Municipal budgetary educational institution
"Secondary school No. 26
with in-depth study of individual subjects"
city of Nizhnekamsk of the Republic of Tatarstan
Math lesson notes
in 8th grade
Solving inequalities with one variable
and their systems
prepared
math teacher
first qualification category
Kungurova Gulnaz Rafaelovna
Nizhnekamsk 2014
Plan summary lesson
Teacher: Kungurova G.R.
Subject: mathematics
Topic: “Solving linear inequalities with one variable and their systems.”
Class: 8B
Date: 04/10/2014
Lesson type: lesson of generalization and systematization of the studied material.
Objective of the lesson: consolidation of practical skills in solving inequalities with one variable and their systems, inequalities containing a variable under the modulus sign.
Lesson objectives:
Educational:
generalization and systematization of students’ knowledge about ways to solve inequalities with one variable;
expansion of the type of inequalities: double inequalities, inequalities containing a variable under the modulus sign, systems of inequalities;
establishing interdisciplinary connections between mathematics, Russian language, and chemistry.
Educational:
activation of attention, mental activity, development of mathematical speech, cognitive interest in students;
mastering the methods and criteria of self-assessment and self-control.
Educational:
fostering independence, accuracy, and the ability to work in a team
Basic methods used in the lesson: communicative, explanatory-illustrative, reproductive, method of programmed control.
Equipment:
computer
computer presentation
monoblocks (performing an individual online test)
handouts (multi-level individual assignments);
self-control sheets;
Lesson plan:
4. Independent work
5. Reflection
6. Lesson summary.
Lesson progress:
1. Organizational moment.
(The teacher tells the students the goals and objectives of the lesson.).
Today we face a very important task. We must summarize this topic. Once again, it will be necessary to work very carefully on theoretical issues, do calculations, and consider the practical application of this topic in our everyday life. And we must never forget about how we reason, analyze, and build logical chains. Our speech should always be literate and correct.
Each of you has a self-control sheet on your desk. Throughout the lesson, remember to mark your contributions to this lesson with a “+” sign.
The teacher asks homework by commenting on it:
№1026(a,b), No.1019(c,d); additionally - No. 1046(a)
2. Updating knowledge, skills and abilities
1) Before we start practical tasks, let's turn to theory.
The teacher announces the beginning of the definition, and the students must complete the formulation.
a) An inequality in one variable is an inequality of the form ax>b, ax<в;
b) Solving an inequality means finding all its solutions or proving that there are no solutions;
c) The solution to an inequality with one variable is the value of the variable that turns it into a true inequality;
d) Inequalities are said to be equivalent if their sets of solutions coincide. If they have no solutions, then they are also called equivalent
2) On the board there are inequalities with one variable, arranged in one column. And next to it, in another column, their solutions are written in the form of numerical intervals. The students' task is to establish correspondence between inequalities and corresponding intervals.
Establish a correspondence between inequalities and numerical intervals:
1. 3x > 6 a) (-∞ ; - 0.2]
2. -5x ≥ 1 b) (- ∞ ; 15)
3. 4x > 3 c) (2; + ∞)
4. 0.2x< 3 г) (0,75; + ∞)
3) Practical work in a self-test notebook.
Students write a linear inequality in one variable on the board. Having completed this, one of the students voices his decision and the mistakes made are corrected)
Solve the inequality:
4 (2x - 1) - 3(x + 6) > x;
8x - 4 - 3x - 18 > x;
8x - 3x – x > 4+18;
4x > 22 ;
x > 5.5.
Answer. (5.5 ; +∞)
3. Practical Application inequalities in everyday life ( chemical experiment)
Inequalities in our daily lives can be good helpers. And besides, of course, there is an inextricable connection between school subjects. Mathematics goes hand in hand not only with the Russian language, but also with chemistry.
(On each desk there is a reference scale for the pH value, ranging from 0 to 12)
If indicator 0 ≤ pH< 7, то среда кислая;
if pH = 7, then the environment is neutral;
if the indicator is 7< pH ≤ 12, то среда щелочная
The teacher pours 3 colorless solutions into different test tubes. From the chemistry course, students are asked to remember the types of solution media (acidic, neutral, alkaline). Next, experimentally, involving students, the environment of each of the three solutions is determined. To do this, a universal indicator is lowered into each solution. What happens is that each indicator is colored accordingly. And according to the color scheme, thanks to the standard scale, students establish the environment of each of the proposed solutions.
Conclusion:
1 indicator turns red, indicator 0 ≤ pH< 7, значит среда первого раствора кислая, т.е. имеем кислоту в 1пробирке
2 indicator turns green, pH = 7, which means the medium of the second solution is neutral, i.e. we had water in test tube 2
3 indicator turns blue, indicator 7< pH ≤ 12 , значит среда третьего раствора щелочная, значит в 3 пробирке была щелочь
Knowing the pH limits, you can determine the acidity level of soil, soap, and many cosmetics.
Continued updating of knowledge, skills and abilities.
1) Again, the teacher begins to formulate definitions, and students must complete them
Continue definitions:
a) Solving a system of linear inequalities means finding all its solutions or proving that there are none
b) The solution to a system of inequalities with one variable is the value of the variable for which each of the inequalities is true
c) To solve a system of inequalities with one variable, you need to find a solution to each inequality, and find the intersection of these intervals
The teacher again reminds students that the ability to solve linear inequalities with one variable and their systems is the basis, the basis for more complex inequalities, which will be studied in higher grades. A foundation of knowledge is laid, the strength of which will have to be confirmed at the OGE in mathematics after 9th grade.
Students solve systems of linear inequalities with one variable in their notebooks. (2 students complete these tasks on the board, explain their solution, voice the properties of the inequalities used in solving the systems).
№1012(d). Solve system of linear inequalities
0.3 x+1< 0,4х-2;
1.5 x-3 > 1.3 x-1. Answer. (30; +∞).
№1028(d). Solve the double inequality and list all the integers that are its solution
1 < (4-2х)/3 < 2 . Ответ. Целое число: 0
2) Solving inequalities containing a variable under the modulus sign.
Practice shows that inequalities containing a variable under the modulus sign cause anxiety and self-doubt in students. And often students simply do not take on such inequalities. And the reason for this is a poorly laid foundation. The teacher encourages students to work on themselves in a timely manner and to consistently learn all the steps to successfully fulfill these inequalities.
Oral work is carried out. (Front survey)
Solving inequalities containing a variable under the modulus sign:
1. The modulus of a number x is the distance from the origin to the point with coordinate x.
| 35 | = 35,
| - 17 | = 17,
| 0 | = 0
2. Solve inequalities:
a) | x |< 3 . Ответ. (-3 ; 3)
b) | x | > 2. Answer. (- ∞; -2) U (2; +∞)
The progress of solving these inequalities is displayed in detail on the screen and the algorithm for solving inequalities containing a variable under the modulus sign is spelled out.
4. Independent work
In order to control the degree of mastery of this topic, 4 students take seats at the monoblocks and take thematic online testing. Testing time is 15 minutes. After completion, a self-test is carried out both in points and as a percentage.
The rest of the students at their desks do independent work in variants.
Independent work (completion time 13min)
Option 1
Option 2
1. Solve the inequalities:
a) 6+x< 3 - 2х;
b) 0.8(x-3) - 3.2 ≤ 0.3(2 - x).
3(x+1) - (x-2)< х,
2 > 5x - (2x-1) .
-6 < 5х - 1 < 5
4*. (Additionally)
Solve the inequality:
| 2- 2x | ≤ 1
1. Solve the inequalities:
a) 4+x< 1 - 2х;
b) 0.2(3x - 4) - 1.6 ≥ 0.3(4-3x).
2. Solve the system of inequalities:
2(x+3) - (x - 8)< 4,
6x > 3(x+1) -1.
3. Solve double inequality:
-1 < 3х - 1 < 2
4*. (Additionally)
Solve the inequality:
| 6x-1 | ≤ 1
After execution independent work Students hand in their notebooks for checking. Students who worked on monoblocks also hand over their notebooks to the teacher for checking.
5. Reflection
The teacher reminds students of the self-control sheets, on which they had to evaluate their work with a “+” throughout the lesson, at its various stages.
But students will have to give the main assessment of their activities only now, after voicing one ancient parable.
Parable.
A sage was walking, and 3 people met him. They carried carts with stones under the hot sun for the construction of the temple.
The sage stopped them and asked:
- What did you do all day?
“I carried the damned stones,” answered the first.
“I did my job conscientiously,” answered the second.
“And I took part in the construction of the temple,” the third answered proudly.
In the self-control sheets, in point No. 3, students must enter a phrase that would correspond to their actions in this lesson.
Self-control sheet __________________________________________
№ n /n
Lesson steps
Grade educational activities
Oral work in class
Solving inequalities with one variable;
solving systems of inequalities;
solving double inequalities;
solving inequalities with modulus sign
Reflection
In paragraphs 1 and 2, mark the correct answers in the lesson with a “+” sign;
in paragraph 3, evaluate your work in class according to the instructions
6. Lesson summary.
The teacher, summing up the lesson, notes successful moments and problems on which additional work remains to be done.
Students are asked to evaluate their work according to self-control sheets, and students receive one more mark based on the results of independent work.
At the end of the lesson, the teacher draws the students’ attention to the words of the French scientist Blaise Pascal: “The greatness of a person lies in his ability to think.”
References:
1 . Algebra. 8th grade. Yu.N.Makarychev, N.G. Mindyuk, K.E. Neshkov, I.E. Feoktistov.-M.:
Mnemosyne, 2012
2. Algebra.8th grade. Didactic materials. Methodical recommendations/ I.E. Feoktistov.
2nd edition., St.-M.: Mnemosyne, 2011
3. Testing and measuring materials. Algebra: 8th grade / Compiled by L.I. Martyshova.-
M.: VAKO, 2010
Internet resources:
Lesson topic: Solving a system of linear inequalities with one variable
Date: _______________
Class: 6a, 6b, 6c
Lesson type: learning new material and primary consolidation.
Didactic goal: create conditions for awareness and comprehension of a block of new educational information.
Goals: 1) Educational: introduce the concepts: solution of systems of inequalities, equivalent systems of inequalities and their properties; teach how to apply these concepts when solving simple systems of inequalities with one variable.
2) Developmental: promote the development of elements of creative, independent activity of students; develop speech, the ability to think, analyze, generalize, express your thoughts clearly and concisely.
3) Educational: fostering a respectful attitude towards each other and a responsible attitude towards educational work.
Tasks:
repeat the theory on the topic of numerical inequalities and numerical intervals;
give an example of a problem that can be solved by a system of inequalities;
consider examples of solving systems of inequalities;
do independent work.
Forms of organizing educational activities:- frontal – collective – individual.
Methods: explanatory - illustrative.
Lesson plan:
1. Organizational moment, motivation, goal setting
2. Updating the study of the topic
3. Learning new material
4. Primary consolidation and application of new material
5. Doing independent work
7. Summing up the lesson. Reflection.
Lesson progress:
1. Organizational moment
Inequality can be a good helper. You just need to know when to turn to him for help. The formulation of problems in many applications of mathematics is often formulated in the language of inequalities. For example, many economic problems come down to the study of systems of linear inequalities. Therefore, it is important to be able to solve systems of inequalities. What does it mean to “solve a system of inequalities”? This is what we will look at in today's lesson.
2. Updating knowledge.
Oral work with class, three students work using individual cards.
To review the theory of the topic “Inequalities and their properties,” we will conduct testing, followed by verification and a conversation on the theory of this topic. Each test task requires the answer “Yes” - figure, “No” - figure ____
The result of the test should be some kind of figure.
(answer: ).
Establish a correspondence between inequality and numerical interval
1. (– ; – 0,3)
2. (3; 18)
3. [ 12; + )
4. (– 4; 0]
5. [ 4; 12]
6. [ 2,5; 10)
“Mathematics teaches you to overcome difficulties and correct your own mistakes.” Find the error in solving the inequality, explain why the error was made, write down the correct solution in your notebook.
2x<8-6
x>-1
3. Studying new material.
What do you think is called a solution to a system of inequalities?
(The solution to a system of inequalities with one variable is the value of the variable for which each of the inequalities of the system is true)
What does it mean to “Solve a system of inequalities”?
(Solving a system of inequalities means finding all its solutions or proving that there are no solutions)
What needs to be done to answer the question “is a given number
solution to a system of inequalities?
(Substitute this number into both inequalities of the system, if the inequalities are true, then the given number is a solution to the system of inequalities, if the inequalities are incorrect, then the given number is not a solution to the system of inequalities)
Formulate an algorithm for solving systems of inequalities
1. Solve each inequality of the system.
2. Graphically depict the solutions to each inequality on the coordinate line.
3. Find the intersection of solutions to inequalities on the coordinate line.
4. Write the answer as a number interval.
Consider examples:
Answer: 
Answer: no solutions
4. Securing the topic.
Working with textbook No. 1016, No. 1018, No. 1022
5. Independent work according to options (Task cards for students on the tables)
Independent work
Option 1
Solve the system of inequalities:
1. The concept of inequality with one variable
2. Equivalent inequalities. Theorems on the equivalence of inequalities
3. Solving inequalities with one variable
4. Graphical solution of inequalities with one variable
5. Inequalities containing a variable under the modulus sign
6. Main conclusions
Inequalities with one variable
Offers 2 X + 7 > 10's, x 2 +7x< 2,(х + 2)(2х-3)> 0 are called inequalities with one variable.
IN general view this concept is defined as follows:
Definition. Let f(x) and g(x) be two expressions with variable x and domain X. Then an inequality of the form f(x) > g(x) or f(x)< g(х) называется неравенством с одной переменной. Множество X называется областью его определения.
Variable value x from many X, in which the inequality turns into a true numerical inequality is called decision. Solving an inequality means finding many solutions to it.
Thus, by solving inequality 2 x + 7 > 10 -x, x? R is the number x= 5, since 2 5 + 7 > 10 - 5 is a true numerical inequality. And the set of its solutions is the interval (1, ∞), which is found by performing the transformation of the inequality: 2 x + 7 > 10-x => 3x >3 => x >1.
Equivalent inequalities. Theorems on the equivalence of inequalities
The basis for solving inequalities with one variable is the concept of equivalence.
Definition. Two inequalities are said to be equivalent if their solution sets are equal.
For example, inequalities 2 x+ 7 > 10 and 2 x> 3 are equivalent, since their solution sets are equal and represent the interval (2/3, ∞).
Theorems on the equivalence of inequalities and the consequences from them are similar to the corresponding theorems on the equivalence of equations. Their proof uses the properties of true numerical inequalities.
Theorem 3. Let inequality f(x) > g(x) defined on the set X And h(x) is an expression defined on the same set. Then the inequalities f(x) > g(x) and f(x)+ h(x) > g(x) + h(x) are equivalent on the set X.
Corollaries follow from this theorem, which are often used when solving inequalities:
1) If to both sides of the inequality f(x) > g(x) add the same number d, then we get the inequality f(x) + d > g(x)+ d, equivalent to the original one.
2) If any term (numerical expression or expression with a variable) is transferred from one part of the inequality to another, changing the sign of the term to the opposite, then we obtain an inequality equivalent to the given one.
Theorem 4. Let inequality f(x) > g(x) defined on the set X And h(X X from many X expression h(x) takes positive values. Then the inequalities f(x) > g(x) and f(x) h(x) > g(x) h(x) are equivalent on the set X.
f(x) > g(x) multiply by the same positive number d, then we get the inequality f(x) d > g(x) d, equivalent to this.
Theorem 5. Let inequality f(x) > g(x) defined on the set X And h(X) - an expression defined on the same set, and for all X there are many of them X expression h(X) accepts negative values. Then the inequalities f(x) > g(x) and f(x) h(x) > g(x) h(x) are equivalent on the set X.
A corollary follows from this theorem: if both sides of the inequality f(x) > g(x) multiply by the same negative number d and change the inequality sign to the opposite one, we get the inequality f(x) d > g(x) d, equivalent to this.
Solving inequalities with one variable
Let's solve inequality 5 X - 5 < 2х - 16, X? R, and we will justify all the transformations that we will perform in the solution process.
Solving the inequality X < 7 является промежуток (-∞, 7) и, следовательно, множеством решений неравенства 5X - 5 < 2x + 16 is the interval (-∞, 7).
Exercises
1. Determine which of the following entries are inequalities with one variable:
a) -12 - 7 X< 3x+ 8; d) 12 x + 3(X- 2);
b) 15( x+ 2)>4; e) 17-12·8;
c) 17-(13 + 8)< 14-9; е) 2x 2+ 3x-4> 0.
2. Is the number 3 a solution to the inequality 6(2x + 7) < 15(X + 2), X? R? What about the number 4.25?
3. Are the following pairs of inequalities equivalent on the set of real numbers:
a) -17 X< -51 и X > 3;
b) (3 x-1)/4 >0 and 3 X-1>0;
c) 6-5 x>-4 and X<2?
4. Which of the following statements are true:
a) -7 X < -28 => x>4;
b) x < 6 => x < 5;
V) X< 6 => X< 20?
5. Solve inequality 3( x - 2) - 4(X + 1) < 2(х - 3) - 2 and justify all the transformations that you will perform.
6. Prove that by solving the inequality 2(x+ 1) + 5 > 3 - (1 - 2X) is any real number.
7. Prove that there is no real number that would be a solution to the inequality 3(2 - X) - 2 > 5 - 3X.
8. One side of the triangle is 5 cm, and the other is 8 cm. What can be the length of the third side if the perimeter of the triangle is:
a) less than 22 cm;
b) more than 17 cm?
GRAPHICAL SOLUTION OF INEQUALITIES WITH ONE VARIABLE. To solve the inequality graphically f (x) > g (x) need to build graphs of functions
y = f (x) = g (x) and select those intervals of the abscissa axis on which the graph of the function y = f(x) located above the graph of the function y = g(x).
Example 17.8. Solve graphically the inequality x 2- 4 > 3X.
| Y - x* - 4 |
Solution. Let's construct graphs of functions in one coordinate system
y = x 2 - 4 and y = Zx (Fig. 17.5). The figure shows that the graphs of functions at= x 2- 4 is located above the graph of the function y = 3 X at X< -1 and x > 4, i.e. the set of solutions to the original inequality is the set
(- ¥; -1) È (4; + oo) .
Answer: x О(- oo; -1) and ( 4; + oo).
Schedule quadratic function at= ax 2 + bx + c is a parabola with branches pointing upward if a > 0, and down if A< 0. In this case, three cases are possible: the parabola intersects the axis Oh(i.e. equation ah 2+ bx+ c = 0 has two different roots); parabola touches axis X(i.e. equation ax 2 + bx+ c = 0 has one root); the parabola does not intersect the axis Oh(i.e. equation ah 2+ bx+ c = 0 has no roots). Thus, there are six possible positions of the parabola, which serves as a graph of the function y = ah 2+ b x + c(Fig. 17.6). Using these illustrations, you can solve quadratic inequalities.
Example 17.9. Solve the inequality: a) 2 x g+ 5x - 3 > 0; b) -Zx 2 - 2x- 6 < 0.
Solution, a) The equation 2x 2 + 5x -3 = 0 has two roots: x, = -3, x 2 = 0.5. Parabola serving as a graph of a function at= 2x 2+ 5x -3, shown in Fig. A. Inequality 2x 2+ 5x -3 > 0 is satisfied for those values X, for which the points of the parabola lie above the axis Oh: it will be at X< х х or when X> x g> those. at X< -3 or at x > 0.5. This means that the set of solutions to the original inequality is the set of (- ¥; -3) and (0.5; + ¥).
b) Equation -Зх 2 + 2x- 6 = 0 has no real roots. Parabola serving as a graph of a function at= - 3x 2 - 2x - 6, shown in Fig. 17.6 Inequality -3x 2 - 2x - 6 < О выполняется при тех значениях X, for which the points of the parabola lie below the axis Oh. Since the entire parabola lies below the axis Oh, then the set of solutions to the original inequality is the set R .
INEQUALITIES CONTAINING A VARIABLE UNDER THE MODULE SIGN. When solving these inequalities, it should be kept in mind that:
|f(x) | =
f(x), If f(x) ³ 0,
- f(x), If f(x) < 0,
In this case, the range of permissible values of the inequality should be divided into intervals, at each of which the expressions under the modulus sign retain their sign. Then, expanding the modules (taking into account the signs of the expressions), you need to solve the inequality on each interval and combine the resulting solutions into a set of solutions to the original inequality.
Example 17.10. Solve the inequality:
|x -1| + |2- x| > 3+x.
Solution. Points x = 1 and x = 2 divide the numerical axis (ODZ of inequality (17.9) into three intervals: x< 1, 1 £ х £.2, х >2. Let’s solve this inequality for each of them. If x< 1, то х - 1 < 0 и 2 – х >0; therefore |x -1| = - (x - I), |2 - x | = 2 - x. This means that inequality (17.9) takes the form: 1- x + 2 - x > 3 + x, i.e. X< 0. Таким образом, в этом случае решениями неравенства (17.9) являются все отрицательные числа.
If 1 £ x £.2, then x - 1 ³ 0 and 2 – x ³ 0; therefore | x- 1| = x - 1, |2 - x| = 2 – x. This means that the system holds:
x – 1 + 2 – x > 3 + x,
The resulting system of inequalities has no solutions. Therefore, on the interval [ 1; 2] the set of solutions to inequality (17.9) is empty.
If x > 2, then x - 1 >0 and 2 – x<0; поэтому | х - 1| = х- 1, |2-х| = -(2- х). Значит, имеет место система:
x -1 + x – 2 > 3+x,
x > 6 or
Combining the solutions found on all parts of the ODZ inequality (17.9), we obtain its solution - the set (-¥; 0) È (6; +oo).
Sometimes it is useful to use the geometric interpretation of the modulus of a real number, according to which | a | means the distance of point a of the coordinate line from the origin O, a | a - b | means the distance between points a and b on the coordinate line. Alternatively, you can use the method of squaring both sides of the inequality.
Theorem 17.5. If expressions f(x) and g(x) for any x take only non-negative values, then the inequalities f (x) > g (x) And f (x) ² > g (x) ² are equivalent.
58. Main conclusions § 12
In this section we have defined the following concepts:
Numeric expression;
Meaning numerical expression;
An expression that has no meaning;
Expression with variable(s);
Expression definition area;
Identically equal expressions;
Identity;
Identical transformation of an expression;
Numerical equality;
Numerical inequality;
Equation with one variable;
Root of the equation;
What does it mean to solve an equation;
Equivalent equations;
Inequality with one variable;
Solving Inequalities;
What does it mean to solve inequality;
Equivalent inequalities.
In addition, we examined theorems on the equivalence of equations and inequalities, which are the basis for their solution.
Knowledge of the definitions of all the above concepts and theorems on the equivalence of equations and inequalities is a necessary condition for methodologically competent study of algebraic material with primary school students.
The topic of the lesson is “Solving inequalities and their systems” (mathematics grade 9)
Lesson type: lesson on systematization and generalization of knowledge and skills
Lesson technology: technology for the development of critical thinking, differentiated learning, ICT technologies
Purpose of the lesson: repeat and systematize knowledge about the properties of inequalities and methods for solving them, create conditions for developing the skills to apply this knowledge when solving standard and creative problems.
Tasks.
Educational:
contribute to the development of students’ skills to generalize acquired knowledge, conduct analysis, synthesis, comparisons, and draw the necessary conclusions
organize the activities of students to apply the acquired knowledge in practice
promote the development of skills to apply acquired knowledge in non-standard conditions
Educational:
continue formation logical thinking, attention and memory;
improve skills of analysis, systematization, generalization;
creating conditions that ensure the development of self-control skills in students;
promote the acquisition of the necessary skills for independent learning activities.
Educational:
cultivate discipline and composure, responsibility, independence, critical attitude towards oneself, and attentiveness.
Planned educational results.
Personal: responsible attitude towards learning and communicative competence in communication and cooperation with peers in the process educational activities.
Cognitive: the ability to define concepts, create generalizations, independently select grounds and criteria for classification, build logical reasoning, and draw conclusions;
Regulatory: the ability to identify potential difficulties when solving an educational and cognitive task and find means to eliminate them, evaluate one’s achievements
Communicative: the ability to make judgments using mathematical terms and concepts, formulate questions and answers during the task, exchange knowledge between group members to make effective joint decisions.
Basic terms and concepts: linear inequality, quadratic inequality, system of inequalities.
Equipment
Projector, teacher's laptop, several netbooks for students;
Presentation;
Cards with basic knowledge and skills on the topic of the lesson (Appendix 1);
Cards with independent work (Appendix 2).
Lesson Plan
| Lesson progress |
|||
| Technological stages. Target. | Teacher activities | Student activities | |
| Introductory and motivational component |
|||
| 1.Organizational Goal: psychological preparation for communication. | Hello. Nice to see you all. Sit down. Check if you have everything ready for the lesson. If everything is okay, then look at me. | They say hello. Check accessories. Getting ready for work. | Personal. A responsible attitude towards learning is formed. |
| 2.Updating knowledge (2 min) Goal: identify individual knowledge gaps on a topic | The topic of our lesson is “Solving inequalities with one variable and their systems.” (slide 1) Here is a list of basic knowledge and skills on the topic. Assess your knowledge and skills. Place the appropriate icons. (slide 2) | Evaluate their own knowledge and skills. (Appendix 1) | Regulatory Self-assessment of your knowledge and skills |
| 3.Motivation (2 min) Purpose: to provide activities to determine lesson goals . | IN work of the OGE in mathematics, several questions in both the first and second parts determine the ability to solve inequalities. What do we need to repeat in class to successfully complete these tasks? | They reason and name questions for repetition. | Cognitive. Identify and formulate a cognitive goal. |
| Conception stage (content component) |
|||
| 4.Self-esteem and choice of trajectory (1-2 min) | Depending on how you assessed your knowledge and skills on the topic, choose the form of work in the lesson. You can work with the whole class with me. You can work individually on netbooks, using my consultation, or in pairs, helping each other. | Determined with an individual learning path. If necessary, change places. | Regulatory identify potential difficulties when solving an educational and cognitive task and find means to eliminate them |
| 5-7 Work in pairs or individually (25 min) | The teacher advises students working independently. | Students who know the topic well work individually or in pairs with a presentation (slides 4-10) Complete assignments (slides 6,9). | Cognitive ability to define concepts, create generalizations, build a logical chain Regulatory the ability to determine actions in accordance with the educational and cognitive task Communication ability to organize educational cooperation and joint activities, work with the source of information Personal responsible attitude to learning, readiness and ability for self-development and self-education |
| 5. Solving linear inequalities. (10 min) | What properties of inequalities do we use to solve them? Can you distinguish between linear and quadratic inequalities and their systems? (slide 5) How to solve linear inequality? Follow the solution. (slide 6) The teacher monitors the solution at the board. Check the correctness of the solution. | Name the properties of inequalities; after answering or in case of difficulty, the teacher opens slide 4. Name the distinctive features of inequalities. Using the properties of inequalities. One student solves inequality No. 1 at the board. The rest are in notebooks, following the answerer’s decision. Inequalities No. 2 and 3 are satisfied independently. They check the ready answer. | Cognitive Communication |
| 6. Solving quadratic inequalities. (10 min) | How to solve inequality? What kind of inequality is this? What methods are used to solve quadratic inequalities? Let's remember the parabola method (slide 7). The teacher recalls the stages of solving an inequality. The interval method is used to solve inequalities of the second or more high degrees. (slide 8) To solve quadratic inequalities, you can choose a method that is convenient for you. Solve the inequalities. (slide 9). The teacher monitors the progress of the solution and recalls methods for solving incomplete quadratic equations. The teacher advises individually working students. | Answer: Quadratic inequality We solve using the parabola method or the interval method. Students follow up on the presentation solution. At the board, students take turns solving inequalities No. 1 and 2. They check the answer. (to solve nerve No. 2, you need to remember the method for solving incomplete quadratic equations). Inequality No. 3 is solved independently and checked against the answer. | Cognitive the ability to define concepts, create generalizations, build reasoning from general patterns to particular solutions Communication the ability to present a detailed plan of one’s own activities orally and in writing; |
| 7. Solving systems of inequalities (4-5 min) | Remember the stages of solving a system of inequalities. Solve the system (Slide 10) | Name the stages of the solution The student solves at the board and checks the solution on the slide. | |
| Reflective-evaluative stage |
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| 8.Control and testing of knowledge (10 min) Goal: to identify the quality of learning the material. | Let's test your knowledge on the topic. Solve the problems yourself. The teacher checks the result using ready-made answers. | Perform independent work on options (Appendix 2) Having completed the work, the student reports this to the teacher. The student determines his grade according to the criteria (slide 11). If the work is successfully completed, he can begin an additional task (slide 11) | Cognitive. Build logical chains of reasoning. |
| 9.Reflection (2 min) Goal: adequate self-esteem of one’s capabilities and abilities, advantages and limitations is formed | Is there an improvement in the result? If you still have questions, refer to the textbook at home (p. 120) | Assess their own knowledge and skills on the same piece of paper (Appendix 1). Compare with self-esteem at the beginning of the lesson and draw conclusions. | Regulatory Self-assessment of your achievements |
| 10.Homework (2 min) Goal: consolidation of the studied material. | Determine homework based on the results of independent work (slide 13) | Determine and record individual task | Cognitive. Build logical chains of reasoning. Analyze and transform information. |
List of used literature: Algebra. Textbook for 9th grade. / Yu.N.Makrychev, N.G.Mindyuk, K.I.Neshkov, S.B.Suvorova. - M.: Education, 2014
A program for solving linear, quadratic and fractional inequalities not only gives the answer to the problem, it gives 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 your math or algebra homework done as quickly as possible? In this case, you can also use our programs with detailed solutions.
In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems 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 in the form of a decimal, but also in the form of an ordinary fraction.
Rules for entering decimal fractions.
In decimal fractions, the fractional part can be separated from the whole part 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 a numerical fraction, the numerator is separated from the denominator by a division sign: /
The whole part is separated from the fraction by the ampersand sign: &
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 familiar 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 \(5x \leq 4000\) the unknown number x is 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 $$.
The curly bracket shows that you need to find 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 turn into 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: \(-2 \leq x \leq 3 \).
Solutions to systems of inequalities with one unknown are various numerical sets. These sets have names. Thus, on the 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.
| -2 | 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 turns the given inequality into a true 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:
| -2 | 3 |
The intersection is the segment [-2; 3] - this is the solution to the original system of inequalities.