numbers, among which a and b are not equal to zero.

An example of such an inequality is x+3y>6. To transform the inequality into an equivalent inequality reflecting the dependence of the values ​​of y on the values ​​of x, both sides of the inequality are divided by 3, y remains on one side of the equation, and x is moved to the other. The value x=3 is arbitrarily selected for substitution into the inequality. It is noted that if you substitute this x value into the inequality and replace the inequality sign with an equal sign, you can find the corresponding value y=1. The pair (3;1) will be a solution to the equation y=-(1/3)x+2. If we substitute any values ​​of y greater than 1, then the inequality with a given value of x will be true: (3;2), (3;8), etc. Similar to this process of finding a solution, the general case for finding a set of solutions to a given inequality is considered. The search for a set of solutions to the inequality begins with the substitution of a certain value x 0. On the right side of the inequality we get the expression -(1/3)x 0 +2. A certain pair of numbers (x 0;y 0) is a solution to the equation y=-(1/3)x+2. Accordingly, the solutions to the inequality y>-(1/3)x 0 +2 will be the corresponding pairs of values ​​with x 0, where y is greater than the values ​​of y 0. That is, the solutions to this inequality will be pairs of values ​​(x 0 ; y).

To find the set of solutions to the inequality x+3y>6 on the coordinate plane, the construction of a straight line corresponding to the equation y=-(1/3)x+2 is demonstrated on it. On this line, point M is marked with coordinates (x 0; y 0). It is noted that all points K(x 0 ;y) with ordinates y>y 0, that is, located above this line, will satisfy the conditions of inequality y>-(1/3)x+2. From the analysis it is concluded that this inequality is given by a set of points that are located above the straight line y=-(1/3)x+2. This set of points constitutes a half-plane over a given line. Since the inequality is strict, the straight line itself is not among the solutions. In the figure, this fact is marked with a dotted designation.

Summarizing the data obtained as a result of describing the solution to the inequality x+3y>6, we can say that the straight line x+3y=6 divides the plane into two half-planes, while the half-plane located above reflects the set of values ​​satisfying the inequality x+3y>6, and located below the line - solution to the inequality x+3y<6. Данный вывод является важным для понимания, каким образом решаются неравенства, поэтому выведен на экран отдельно в рамке.

Next, we consider an example of solving a non-strict inequality of the second degree y>=(x-3) 2. To determine the set of solutions, a parabola y = (x-3) 2 is constructed nearby in the figure. The point M(x 0 ; y 0) is marked on the parabola, the values ​​of which will be solutions to the equation y = (x-3) 2. At this point, a perpendicular is constructed, on which a point K(x 0 ;y) is marked above the parabola, which will be the solution to the inequality y>(x-3) 2. We can conclude that the original inequality is satisfied by the coordinates of points located on a given parabola y=(x-3) 2 and above it. In the figure, this solution area is marked by shading.

The next example demonstrating the position on the plane of points that are a solution to an inequality of the second degree is a description of the solution to the inequality x 2 + y 2<=9. На координатной плоскости строится окружность радиусом 3 с центром в начале координат. Отмечается, что решениями уравнения будут точки, сумма квадратов координат которых не превышает квадрата радиуса. Также отмечается, что окружность х 2 +у 2 =9 разбивает плоскость на области внутри окружности и вне круга. Очевидно, что множество точек внутренней части круга удовлетворяют неравенству х 2 +у 2 <9, а внешняя часть - неравенству х 2 +у 2 >9. Accordingly, the solution to the original inequality will be the set of points on the circle and the region inside it.

Next, we consider the solution to the equation xy>8. On the coordinate plane next to the task, a hyperbola is constructed that satisfies the equation xy=8. Mark the point M(x 0;y 0) belonging to the hyperbola and K(x 0;y) above it parallel to the y-axis. It is obvious that the coordinates of point K correspond to the inequality xy>8, since the product of the coordinates of this point exceeds 8. It is indicated that in the same way one can prove the correspondence of points belonging to area B to the inequality xy<8. Следовательно, решением неравенства ху>8 there will be a set of points lying in areas A and C.

The video lesson “Inequalities with two variables” can serve as a visual aid for the teacher in the classroom. The material will also help students who are learning the material on their own. It is useful to use a video lesson during distance learning.

Inequality with two variablesx and y called an inequality of the form:

(or sign)

where is some expression with these variables.

By decision inequalities in two variables are called an ordered pair of numbers under which this inequality turns into a true numerical inequality.

Solve inequality- means finding the set of all its solutions. The solution to an inequality with two variables is a certain set of points on the coordinate plane.

The main method for solving these inequalities is graphic method. It consists in drawing boundary lines (if the inequality is strict, the line is drawn with a dotted line). We obtain the boundary equation if in a given inequality we replace the inequality sign with an equal sign. All lines together divide the coordinate plane into parts. The required set of points that corresponds to a given inequality or system of inequalities can be determined by taking a control point inside each region of the region.

The set of inequalities with two variables has the form

The solution to the population is the union of all solutions to the inequalities.

Example 1. Solve the system

Solution. Let's build in the system Ohoo corresponding lines (Fig. 19):

The equation defines a circle centered at ABOUT¢(0; 1) and R = 2.

The equation defines a parabola with vertex at ABOUT(0; 0).

Let us find solutions to each of the inequalities included in the system. The first inequality corresponds to the area inside the circle and the circle itself (we are convinced of the validity of this if we substitute the coordinates of any point from this area into the inequality). The second inequality corresponds to the area located under the parabola.

The solution to the system is the intersection of the two indicated areas (shown in Fig. 19 by superimposing two hatches).

Tasks

I level

1.1. Solve graphically:

3) ; 4) ;

5) ; 6) ;

7) ;

Level II

2.1. Solve graphically:

1) 2)

2.2. Find the number of integer solutions to the system:

1) 2) 3)

2.3. Find all integer solutions of the system:

1) 2)

3)

2.4. Solve the inequality. In your answer, indicate the number of solutions with two integer coordinates

If in a school course of mathematics and algebra we highlight the topic of “inequality” separately, then most of the time we will learn the basics of working with inequalities that contain a variable in their notation. In this article we will look at what inequalities with variables are, say what their solution is called, and also figure out how solutions to inequalities are written. For clarification, we will provide examples and necessary comments.

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What are inequalities with variables?

For example, if an inequality has no solutions, then they write “no solutions” or use the empty set sign ∅.

When the general solution to an inequality is one number, then it is written that way, for example, 0, −7.2 or 7/9, and sometimes also enclosed in curly brackets.

If the solution to an inequality is represented by several numbers and their number is small, then they are simply listed separated by commas (or separated by a semicolon), or written separated by commas in curly brackets. For example, if the general solution to an inequality with one variable is three numbers −5, 1.5 and 47, then write −5, 1.5, 47 or (−5, 1.5, 47).

And to write solutions to inequalities that have an infinite number of solutions, they use both the accepted designations for sets of natural, integer, rational, real numbers of the form N, Z, Q and R, designations for numerical intervals and sets of individual numbers, the simplest inequalities, and a description of a set through a characteristic property , and all unnamed methods. But in practice, the simplest inequalities and numerical intervals are most often used. For example, if the solution to the inequality is the number 1, the half-interval (3, 7] and the ray, ∪; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 pp.: ill. - ISBN 978-5-09-019243-9.

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Slide captions:

Inequalities with two variables and their systems Lesson 1

Inequalities with two variables Inequalities 3x – 4y  0; and are inequalities with two variables x and y. The solution to an inequality in two variables is a pair of values ​​of the variables that turns it into a true numerical inequality. For x = 5 and y = 3, the inequality 3x - 4y  0 turns into the correct numerical inequality 3  0. The pair of numbers (5;3) is a solution to this inequality. The pair of numbers (3;5) is not its solution.

Is the pair of numbers (-2; 3) a solution to the inequality: No. 482 (b, c) Is not Is

The solution to an inequality is an ordered pair of real numbers that turns the inequality into a true numerical inequality. Graphically, this corresponds to specifying a point on the coordinate plane. Solving an inequality means finding many solutions to it.

Inequalities with two variables have the form: The set of solutions to an inequality is the set of all points of the coordinate plane that satisfy a given inequality.

Solution sets for the inequality F(x,y) ≥ 0 x y F(x,y)≤0 x y

F(x, y)>0 F(x, y)

Trial point rule Construct F(x ; y)=0 Taking a trial point from any area, determine whether its coordinates are a solution to the inequality Draw a conclusion about the solution to the inequality x y 1 1 2 A(1;2) F(x ; y) =0

Linear inequalities with two variables A linear inequality with two variables is called an inequality of the form ax + bx +c  0 or ax + bx +c

Find the error! No. 484 (b) -4 2 x 2 -6 y 6 -2 0 4 -2 - 4

Solve graphically the inequality: -1 -1 0 x 1 -2 y -2 2 2 1 We draw graphs with solid lines:

Let's determine the inequality sign in each of the areas -1 -1 0 x 1 -2 y -2 2 2 1 3 4 - + 1 + 2 - 7 + 6 - 5 +

The solution to the inequality is a set of points from the areas containing the plus sign and solutions to the equation -1 -1 0 x 1 -2 y -2 2 2 1 3 4 - + 1 + 2 - 7 + 6 - 5 +

Let's solve together No. 485 (b) No. 486 (b, d) No. 1. Set the inequality and draw on the coordinate plane the set of points for which: a) the abscissa is greater than the ordinate; b) the sum of the abscissa and ordinate is greater than their double difference.

Let's solve together No. 2. Define by inequality an open half-plane located above the straight line AB passing through points A(1;4) and B(3;5). Answer: y  0.5x +3.5 No. 3. For what values ​​of b does the set of solutions to the inequality 3x – b y + 7  0 represent an open half-plane located above the straight line 3x – b y + 7 = 0. Answer: b  0.

Homework P. 21, No. 483; No. 484(c,d); No. 485(a); No. 486(c).

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Inequalities with two variables and their systems Lesson 2

Systems of inequalities with two variables

The solution to a system of inequalities with two variables is a pair of values ​​of variables that turns each of the inequalities of the system into a true numerical inequality. No. 1. Draw the set of solutions to systems of inequalities. No. 496 (oral)

a) x y 2 2 x y 2 2 b)

Let’s solve together No. 1. At what values ​​of k does the system of inequalities define a triangle on the coordinate plane? Answer: 0

We solve together x y 2 2 2 2 No. 2. The figure shows a triangle with vertices A(0;5), B(4;0), C(1;-2), D(-4;2). Define this quadrilateral with a system of inequalities. A B C D

Let's solve together No. 3. For what k and b is the set of points of the coordinate plane defined by the system of inequalities: a) strip; b) angle; c) empty set. Answer: a) k= 2,b  3; b) k ≠ 2, b – any number; c) k = 2; b

Let's solve number 4 together. What figure is given by the equation? (orally) 1) 2) 3) No. 5. Draw on the coordinate plane the set of solutions of points specified by the inequality.

Let's solve together No. 497 (c, d), 498 (c)

Homework P.22 No. 496, No. 497 (a, b), No. 498 (a, b), No. 504.

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Inequalities with two variables and their systems Lesson 3

Find the error! -4 2 x 2 -6 y 6 -2 0 4 -2 - 4

Find the error! | | | | | | | | | | | | | | | | | | 1 x y 2

Determine the inequality 0 - 6 - 1 5 3 1 2 y x - 3 - 2 1 -3 4

0 - 6 - 1 5 3 1 2 y x - 3 - 2 1 -3 4 Determine the inequality

0 - 3 - 1 5 3 1 2 y x - 3 - 2 1 Determine the inequality sign ≤

Solve graphically the system of inequalities -1 -1 0 x 1 -2 y -2 2 2 1

Inequalities and systems of inequalities of higher degrees with two variables No. 1. Draw on the coordinate plane the set of points specified by the system of inequalities

Inequalities and systems of inequalities of higher degrees with two variables No. 2. Draw on the coordinate plane the set of points specified by the system of inequalities

Inequalities and systems of inequalities of higher degrees with two variables No. 3. Draw on the coordinate plane the set of points specified by the system of inequalities. Let us transform the first inequality of the system:

Inequalities and systems of inequalities of higher degrees with two variables We obtain an equivalent system

Inequalities and systems of inequalities of higher degrees with two variables No. 4. Draw on the coordinate plane the set of points specified by the system of inequalities

Let's decide together No. 502 Collection of Galitsky. No. 9.66 b) y ≤ |3x -2| 0 - 6 - 1 5 3 1 2 y x - 3 - 2 1 -3 4

. No. 9.66(c) Solve together 0 - 6 - 1 5 3 1 2 y x - 3 - 2 1 -3 4 |y| ≥ 3x - 2

We solve together No. 9.66(g) 0 - 6 - 1 5 3 1 2 y x - 3 - 2 1 -3 4 |y|

Solve the inequality: x y -1 -1 0 1 -2 -2 2 2 1

0 - 6 - 1 5 3 1 2 y x - 3 - 2 1 -3 4 Write down the system of inequalities

11:11 3) What figure is determined by the set of solutions to the system of inequalities? Find the area of ​​each figure. 6) How many pairs of natural numbers are solutions to the system of inequalities? Calculate the sum of all such numbers. Solution of training exercises 2) Write down a system of inequalities with two variables, the set of solutions of which is shown in Figure 0 2 x y 2 1) Draw the set of solutions of the system on the coordinate plane: 4) Define the ring shown in the figure as a system of inequalities. 5) Solve the system of inequalities y x 0 5 10 5 10

Solution of training exercises 7) Calculate the area of ​​the figure given by the set of solutions to the system of inequalities and find the greatest distance between the points of this figure 8) At what value of m does the system of inequalities have only one solution? 9) Indicate some values ​​of k and b at which the system of inequalities defines on the coordinate plane: a) a strip; b) angle.

This is interesting. The English mathematician Thomas Harriot (Harriot T., 1560-1621) introduced the familiar inequality sign, arguing it as follows: “If two parallel segments serve as a symbol of equality, then intersecting segments must be a symbol of inequality.” In 1585, young Harriot was sent by the Queen of England on an exploring expedition to North America. There he saw a tattoo popular among Indians in the form. This is probably why Harriot proposed the inequality sign in two of its forms: “>” is greater than... and “

This is interesting. The symbols ≤ and ≥ for non-strict comparison were proposed by Wallis in 1670. Originally, the line was above the comparison sign, and not below it, as it is now. These symbols became widespread after the support of the French mathematician Pierre Bouguer (1734), from whom they acquired their modern form.

Lesson topic: Inequalities with two variables.

The purpose of the lesson: Teach students how to solve inequalities in two variables.

Lesson objectives:

1. Introduce the concept of inequality with two variables. Teach students how to solve inequalities. To develop skills in using the graphical method when solving inequalities, the ability to show the solution on a coordinate plane.

2.Develop students’ thinking, develop students’ practical skills.

3. To instill in students hard work, independence, a responsible attitude to business, initiative and independent decision-making.

Textbook/literature: Algebra 9, teaching materials.

During the classes:

1. The concept of inequality with two variables and its solution.

2. Linear inequality with two variables.

Let's consider the inequalities: 0.5x 2 -2y+l 20 - inequality with two variables.

Consider the inequality 0.5x 2 -2y+l

When x=1, y=2. We obtain the correct inequality 0.5 1 - 2 2 + 1

A pair of numbers (1; 2), in which the value x is in the first place, and the value y is in the second place, is called the solution to the inequality 0.5x 2 -2y+l

Definition. A solution to an inequality in two variables is a pair of values ​​of these variables that turns the inequality into a true numerical inequality.

If each solution to an inequality with two variables is represented by a point in the coordinate plane, then a graph of this inequality will be obtained. He is some kind of figure. This figure is said to be given or described by an inequality.

Let's consider linear inequalities with two variables.

Definition. A linear inequality with two variables is an inequality of the form ax + by c, where x and y are variables, a, b and c are some numbers.

In a linear inequality with two variables, if you replace the inequality sign with an equal sign, you get a linear equation. The graph of a linear equation ax + by = c, in which a or b is not equal to zero, is a straight line. It divides the set of points of the coordinate plane that do not belong to it into two regions representing open half-planes.

Using examples, we will consider how the set of solutions to an inequality with two variables is depicted on the coordinate plane.

Example 1. Let us depict on the coordinate plane the set of solutions to the inequality 2y+3x≤6.

We build a straight line 2y+3x=6, y=3-1.5x

A straight line divides the set of all points of the coordinate plane into points located below it and points located above it. Let's take a control point from each area: A(1;1), B(1;3).

The coordinates of point A satisfy this inequality 2y+3x≤6, 2·1+3·1≤6, 5≤6

The coordinates of point B do not satisfy this inequality: 2y+3x≤6, 2·3+3·1≤6.

This inequality is satisfied by the set of points in the region where point A is located. Let us shade this region. We have depicted the set of solutions to the inequality 2y+3x≤6.

To depict a set of solutions to inequalities on the coordinate plane, proceed as follows:

1. We build a graph of the function y = f(x), which divides the plane into two regions.

2. Select any of the resulting areas and consider an arbitrary point in it. We check the feasibility of the original inequality for this point. If the test results in a correct numerical inequality, then we conclude that the original inequality is satisfied in the entire region to which the selected point belongs. Thus, the set of solutions to the inequality is the region to which the selected point belongs. If the result of the check is an incorrect numerical inequality, then the set of solutions to the inequality will be the second region to which the selected point does not belong.

3. If the inequality is strict, then the boundaries of the region, that is, the points of the graph of the function y = f(x), are not included in the set of solutions and the boundary is depicted with a dotted line. If the inequality is not strict, then the boundaries of the region, that is, the points of the graph of the function y = f(x), are included in the set of solutions to this inequality and the boundary in this case is depicted as a solid line.

Conclusion: - solution of the inequality f(x,y)˃0, )