Preview:
MINISTRY OF EDUCATION OF THE MOSCOW REGION
State Educational Institution NPO Vocational School No. 37
PROJECT:
QUADRATE EQUATIONS AND INEQUALITIES WITH PARAMETERS"
Performed -
Matsuk Galina Nikolaevna,
Mathematics teacher, State Educational Institution NPO
vocational school No. 37 MO.
G. Noginsk, 2011
1. Introduction
4. Methodology for solving quadratic equations under initial conditions.
6. Methodology for solving quadratic inequalities with parameters in general form.
7. Methodology for solving quadratic inequalities under initial conditions.
8. Conclusion.
9.Literature.
- Introduction.
The main task of teaching mathematics in a vocational school is to ensure students’ strong and conscious mastery of the system of mathematical knowledge and skills necessary in everyday life and work, sufficient for studying related disciplines and continuing education, as well as in professional activities that require a sufficiently high mathematical culture.
Profiled mathematics training is carried out through solving applied problems related to the professions of metalworking, electrical installation work, and woodworking. For life in modern society, it is important to develop a mathematical communication style, which manifests itself in certain mental skills. Problems with parameters have diagnostic and prognostic value. With their help, you can test your knowledge of the main sections of elementary mathematics, the level of logical thinking, and initial research skills.
Teaching tasks with parameters requires students to have great mental and volitional efforts, developed attention, and the cultivation of such qualities as activity, creative initiative, and collective cognitive work. Problems with parameters are oriented for study during general repetition in the 2nd year in preparation for the final state certification and in the 3rd year in additional classes in preparation for students who have expressed a desire to take final exams in the form of the Unified State Exam.
The main direction of modernization of mathematics education is the development of mechanisms for final certification through the introduction of the Unified State Exam. In recent years, problems with parameters have been introduced in mathematics assignments. Such tasks are required for university entrance exams. The appearance of such problems is very important, since with their help they test the applicant’s mastery of formulas of elementary mathematics, methods of solving equations and inequalities, the ability to build a logical chain of reasoning, and the level of logical thinking of the applicant. An analysis of previous Unified State Exam results over several previous years shows that graduates have great difficulty solving such tasks, and many do not even start them. Most either cannot cope with such tasks at all, or provide cumbersome calculations. The reason for this is the lack of a system of assignments on this topic in school textbooks. In this regard, there was a need to conduct special topics in graduate groups in preparation for exams on solving problems with parameters and problems of an applied nature related to professional orientation.
The study of these topics is intended for 3rd year students who want to learn how to solve problems of an increased level of complexity in algebra and the beginnings of analysis. Solving such problems causes them significant difficulties. This is due to the fact that each equation or inequality with parameters represents a whole class of ordinary equations and inequalities, for each of which a solution must be obtained.
In the process of solving problems with parameters, the arsenal of techniques and methods of human thinking naturally includes induction and deduction, generalization and specification, analysis, classification and systematization, and analogy. Since the curriculum in vocational schools provides for consultations in mathematics, which are included in the schedule of classes, for students who have sufficient mathematical training, show interest in the subject being studied, and have the further goal of entering a university, it is advisable to use the specified hours to solve problems with parameters for preparing for olympiads, mathematical competitions, various types of exams, in particular the Unified State Exam. The solution of such problems is especially relevant for applied and practical purposes, which will help in conducting various studies.
2. Goals, main tasks, methods, technologies, knowledge requirements.
Project goals:
- Formation of abilities and skills in solving problems with parameters, which boil down to the study of quadratic equations and inequalities.
- Forming interest in the subject, developing mathematical abilities, preparing for the Unified State Exam.
- Expanding mathematical understanding of techniques and methods for solving equations and inequalities.
- Development of logical thinking and research skills.
- Involvement in creative, research and educational activities.
- Providing conditions for independent creative work.
- Fostering students' mental and volitional efforts, developed attention, activity, creative initiative, and skills of collective cognitive work.
Main objectives of the project:
- To provide students with the opportunity to realize their interest in mathematics and individual opportunities for its development.
- Promote the acquisition of factual knowledge and skills.
- Show the practical significance of problems with parameters in the field of applied research.
- Teach methods for solving standard and non-standard equations and inequalities.
- To deepen knowledge in mathematics, providing for the formation of a sustainable interest in the subject.
- Identify and develop students’ mathematical abilities.
- Provide preparation for entering universities.
- Provide preparation for professional activities requiring high mathematical culture.
- Organize research and project activities that promote the development of intellectual and communication skills.
Methods used during classes:
- Lecture – to convey theoretical material, accompanied by a conversation with students.
- Seminars - to consolidate material on discussing theory.
- Workshops – for solving mathematical problems.
- Discussions – to provide arguments for your solutions.
- Various forms of group and individual activities.
- Research activities, which are organized through: work with didactic material, preparation of messages, defense of abstracts and creative works.
- Lectures – presentations using a computer and projector.
Technologies used:
- Lecture-seminar training system.
- Information and communication technologies.
- A research method in teaching aimed at developing thinking abilities.
- Problem-based learning, which provides motivation for research by posing a problem, discussing various options for the problem.
- Activity method technology that helps develop the cognitive interests of students.
Requirements for students' knowledge.
As a result of studying various ways of solving quadratic equations and inequalities with parameters, students should acquire the skills:
- Firmly grasp the concept of a parameter in a quadratic equation and quadratic inequality;
- Be able to solve quadratic equations with parameters.
- Be able to solve quadratic inequalities with parameters.
- Find the roots of a quadratic function.
- Build graphs of quadratic functions.
- Explore quadratic trinomial.
- Apply rational methods of identity transformations.
- Use the most commonly used heuristic techniques.
- Be able to apply the acquired knowledge when working on a personal computer.
Forms of control.
- Lessons – self-assessments and assessments of comrades.
- Presentation of educational projects.
- Testing.
- Rating – table.
- Homework problems from previous years' Unified State Exam collections.
- Test papers.
3. Methodology for solving quadratic equations with parameters in general form.
Don't be afraid of problems with parameters. First of all, when solving equations and inequalities with parameters, you need to do what is done when solving any equation and inequality - reduce the given equations or inequalities to a simpler form, if possible: factorize the rational expression, reduce it, put the factor out of brackets, etc. .d. There are problems that can be divided into two large classes.
The first class includes examples in which it is necessary to solve an equation or inequality for all possible values of a parameter.
The second class includes examples in which it is necessary to find not all possible solutions, but only those that satisfy some additional conditions. The class of such problems is inexhaustible.
The most understandable way for students to solve such problems is to first find all the solutions and then select those that satisfy additional conditions.
When solving problems with parameters, it is sometimes convenient to construct graphs in the usual plane (x, y), and sometimes it is better to consider graphs in the plane (x, a), where x is the independent variable and “a” is the parameter. This is primarily possible in a problem where you have to construct familiar elementary graphs: straight lines, parabolas, circles, etc. In addition, sketches of graphs sometimes help to clearly see the “progress” of the solution.
When solving the equations f (x,a) = 0 and the inequalities f (x,a) › 0, we must remember that first of all the solution is considered for those values of the parameter at which the coefficient at the highest power x of the square trinomial f (x ,a), thereby reducing the degree. Quadratic equation A(a) x 2 + B(a) x + C(a) = 0 at A(a) = 0 turns into linear if B(a) ≠ 0, and the methods for solving quadratic and linear equations are different.
Let us recall the basic formulas for working with quadratic equations.
Equation of the form ah 2 + in + c = 0, where x R are unknowns, a, b, c are expressions that depend only on parameters, and a ≠ 0 is called a quadratic equation, and D = b 2 – 4ac is called the discriminant of a quadratic trinomial.
If D
If D > 0, then the equation has two different roots
x 1 = , x 2 = , and then ax 2 + in + c = a (x – x 1) (x – x 2).
These roots are related through the coefficients of the equation by Vieta’s formulas
If D = 0, then the equation has two coinciding roots x 1 = x 2 = , and then ax 2 + in + c = a (x – x 1) 2 . In this case, the equation is said to have one solution.
When, i.e. = 2k, the roots of the quadratic equation are determined by the formula x 1,2 = ,
To solve the reduced quadratic equation x 2 + px + q = 0
The formula used is x 1,2 = - , as well as Vieta's formulas
Examples. Solve equations:
Example 1. + =
Solution:
For a ≠ - 1, x ≠ 2 we get x 2 + 2ax – 3b + 4 = 0 and roots
x 1 = - a - , x 2 = -a + , existing at
A 2 + 2a – 4 0, i.e. at
Now let's check whether there are any a such that either x 1 or x 2 is equal to 2. Substitute x = 2 into the quadratic equation, and we get a = - 8.
The second root in this case is equal to(according to Vieta’s theorem) and for a = - 8 is equal to 14.
Answer: for a = - 8, the only solution is x = 14;
If a (- ∞; - 8) (- 8; - 4) (1; + ∞) – two roots x 1 and x 2;
If a = - the only solution x =respectively;
If a (- 4; 1), then x .
Sometimes equations with fractional terms are reduced to quadratic ones. Consider the following equation.
Example 2. - =
Solution: When a = 0 it does not make sense, the value x must satisfy the conditions: x -1, x -2. Multiplying all terms of the equation by a (x + 1) (x +2) 0,
We get x 2 – 2(a – 1)x + a 2 – 2a – 3 = 0, equivalent to this. Its roots:
x 1 = a + 1, x 2 = - 3. Let us select extraneous roots from these roots, i.e. those that are equal to – 1 and – 2:
X 1 = a + 1 = - 1, a = - 2, but with a = - 2 x 2 = - 5;
X 1 = a + 1 = - 2, a = - 3, but with a = - 3 x 2 = - 6;
X 2 = a - 3 = - 1, a = 2, but with a = 2 x 1 = 3;
X 2 = a - 3 = - 2, a = 1, but with a = 1 x 1 = 2.
Answer: for a ≠ 0, a ≠ 2, a ≠ - 3, a ≠ 1 x 1 = a + 1, x 2 = a – 3;
When a = - 2 x = - 5; when a = - 3 x = - 6.
4. Methodology for solving quadratic equations under initial conditions.
The conditions for parametric quadratic equations are varied. For example, you need to find the value of a parameter for which the roots are: positive, negative, have different signs, greater or less than a certain number, etc. To solve them, you should use the properties of the roots of the quadratic equation ax 2 + in + c = 0.
If D > 0, a > 0, then the equation has two real different roots, the signs of which for c > 0 are the same and opposite to the sign of the coefficient b, and for c
If D = 0, a > 0, then the equation has real and equal roots, the sign of which is opposite to the sign of the coefficient b.
If D 0, then the equation has no real roots.
Similarly, we can establish the properties of the roots of the quadratic equation for a
- If in a quadratic equation we swap the coefficients a and c, we get an equation whose roots are the inverse of the roots of the given one.
- If in a quadratic equation we change the sign of the coefficient b, we obtain an equation whose roots are opposite to the roots of the given one.
- If in a quadratic equation the coefficients a and c have different signs, then it has real roots.
- If a > 0 and D = 0, then the left side of the quadratic equation is a complete square, and vice versa, if the left side of the equation is a complete square, then a > 0 and D = 0.
- If all the coefficients of the equation are rational and the discriminant expresses a perfect square, then the roots of the equation are rational.
- If we consider the location of the roots relative to zero, then we apply Vieta’s theorem.
Selection of roots of a quadratic trinomial according to conditions and location of zeros of a quadratic function on the number line.
Let f (x) = ax 2 + in + c, a 0, roots x 1 ˂ x 2, ˂ .
The location of the roots on the number line. | Necessary and sufficient condition. |
|
x 1, x 2 | and f ( ) > 0, D 0, x 0 |
|
x 1, x 2 > | and f ( ) > 0, D 0, x 0 > |
|
x 1 2 | and f ( ) |
|
1 ,x 2 . | and f ( ) > 0, D 0, and f ( ) > 0 0 . |
|
1 2 | and f ( ) > 0, and f ( ) |
|
x 1 2 | and f ( ) ) > 0 |
|
x 1 2 | and f ( ) ) |
Example 3. Determine at what values of a the equation
x 2 – 2 (a – 1) x + 2a + 1 = 0
- has no roots:
necessary and sufficient condition D
D = (a – 1) 2 – 2a – 1 = a 2 – 4a
- has roots:
D 0, D = (a – 1) 2 – 2a – 1 0, a
- has one root:
- has two roots:
D > 0, i.e. a
- has positive roots:
2(a – 1) > 0 a 4
If the question is “has two positive roots,” then the system should replace D > 0;
- has negative roots:
2(a – 1)
- has roots of different signs, i.e. one is positive and the other is negative:
a ;
Condition It is not necessary to use it, x is enough 1 x 2
- has one of the roots equal to 0:
a necessary sufficient condition is that the free term of the equation is equal to zero, i.e. 2a + 1 = 0, a = -1/2.
The sign of the second root is determined either by substituting a = -1/2 into the original equation, or, more simply, by Vieta’s theorem x 1 + x 2 = 2 (a – 1), and after substituting a = -1/2 we get x 2 = - 3, i.e. for a = -1/2 two roots: x 1 = 0, x 2 = - 3.
Example 4 . At what values of the parameter a does the equation
(a – 2) x 2 – 4ax +3 -2a = 0 has a unique solution that satisfies the inequality x
Solution.
Discriminant 2 – (a – 2)(3 – 2a)
4a 2 – 3a + 6 + 2a 2 – 4a = 6a 2 – 7a + 6
Since 49 – 144 = - 95 and the first coefficient is 6 then 6a 2 – 7a + 6 for all x R.
Then x 1.2 = .
According to the conditions of the problem x2, then we get the inequality
We have:
true for all a R.
6a 2 – 7a + 6 6a 2 – 7a - 10 2
A 1.2 = 1/12 (7 17), and 1 = 2, and 2 = - 5/6.
Therefore -5/6
Answer: -
5. Parameter as an equal variable.
In all analyzed tasksthe parameter was treated as a fixed but unknown number. Meanwhile, from a formal point of view, a parameter is a variable, and “equal” to others present in the example. For example, with this view of the form parameter f (x; a), functions are defined not with one (as before), but with two variables. Such an interpretation naturally forms another type (or rather, a solution method that defines this type) of problems with parameters. Let us show an analytical solution of this type.
Example 5. On the xy plane, indicate all the points through which none of the curves of the family y = x passes 2 – 4рх + 2р 2 – 3, where p is a parameter.
Solution: If (x 0;y 0 ) is a point through which none of the curves of a given family passes, then the coordinates of this point do not satisfy the original equation. Consequently, the problem boiled down to finding a relationship between x and y such that the equation given in the condition would have no solutions. It is easy to obtain the desired dependence by focusing not on the variables x and y, but on the parameter p. In this case, a productive idea arises: consider this equation as quadratic with respect to p. We have
2р 2 – 4рх+ x 2 – y – 3 = 0. Discriminant= 8x 2 + 8y + 24 must be negative. From here we get y ˂ - x 2 – 3, therefore, the required set is all the points of the coordinate plane lying “under” the parabola y = - x 2 – 3.
Answer: y 2 – 3
6. Methodology for solving quadratic inequalities with parameters
In general.
Quadratic (strict and non-strict) inequalities of the form
Acceptable values are those parameter values for which a, b, c are valid. It is convenient to solve quadratic inequalities either analytically or graphically. Since the graph of a quadratic function is a parabola, then for a > 0 the branches of the parabola are directed upward, for a
Different positions of the parabola f (x) = ax 2 + in + s, a 0 for a > 0 is shown in Fig. 1
A) b) c)
a) If f (x) > 0 and D R;
b) If f (x) > 0 and D = 0, then x ;
c) If f (x) > 0 and D > 0, then x (- ; x 1 ) (x 2 ; + ).
The positions of the parabola are considered similarly for a
For example, one of the three cases when
for a 0 and f (x) > 0 x (x 1; x 2);
for a 0 and f (x) (- ; x 1 ) (x 2 ; + ).
As an example, consider solving an inequality.
Example 6. Solve inequality x 2 + 2x + a > 0.
Let D be the discriminant of the trinomial x 2 + 2x + a > 0. For D = 0, for a = 1, the inequality takes the form:
(x + 1) 2 > 0
It is true for any real values of x except x = - 1.
For D > 0, i.e. at x, trinomial x 2 + 2x + a has two roots: - 1 – And
1 + and the solution to the inequality is the interval
(- ; - 1 – ) (- 1 + ; + )
This inequality is easy to solve graphically. To do this, let us represent it in the form
X 2 + 2x > - a
and build a graph of the function y = x 2 + 2x
The abscissas of the points of intersection of this graph with the line y = - a are the roots of the equation x 2 + 2x = - a.
Answer:
for –a > - 1, i.e. at a, x (- ; x 1 ) (x 2 ;+ );
at – a = - 1, i.e. for a = 1, x is any real number except - 1;
at – a , that is, for a > 1, x is any real number.
Example 7 . Solve inequality cx 2 – 2 (s – 1)x + (s + 2)
When c = 0 it takes the form: 2x + 2the solution will be x
Let us introduce the notation f (x) = cx 2 – 2 (s – 1)x + (s + 2) where c ≠ 0.
In this case the inequality f(x)
Let D be the discriminant of f(x). 0.25 D = 1 – 4s.
If D > 0, i.e. if with> 0.25, then the sign of f (x) coincides with the sign of c for any real values of x, i.e. f(x)> 0 for any x R, which means for c > 0.25 inequality f(x)
If D = 0, i.e. c = 0.25, then f (x) = (0.25 x + 1.5) 2, i.e. f (x) 0 for any
X R. Therefore, for c = 0.25 the inequality f (x)
Consider the case D 0). f (x) = 0 for two real values of x:
x 1 = (c – 1 – ) and x 2 = (c – 1 + ).
Two cases may arise here:
Solve inequality f(x)
f(x) coincides with the sign of c. To answer this question, note that - , i.e. s – 1 – ˂ s – 1 + , but since s (s – 1 – ) (s – 1 + ) and therefore the solution to the inequality will be:
(- ; (s – 1 – )) ( (s – 1 + ); + ).
Now, to solve the inequality, it is enough to indicate those values of c for which the sign of f (x) is opposite to the sign of c. Since at 0 1 2, then x (x 1; x 2).
Answer: when c = 0 x R;
With (- ; x 2 ) (x 1 ; + );
At 0 (x 1; x 2);
For c 0.25 there are no solutions.
The view of a parameter as an equal variable is reflected in graphical methods for solving and quadratic inequalities. In fact, since the parameter is “equal in rights” to the variable, it is natural that it can be “allocated” to its own coordinate axis. Thus, a coordinate plane (x; a) arises. Such a minor detail as abandoning the traditional choice of letters x and y to denote the axes determines one of the most effective methods for solving problems with parameters.
It is convenient when the problem involves one parameter a and one variable x. The solution process itself looks schematically like this. First, a graphic image is constructed, then, intersecting the resulting graph with straight lines perpendicular to the parametric axis, we “remove” the necessary information.
The rejection of the traditional choice of the letters x and y to designate the axes determines one of the most effective methods for solving problems with parameters - the “domain method”
- Methodology for solving quadratic inequalities under initial conditions.
Let us consider an analytical solution to a quadratic inequality with parameters, the results of which are considered on the number line.
Example 8.
Find all values of x, for each of which the inequality
(2x)a 2 +(x 2 -2x+3)a-3x≥0
is satisfied for any value of a belonging to the interval [-3;0].
Solution. Let us transform the left side of this inequality as follows:
(2-x)a 2 + (x 2 -2x+3)a-3x=ax 2 - a 2 x - 2ax + 2a 2 + 3a - 3x =
Ax (x - a)-2a(x - a)- 3(x-a) = (x - a)(ax- 2a - 3).
This inequality will take the form: (x - a) (ax - 2a - 3) ≥ 0.
If a = 0, we get - Zx ≥ 0 x ≤ 0.
If a ≠ 0, then -3 a
Because A 0, then the solution to this inequality will be the interval of the numerical axis located between the roots of the equation corresponding to the inequality.
Let's find out the relative position of the numbers a and , taking into account the condition - 3 ≤ a
3 ≤a
A = -1.
Let us present in all considered cases the solutions to this inequality depending on the parameter values:
We find that only x = -1 is a solution to this inequality for any value of parameter a .
Answer: -1
- Conclusion.
Why did I choose a project on the topic “Development of methodological recommendations for solving quadratic equations and inequalities with parameters”? Since when solving any trigonometric, exponential, logarithmic equations, inequalities, systems, we most often come to consider sometimes linear, and most often quadratic equations and inequalities. When solving complex problems with parameters, most tasks are reduced, using equivalent transformations, to the choice of solutions of the type: a (x – a) (x – c) > 0 (
We reviewed the theoretical basis for solving quadratic equations and inequalities with parameters. We remembered the necessary formulas and transformations, looked at the different arrangements of graphs of a quadratic function depending on the value of the discriminant, on the sign of the leading coefficient, on the location of the roots and vertices of the parabola. We identified a scheme for solving and selecting results and compiled a table.
The project demonstrates analytical and graphical methods for solving quadratic equations and inequalities. Students in a vocational school need visual perception of the material for better assimilation of the material. It is shown how the variable x can be changed and the parameter accepted as an equal value.
For a clear understanding of this topic, the solution to 8 problems with parameters is considered, 1 – 2 for each section. In example No. 1, the number of solutions for various parameter values is considered; in example No. 3, the solution of a quadratic equation is analyzed under a variety of initial conditions. A graphic illustration has been made to solve quadratic inequalities. In example No. 5, the method of replacing a parameter as an equal value is used. The project includes a consideration of example No. 8 from the tasks included in section C for intensive preparation for passing the Unified State Exam.
For high-quality training of students in solving problems with parameters, it is recommended to fully use multimedia technologies, namely: use presentations for lectures, electronic textbooks and books, and your own developments from the media library. Binary lessons in mathematics + computer science are very effective. The Internet is an indispensable assistant for teachers and students. The presentation requires imported objects from existing educational resources. The most convenient and acceptable to work with is the “Using Microsoft Office at School” center.
The development of methodological recommendations on this topic will facilitate the work of young teachers who come to work at the school, will add to the teacher’s portfolio, will serve as a model for special subjects, and sample solutions will help students cope with complex tasks.
- Literature.
1. Gornshtein P.I., Polonsky V.B., Yakir M.S. Problems with parameters. “Ilexa”, “Gymnasium”, Moscow - Kharkov, 2002.
2. Balayan E.N. A collection of problems in mathematics for preparing for the Unified State Exam and Olympiads. 9-11 grades. "Phoenix", Rostov-on-Don, 2010.
3. Yastrebinetsky G.A. Problems with parameters. M., "Enlightenment", 1986.
4. Kolesnikova S.I. Mathematics. Solving complex problems of the Unified State Exam. M. "IRIS - press", 2005.
5. Rodionov E.M., Sinyakova S.L. Mathematics. A guide for applicants to universities. Training center "Orientir" MSTU named after. N.E. Bauman, M., 2004.
6. Skanavi M.I. Collection of problems in mathematics for those entering universities: In 2 books. Book 1, M., 2009.
Job type: 18
Condition
For what values of the parameter a does the inequality
\log_(5)(4+a+(1+5a^(2)-\cos^(2)x) \cdot\sin x - a \cos 2x) \leq 1 is satisfied for all values of x?
Show solutionSolution
This inequality is equivalent to the double inequality 0 < 4+a+(5a^{2}+\sin^{2}x) \sin x+ a(2 \sin^(2)x-1) \leq 5 .
Let \sin x=t , then we get the inequality:
4 < t^{3}+2at^{2}+5a^{2}t \leq 1 \: (*) , which must be executed for all values of -1 \leq t \leq 1 . If a=0, then inequality (*) holds for any t\in [-1;1] .
Let a \neq 0 . The function f(t)=t^(3)+2at^(2)+5a^(2)t increases on the interval [-1;1] , since the derivative f"(t)=3t^(2)+4at +5a^(2) > 0 for all values of t \in \mathbb(R) and a \neq 0 (discriminant D< 0 и старший коэффициент больше нуля).
Inequality (*) will be satisfied for t \in [-1;1] under the conditions
\begin(cases) f(-1) > -4, \\ f(1) \leq 1, \\ a \neq 0; \end(cases)\: \Leftrightarrow \begin(cases) -1+2a-5a^(2) > -4, \\ 1+2a+5a^(2) \leq 1, \\ a \neq 0; \end(cases)\: \Leftrightarrow \begin(cases) 5a^(2)-2a-3< 0, \\ 5a^{2}+2a \leq 0, \\ a \neq 0; \end{cases}\: \Leftrightarrow -\frac(2)(5)\leq a< 0 .
So, the condition is satisfied when -\frac(2)(5) \leq a \leq 0 .
Answer
\left [ -\frac(2)(5); 0\right ]
Source: “Mathematics. Preparation for the Unified State Exam 2016. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Job type: 18
Topic: Inequalities with a parameter
Condition
Find all values of the parameter a, for each of which the inequality
x^2+3|x-a|-7x\leqslant -2a
has a unique solution.
Show solutionSolution
Inequality is equivalent to a set of systems of inequalities
\left[\!\!\begin(array)(l) \begin(cases) x \geqslant a, \\ x^2+3x-3a-7x+2a\leqslant0; \end(cases) \\ \begin(cases)x
In the Oxa coordinate system, we will construct graphs of functions a=x, a=x^2-4x, a=-\frac(x^2)(5)+2x.
The resulting set is satisfied by the points enclosed between the graphs of the functions a=x^2-4x, a=-\frac(x^2)(5)+2x on the interval x\in (shaded area).
From the graph we determine: the original inequality has a unique solution for a=-4 and a=5, since in the shaded area there will be a single point with ordinate a equal to -4 and equal to 5.
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Solving inequalities with a parameter.
Inequalities that have the form ax > b, ax< b, ax ≥ b, ax ≤ b, где a и b – действительные числа или выражения, зависящие от параметров, а x – неизвестная величина, называются linear inequalities.
The principles for solving linear inequalities with a parameter are very similar to the principles for solving linear equations with a parameter.
Example 1.
Solve the inequality 5x – a > ax + 3.
Solution.
First, let's transform the original inequality:
5x – ax > a + 3, let’s take x out of brackets on the left side of the inequality:
(5 – a)x > a + 3. Now consider possible cases for parameter a:
If a > 5, then x< (а + 3) / (5 – а).
If a = 5, then there are no solutions.
If a< 5, то x >(a + 3) / (5 – a).
This solution will be the answer to the inequality.
Example 2.
Solve the inequality x(a – 2) / (a – 1) – 2a/3 ≤ 2x – a for a ≠ 1.
Solution.
Let's transform the original inequality:
x(a – 2) / (a – 1) – 2x ≤ 2a/3 – a;
Ах/(а – 1) ≤ -а/3. Multiplying both sides of the inequality by (-1), we get:
ax/(a – 1) ≥ a/3. Let us explore possible cases for parameter a:
1 case. Let a/(a – 1) > 0 or a € (-∞; 0)ᴗ(1; +∞). Then x ≥ (a – 1)/3.
Case 2. Let a/(a – 1) = 0, i.e. a = 0. Then x is any real number.
Case 3. Let a/(a – 1)< 0 или а € (0; 1). Тогда x ≤ (а – 1)/3.
Answer: x € [(a – 1)/3; +∞) for a € (-∞; 0)ᴗ(1; +∞);
x € [-∞; (a – 1)/3] for a € (0; 1);
x € R for a = 0.
Example 3.
Solve the inequality |1 + x| ≤ ax relative to x.
Solution.
It follows from the condition that the right-hand side of the inequality ax must be non-negative, i.e. ax ≥ 0. By the rule of revealing the module from the inequality |1 + x| ≤ ax we have a double inequality
Ax ≤ 1 + x ≤ ax. Let's rewrite the result in the form of a system:
(ax ≥ 1 + x;
(-ax ≤ 1 + x.
Let's transform it to:
((a – 1)x ≥ 1;
((a + 1)x ≥ -1.
We study the resulting system on intervals and at points (Fig. 1):
For a ≤ -1 x € (-∞; 1/(a – 1)].
At -1< а < 0 x € [-1/(а – 1); 1/(а – 1)].
When a = 0 x = -1.
At 0< а ≤ 1 решений нет.
Graphical method for solving inequalities
Plotting graphs greatly simplifies solving equations containing a parameter. Using the graphical method when solving inequalities with a parameter is even clearer and more expedient.
Graphically solving inequalities of the form f(x) ≥ g(x) means finding the values of the variable x for which the graph of the function f(x) lies above the graph of the function g(x). To do this, it is always necessary to find the intersection points of the graphs (if they exist).
Example 1.
Solve the inequality |x + 5|< bx.
Solution.
We build graphs of functions y = |x + 5| and y = bx (Fig. 2). The solution to the inequality will be those values of the variable x for which the graph of the function y = |x + 5| will be below the graph of the function y = bx. 
The picture shows:
1) For b > 1 the lines intersect. The abscissa of the intersection point of the graphs of these functions is the solution to the equation x + 5 = bx, whence x = 5/(b – 1). The graph y = bx is located above at x from the interval (5/(b – 1); +∞), which means this set is the solution to the inequality.
2) Similarly, we find that at -1< b < 0 решением является х из интервала (-5/(b + 1); 5/(b – 1)).
3) For b ≤ -1 x € (-∞; 5/(b – 1)).
4) For 0 ≤ b ≤ 1, the graphs do not intersect, which means that the inequality has no solutions.
Answer: x € (-∞; 5/(b – 1)) for b ≤ -1;
x € (-5/(b + 1); 5/(b – 1)) at -1< b < 0;
there are no solutions for 0 ≤ b ≤ 1; x € (5/(b – 1); +∞) for b > 1.
Example 2.
Solve the inequality a(a + 1)x > (a + 1)(a + 4).
Solution.
1) Let’s find the “control” values for parameter a: a 1 = 0, and 2 = -1.
2) Let’s solve this inequality on each subset of real numbers: (-∞; -1); (-1); (-10); (0); (0; +∞).
a) a< -1, из данного неравенства следует, что х >(a + 4)/a;
b) a = -1, then this inequality will take the form 0 x > 0 – there are no solutions;
c) -1< a < 0, из данного неравенства следует, что х < (a + 4)/a;
d) a = 0, then this inequality has the form 0 x > 4 – there are no solutions;
e) a > 0, from this inequality it follows that x > (a + 4)/a.
Example 3.
Solve the inequality |2 – |x||< a – x.
Solution.
We build a graph of the function y = |2 – |x|| (Fig. 3) and consider all possible cases of the location of the straight line y = -x + a. 
Answer: the inequality has no solutions for a ≤ -2;
x € (-∞; (a – 2)/2) for a € (-2; 2];
x € (-∞; (a + 2)/2) for a > 2.
When solving various problems, equations and inequalities with parameters, a significant number of heuristic techniques are discovered, which can then be successfully applied in any other branches of mathematics.
Problems with parameters play an important role in the formation of logical thinking and mathematical culture. That is why, having mastered the methods of solving problems with parameters, you will successfully cope with other problems.
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