Mathematics teacher, secondary school No. 23, Astrakhan

Novakova S.A.

LESSON TOPIC: RATIONAL INEQUALITIES

Grade 9

The purpose of the lesson: to consolidate and deepen the knowledge of students in the process of solving various exercises on a given topic; to promote the development of mutual assistance and mutual assistance, the ability to conduct a cultural discussion.

Lesson objectives:

1. consolidate the ability to solve rational inequalities by the interval method; consider rational inequalities of various levels of complexity; to test the ability of students to solve rational inequalities;
2. create conditions for the development of skills and abilities to apply knowledge in new situations; for the development of the qualities of thinking: flexibility, purposefulness, rationality, criticality, taking into account individual characteristics.

Lesson type : general lesson; consolidation and improvement of knowledge and skills.

Forms of organizing activities in the lesson:

1. frontal
2. individual
3. collective

Lesson structure:

1. Organizing time;
2. motivational conversation;
3. updating knowledge;
4. individual or collective work with assignments;
5. summarizing.

Methods:

1. verbal;
2. visual;
3. practical.

Equipment:

1. computers;
2. multimedia projector;
3. personal cards.

Predicted result:consolidation of skills and abilities to solve rational inequalities; the formation of the ability to plan their work; achievement by each student of the level of skills that he needs:

I level - to solve the simplest rational inequalities; solve inequalities according to a given algorithm;

Level II - solve rational inequalities, independently choosing a solution method;

Level III - apply the acquired knowledge in a non-standard situation.

DURING THE CLASSES.

1. Organization. Setting goals.
2. Updating of basic knowledge. oral exercises.(Slide 2-4)

1) Are the following inequalities equivalent?

a) and (no)

b) and (yes)

2) Determine the method for solving the equation:

3) Determine the course of solving the inequality:

b) ﴾2х 2 +11х+6)﴾2х 2 +11х+13)

1. Repeat the algorithm for solving a rational inequality using the interval method:(Slide 5)

1. In each factor, the coefficient at the highest degree of the variable must be positive, for this it is necessary to take out the minus from all factors in which the coefficient at the highest degree is negative, and if there is still a minus sign in front of the expression, then the whole inequality must be multiplied by (-1).

Get the roots of the numeratorand discontinuity points of the denominator.

1. On the number line, we plot all the obtained values ​​​​and draw a curve of signs.

1. Problem solving.(Slide 6, 7)

1. Solve the inequality.

Answer:

2. Solve the inequality.
Answer:

3. Find the difference between the integer largest and smallest solutions of the inequality

Answer: 4.

4. Solve the inequality.
Answer:

5. Find the product of the largest negative integer and the smallest positive integer solution of the inequality

Answer: -42.

6. Find the smallest integer solution to the inequality.

7. How many prime numbers are solutions to the inequality?

Answer: 1.

1. Personal cards for verification work.

Card number 1.

1. Solve the inequality:

≤ .

a) [-4; -2) ∪ (0;5],

b) (–1, 0] ∪ ,

d) there are no solutions.

2. Find the largest integer x satisfying the inequality:

- > 1.

a) x ∈ (- ∞ ; -3.5),

B) -3,

at 4,

d) there are no solutions.

Card number 2.

1. Find the largest integer x satisfying the inequality:

- > -.

a)5,

b) -3,

at 4,

d) there are no solutions.

2. Solve the inequality:

a) (-9; -5) ∪ (0; 8),

B) (–8, -7) ∪ (1; 3),

B) (- ∞ ; -7) ∪ (1; 3),

D) there are no solutions.

Card number 3.

1. Solve the inequality:

a) (- ∞ ; -3) ∪ (0; 3,

B) (–3, 0) ∪ (0; ∞ ),

C) (5; 7),

D) there are no solutions.

2. Find integer solutions of inequalities:

a) 0, 1, 2,

B) 4, 5,

AT 7,

D) there are no solutions.

Card number 4.

1. Solve the inequality:

a) (- ∞ ; -3/25) ∪ (0; ∞ ),

b) (–12, 0) ∪ (7;9),

B) (- ∞ ;) ∪ (; 5),

D) there are no solutions.

2. Find the sum of integer solutions of the inequality

a) 2,

b) 4,

c) 0,

d) 1,

e) 3.

1. Summarizing.

During the lesson, students consolidated the ability to solve rational inequalities, considered the solution of rational inequalities of various levels of complexity. Students in practice showed the ability to apply the method of intervals in solving rational inequalities. Particular attention should be paid to solving non-strict rational inequalities.

1. Homework.(Slide 8)

1. Find the smallest integer negative solution to the inequality

2. Solve the inequality.
3. Find the sum of the largest and smallest integer solutions of the inequality

.

1. Bibliography:

1. Algebra: Proc. For 9 cells. general education institutions. / S.M. Nikolsky, M.K. Potapov, N.N. Reshetnikov, A.V. Shevkin. - 2nd ed. – M.: Enlightenment, 2003. – 255 p.
2. Algebra 8th grade. Tasks for the training and development of students. / Belenkova E.Yu., Lebedintseva E.A. - M.: Intellect - center, 2003. - 176 p.
3. "Small USE" in mathematics: Grade 9: Preparing students for final certification / M.N. Kochagin, V.V. Kochagin. – M.: Eksmo, 2008. – 192 p.

Synopsis of an algebra lesson in grade 9 on the topic "Solution of rational inequalities" (TMK S.M. Nikolsky).

Compiled by Karachun V.V., teacher of mathematics and computer science, MBOU Kutulik secondary school

Lesson type : "Discovery" of new knowledge.

Goals:

subject : introduce the concept of rational inequality with one variable; create conditions for the formation of ideas about the algorithm for solving rational inequalities; teach how to apply the interval method to solving rational inequalities; promote the development of mathematical speech; to cultivate a culture of behavior in frontal work, work in groups, individual work.

Communicative : to be able to negotiate and come to a common decision in joint activities, including in a situation of conflict of interest, to participate in a collective discussion of problems.

Regulatory: distinguish between the method and result of an action, evaluate the correctness of the action, the ability to learn and the ability to organize one's activities; create conditions for the development of the ability to analyze, generalize the studied facts, reflection of the methods and conditions of action.

cognitive : to search for the necessary information to complete educational tasks using educational literature; master the general technique for solving rational inequalities,

Personal : formation of cognitive interest.

Means that provide the educational process in the classroom: computer, projector, presentation, task cards for groups.

Lesson plan:

1. Organizational moment: greeting, readiness check.

3. Goal setting.

4. "Discovery" of new knowledge.

Fizminutka (conducted by a student of the class).

5. Fixing a new algorithm of action (work in groups).

6. Independent work.

7. The results of the lesson. (Reflection of activity).

8. Homework.

During the classes.

Teacher activity	Student activities			UUD
1. Organizational moment. Purpose of the stage: engaging students in activities.
Hello guys! Sit down. An ancient Chinese proverb says: "I hear - I forget, I see - I remember, I do - I understand." And today I urge you to follow this wisdom. "I hear - I see - I do"slide 1.	Teachers greet, prepare for the lesson.			Mobilization of attention, respect for others(L)
2. Actualization of students' knowledge. Creation of a problem situation. Purpose of the stage: To form an interest in the process of educational activity by creating a situation of "intellectual conflict"
Solve inequalities: 1.(x-1)(x-2)(x-3)>0 2.(x-1)³(x-2)²(x-4)˂0 4. ˂0	Students solve inequalities #1 and #2. Difficulties arise with solving 3 and 4 inequalities.				Self-determination, learning motivation(L) They are able to complete the training task; fix individual difficulty in a trial educational action(R) Accept and solve educational and cognitive tasks(P) Express their thoughts clearly(TO)
3. Goal setting. Purpose of the stage: Formulation of the topic of the lesson; setting a learning task.
What do you think inequalities #3 and #4 are called? Formulate the topic of the lesson.Slide 2. What will we do in class?	These inequalities are called rational. Solution of rational inequalities. Learn to solve rational inequalities.						Determine and formulate the purpose of the activity(R) Summarize knowledge and draw conclusions(P) Planning for Learning Collaboration(TO)
4. "Discovery" of new knowledge. Purpose of the stage: ensuring the perception, comprehension and primary consolidation by students of a new topic.
Slide 3: Definition of a rational inequality with one unknown. Slide 4: Examples of rational inequalities. Slide 5: What does it mean to solve an inequality? Slide 6: Justification of the equivalence of inequalities > 0 and A(x)B(x)>0 Guys, I suggest you complete the project “Solving Rational Inequalities. Handbook for students of the 9th grade. The class is divided into 5 groups of 4 people. Each group was given a card with tasks: Solve a typical example No. 1-No. 5 pp. 46-48 (one for each group; Appendix 1) Determine the type of this inequality. Write an algorithm for solving the inequality. Choose and solve a "similar" inequality for homework. Choose a "similar" inequality for independent work in two versions.	Give "their" examples of rational inequalities. The guys work with the text of the textbook (item 3.2) and didactic materials on algebra for grade 9 (M.K. Potapov, A.V. Shevkin). Responsibilities in groups are distributed: solution of typical rational inequality by all students of the group; explanation of the solution of the inequality at the blackboard; creation of an algorithm for solving inequality; selection of inequality for homework; formulating assignments for independent work.					self-determination(L) Analysis of objects in order to highlight features; summing up the concept; goal setting(P) Performing a trial educational action; fixing an individual difficulty; self-regulation in difficult situations(R) Expressing your thoughts; argumentation of one's opinion; taking into account different opinions(TO)
Fixing a new algorithm of action. Purpose of the stage : Creation of a new educational product: an algorithm for solving rational inequalities.
Project protection. Emphasizes students' attention to the competent design of solutions to rational inequalities. Answers questions that arise.	All students of the group work in accordance with the distribution of responsibilities: 1st student broadcasts the solution on the screen and explains his solution; 2nd student writes down an algorithm for solving the inequality; 3rd student writes down homework; The 4th student writes down tasks for independent work on the back of the board. The rest of the students write down the solutions of the proposed inequalities in a notebook, ask questions.							Kindness, diligence, diligence(L) Work according to the algorithm, mastering the methods of control and self-control of mastering the studied(R) Applying new knowledge in practice(P) Implementation of mutual control and mutual assistance(TO)
Conclusion of the work of groups. Slide 7. Algorithm for solving rational inequalities. ( A(x)B(x)>0>0 >0
Independent work. Purpose of the stage : check the quality of assimilation of the studied material.
On the reverse side of the board is written independent work in two versions. I option II option 2.

In this lesson, we will recall all the material covered on the topic and will solve examples with different types of inequalities. Let us first repeat the method of intervals and the operations of intersection and union of sets. Next, we will solve examples using standard solution techniques.

Topic: Rational inequalities and their systems

Lesson: Overview lesson on the topic: "Rational inequalities and their systems"

We dosed increased the complexity of systems of inequalities: first we solved linear systems, then we added quadratic inequalities, rational inequalities, themselves constituted systems, and thus we developed a methodology for solving systems of inequalities.

It includes important elements:

1.Spacing Method as a method for solving individual inequalities.

2. The operation of intersection and union of numerical sets.

Let's take a look at these elements. Recall the interval method in an example:

Consider the function

Find the roots of a square trinomial

Find the roots using Vieta's theorem

Let's single out intervals of sign constancy.

When passing through m.-1, the function does not change sign, because parenthesis to an even degree.

We made a mistake by not providing an isolated solution.

Answer:

Let's draw a sketch of the graph of the function.

The interval method is the most important element in solving rational inequalities and systems.

The meaning of the operations of intersection and union of sets, including numerical ones, helps to understand the following picture:

Intersection of many.

We have a set A of some elements and a set B. Some of these elements simultaneously fall into both set A and set B, and it is called the intersection of A and B (Fig. 3).

For example:

2.

Their intersection gives the following set:

Union of sets.

There are elements that are only in set A, there are elements that are only in set B. There are those that are included both there and there - these elements form the intersection of sets.

And all the elements from A and the missing elements from B form a union of sets (Fig. 5).

For example:

(Rice. 6).

The solution to the inequality is the union of two sets:

One more example.

Find the intersection and union of sets.

Intersection of many:

Union of sets:

The solution is any number

5.

Solve a system of simple inequalities.

Answer:

We repeated the method of intervals, the operations of union and intersection of sets. Now consider the inverse problem, which will allow us to better understand the meaning of solving inequalities.

Given a solution to an inequality, you need to come up with at least one inequality for which it is true.

6. Find an inequality whose solution is the given union of sets.

It can be a solution to a quadratic inequality. The graph of the corresponding quadratic function is a parabola passing through points 2 and 4.

Consider tasks with a module.

Consider the first inequality. What ? This is the distance from the point with coordinates x to point 3. A means that the distance between these points is no more than 2. Let's graph it:

Let's solve the second inequality.

Consider the function

The graph is a parabola, the branches are directed upwards.

Let's get back to the system.

Answer:

related tasks.

Find the smallest solution. Answer: There is no smallest solution to this system.

Find the greatest solution. Answer:

We have reviewed the solution of systems of rational inequalities. We have considered the main elements that ensure the success of the technique of solving inequalities. What does it take to solve the inequality? interval method. What is needed to obtain a solution of typical systems? You need to imagine the operations of intersection and union.

We will need inequalities in what follows.

1. Mordkovich A.G. and others. Algebra 9th grade: Proc. For general education Institutions. - 4th ed. - M.: Mnemosyne, 2002.-192 p.: ill.

2. Mordkovich A.G. et al. Algebra Grade 9: Taskbook for students of educational institutions / A. G. Mordkovich, T. N. Mishustina et al. - 4th ed. — M.: Mnemosyne, 2002.-143 p.: ill.

3. Yu. N. Makarychev, Algebra. Grade 9: textbook. for general education students. institutions / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, I. E. Feoktistov. - 7th ed., Rev. and additional - M .: Mnemosyne, 2008.

4. Alimov Sh.A., Kolyagin Yu.M., Sidorov Yu.V. Algebra. Grade 9 16th ed. - M., 2011. - 287 p.

5. Mordkovich A. G. Algebra. Grade 9 At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich, P. V. Semenov. - 12th ed., erased. — M.: 2010. — 224 p.: ill.

6. Algebra. Grade 9 At 2 hours. Part 2. Task book for students of educational institutions / A. G. Mordkovich, L. A. Aleksandrova, T. N. Mishustina and others; Ed. A. G. Mordkovich. - 12th ed., Rev. — M.: 2010.-223 p.: ill.

1. Portal of Natural Sciences ().

2. Portal of Natural Sciences ().

3. Portal of Natural Sciences ().

4. Portal of Natural Sciences ().

5. Electronic educational and methodological complex for preparing grades 10-11 for entrance exams in computer science, mathematics, Russian language ().

7. Center for Education "Technology of Education" ().

8. Education Center "Teaching Technology" ().

9. Education Center "Technology of Education" ().

10. College.ru section on mathematics ().

1. Mordkovich A.G. et al. Algebra Grade 9: Taskbook for students of educational institutions / A. G. Mordkovich, T. N. Mishustina et al. - 4th ed. - M .: Mnemosyne, 2002.-143 p.: ill. No. 82 - 84; Home test number 1.

The material of this lesson is intended to repeat the solution of linear inequalities; formation of the concept of "system of rational inequalities", "solution of rational inequalities"; formation of skills to solve systems of linear inequalities of any complexity.

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Abstract of a math lesson in grade 9

on the topic: "Systems of rational inequalities"

Lesson Objectives:

• repeat the solution of linear inequalities;
• to derive the concepts of "system of rational inequalities", "solution of rational inequalities";
• explain the solution of the simplest systems of linear inequalities;
• to form the ability to solve systems of linear inequalities of any complexity.

During the classes:

1. Organizational moment

2. Work on cards

Card number 1.

Solve the inequality:

a) 5x+4

Card number 2.

Solve the inequality:

a) 8x+9≤ -4x+3 b) x²-2x-24≥0

Card number 3.

1. The set (-10.3; -7; 0; 2.6; 3) is given. Make up its subset consisting of non-negative numbers.
2. Set A consists of divisors of 12, and set B consists of divisors of 18. Find the intersection and union of these sets.

Card number 4.

1. The set (-1.3; 0; 2; 3.8; 6; 11) is given. Make up its subset consisting of natural numbers.

2. Set A consists of divisors of the number 30, and set B consists of divisors of the number 45. Find the intersection and union of these sets.

(Cards are offered to 4 students, and at this time the class performs a mathematical dictation)

Mathematical dictation. (Slide 2)

Inequality	Picture	Gap
x≤9
		(7;9]

For verification, the following table is provided (slide 3):

Inequality	Picture	Gap
x>7		(7;+∞)
x≤9		(-∞; 9]
		(7;9]

3. Preparation for the introduction of new material. Definition of the topic and objectives of the lesson.

The teacher asks questions and the students answer them.

1. What is a system of equations?
2. What is the solution to a system of equations?
3. What does it mean to solve a system of equations?

Solve the system of equations (slide 4): x-y = 5

X+y=7 (6;1)

4) What is rational inequality?

5) What does it mean to solve an inequality?

Let us consider two examples, the solution of which, as we will see, will lead us to a new mathematical model. In these examples, we need to find the scope of expressions. (students decide on their own and check by key) (slide 5)

Example 1. √2x-4

Example 2. √8-x

Now consider the expression √2x-4 + √8-x. (slide 6)

How to find its domain of definition?

Yes, it exists when the first and second roots exist at the same time. What does this remind you of? (children's answers)

So we came to a new mathematical model - a system of inequalities.

What is the topic of today's lesson? (student answers)

Yes. The theme of our lesson: "Systems of rational inequalities." (slide 7)

What questions do you think may arise when studying this topic?

From your answers we have the objectives of the lesson. (slide 8)

What will help us achieve our goals?

4. Learning new material.

Let's return to our expression: √2x-4 + √8-x (slide 9). We said that the domain of definition of a given expression exists when the first and second roots exist at the same time. In this case, we say that we need to solve the system of inequalities

2x - 4 ≥ 0

8 – x ≥ 0.

What is a system of inequalities?

Let's read the definition in the textbook (p. 41) and compare it with the one you voiced.

We solved each inequality separately. And now, to find the general solution, we proceed as follows: on the number line Oh first we mark the solution of the first inequality x ≥ 2, and then on the same line we mark the solution of the second inequality - x ≤ 8. They intersect in the segment . (The record is played on the board) Therefore, the solution to this system will be the segment.

So what is the solution to the system of inequalities? What does it mean to solve a system of inequalities? (student answers)

Let's look at the simplest, but very important basic knowledge. Let's solve systems of inequalities:

X > 7 Answer: x > 10

X > 10

X > 7 Answer: (7; 10]

X ≤ 10

X ≤ 7 Answer: x ≤ 7

X ≤ 10

X ≥ 1 Answer: )