Our "Reshebnik" contains answers to all tasks and exercises from the "Didactic Materials for Algebra Grade 8"; the methods and methods of their solution are analyzed in detail. "Reshebnik" is addressed exclusively to parents of students, to check homework and help in solving problems.
In a short time, parents can become quite effective home tutors.

Option 1 4

to polynomial (repetition) 4

C-2. Factoring (review) 5

C-3. Integer and fractional expressions 6

C-4. Basic property of a fraction. Fraction reduction. 7

C-5; Fraction Reduction (continued) 9

with the same denominators 10

with different denominators 12

denominators (continued) 14

C-9. Multiplication of fractions 16

C-10. Division of fractions 17

C-11. All actions with fractions 18

C-12. Feature 19

C-13. Rational and irrational numbers 22

C-14. Arithmetic square root 23

C-15. Solution of equations of the form x2=a 27

C-16. Finding approximate values

square root 29

C-17. Function y=d/x 30

Root product 31

Private Roots 33

S-20. Square root of 34

C-21. Factoring out the root sign Factoring in the root sign 37

C-23. Equations and their roots 42

Incomplete quadratic equations 43

S-25. Solving quadratic equations 45

(continued) 47

C-27. Vieta's theorem 49

C-28. Solving problems with

quadratic equations 50

factors. Biquadratic Equations 51

S-30. Fractional rational equations 53

C-31. Solving problems with

rational equations 58

S-32. Number comparison (review) 59

C-33. Properties of numerical inequalities 60

S-34. Addition and multiplication of inequalities 62

S-35. Proof of inequalities 63

S-36. Expression value evaluation 65

C-37. Approximation Error Estimation 66

S-38. Rounding Numbers 67

S-39. Relative error 68

S-40. Intersection and Union of Sets 68

C-41. Number gaps 69

S-42. Solving inequalities 74

C-43. Solving inequalities (continued) 76

C-44. Solving systems of inequalities 78

S-45. Solving inequalities 81

variable under modulo sign 83

C-47. Degree with integer exponent 87

degree with integer exponent 88

C-49. Standard form of the number 91

S-50. Recording approximate values ​​92

S-51. Elements of statistics 93

(repeat) 95

S-53. Definition of a quadratic function 99

S-54. Function y=ax2 100

S-55. Graph of the function y \u003d ax2 + bx + c 101

S-56. Solving quadratic inequalities 102

S-57. Spacing method 105

Option 2 108

C-1. Converting an Integer Expression

to polynomial (repetition) 108

C-2. Factoring (review) 109

C-3. Integer and fractional expressions 110

C-4. Basic property of a fraction.

Fraction reduction 111

C-5. Fraction Reduction (continued) 112

C-6. Addition and subtraction of fractions

with the same denominators 114

C-7. Addition and subtraction of fractions

e different denominators 116

C-8. Addition and subtraction of fractions with different

denominators (continued) 117

C-9. Multiplication of fractions, 118

C-10. Division of fractions 119

C-11. All actions with fractions 120

C-12. Feature 121

C-13. Rational and irrational numbers 123

C-14. Arithmetic square root 124

C-15. Solution of equations of the form x2-a 127

C-16. Finding Approximate Square Roots 129
C-17. Function y=\/x" 130

C-18. The square root of the product.

Root product 131

C-19. The square root of a fraction.

Private roots 133

S-20. Square root of 134

C-21. Taking the multiplier out from under the sign of the root

Entering a factor under the sign of the root 137

C-22. Expression conversion,

C-23. Equations and their roots 141

S-24. Definition of a quadratic equation.

Incomplete Quadratic Equations 142

S-25. Solving quadratic equations 144

C-26. Solving quadratic equations

(continued) 146

C-27. Vieta's theorem 148

C-28. Solving problems with

quadratic equations 149

C-29. Decomposition of a square trinomial into

factors. Biquadratic Equations 150

S-30. Fractional rational equations 152

C-31. Solving problems with

rational equations 157

S-32. Number comparison (review) 158

C-33. Properties of numerical inequalities 160

S-34. Addition and multiplication of inequalities 161

S-35. Proof of inequalities 162

S-36. Expression Value Evaluation 163

C-37. Approximation Error Estimation 165

S-38. Rounding Numbers 165

S-39. Relative error 166

S-40. Intersection and Union of Sets 166

C-41. Number gaps 167
S-42. Solving inequalities 172

C-43. Solving inequalities (continued) 174

C-44. Solving systems of inequalities 176

S-45. Solving inequalities 179

S-46. Equations and inequalities containing

variable under modulo sign 181

C-47. Degree with integer exponent 185

C-48. Converting expressions containing

degrees with an integer exponent 187

C-49. Standard form of the number 189

S-50. Recording approximate values ​​190

S-51. Elements of statistics 192

S-52. The concept of a function. Function Graph

(repeat) 193

S-53. Definition of a quadratic function 197

S-54. Function y=ax2 199

S-55. Graph of the function y \u003d ax24-bzh + c 200

S-56. Solving quadratic inequalities 201

S-57. Spacing method 203

Examinations 206

Option 1 206

K-10 (final) 232

Option 2 236

K-2A 238
K-ZA 242

K-9A (final) 257

Final repetition by topic 263

Autumn Olympics 274

Spring Olympics 275

summary of other presentations

"Addition and subtraction of algebraic fractions" - Algebraic fractions. 4a?b. Exploring a new topic. Goals: Remember! Kravchenko G. M. Examples:

"Degrees with an integer indicator" - Feoktistov Ilya Evgenievich Moscow. 3. Degree with an integer indicator (5 hours) p.43. Teaching algebra in grade 8 with in-depth study of mathematics. Delayed introduction of an exponent with an integer negative exponent… Know the definition of an exponent with an integer negative exponent. 2.

"Types of quadratic equations" - Incomplete quadratic equations. Questions... Complete quadratic equations. Quadratic equations. Definition of a quadratic equation Types of quadratic equations Solution of quadratic equations. Methods for solving quadratic equations. Group "Discriminant": Mironov A., Migunov D., Zaitsev D., Sidorov E, Ivanov N., Petrov G. The reduced quadratic equation. Completed: students of the 8th "in" class. Full square selection method. Types of quadratic equations. Let be. Graphic way.

"Numerical inequalities grade 8" - A-c> 0. Inequalities. BUT<0 означает, что а – отрицательное число. >= "Greater than or equal to." b>c. Write a>b or a 0. B-c>0. Numerical inequalities. Non-strict. Properties of numerical inequalities. Examples: If a b, then a-5>b-5. А>0 means that a is a positive number;

"The solution of quadratic equations Vieta's theorem" - One of the roots of the equation is 5. Task number 1. MOU "Kislovskaya secondary school". Supervisor: mathematics teacher Barannikova E. A. Kislovka - 2008 (Presentation for an algebra lesson in grade 8). Find x2 and k. The work was done by: 8th grade student Slinko V. Solving quadratic equations using Vieta's theorem.

M.: 2014 - 288s. M.: 2012 - 256s.

"Reshebnik" contains answers to all tasks and exercises from "Didactic materials for algebra grade 8"; the methods and methods of their solution are analyzed in detail. "Reshebnik" is addressed exclusively to parents of students, to check homework and help in solving problems. In a short time, parents can become quite effective home tutors.

Format: pdf (201 4 , 28 8s., Erin V.K.)

The size: 3.5 MB

Watch, download: drive.google

Format: pdf (2012 , 256 p., Morozov A.V.)

The size: 2.1 MB

Watch, download: links removed (see note!!)

Format: pdf(2005 , 224p., Fedoskina N.S.)

The size: 1.7 MB

Watch, download: drive.google

Table of contents
Independent work 4
Option 1 4

to polynomial (repetition) 4
C-2. Factoring (review) 5
C-3. Integer and fractional expressions 6
C-4. Basic property of a fraction. Fraction reduction 7
C-5. Fraction Reduction (continued) 9

with the same denominators 10

with different denominators 12

denominators (continued) 14
C-9. Multiplication of fractions 16
C-10. Division of fractions 17
C-11. All actions with fractions 18
C-12. Feature 19
C-13. Rational and irrational numbers 22
C-14. Arithmetic square root 23
C-15. Solution of equations of the form x2=a 27

square root 29
C-17. Function y=\/x 30

Root product 31

Private Roots 33
S-20. Square root of 34

Entering a factor under the sign of the root 37

containing square roots 39
C-23. Equations and their roots 42

Incomplete quadratic equations 43
S-25. Solving quadratic equations 45

(continued) 47
C-27. Vieta's theorem 49

quadratic equations 50

factors. Biquadratic Equations 51
S-30. Fractional rational equations 53

rational equations 58
S-32. Number comparison (review) 59
C-33. Properties of numerical inequalities 60
S-34. Addition and multiplication of inequalities 62
S-35. Proof of inequalities 63
S-36. Expression value evaluation 65
C-37. Approximation Error Estimation 66
S-38. Rounding Numbers 67
S-39. Relative error 68
S-40. Intersection and Union of Sets 68
C-41. Number gaps 69
S-42. Solving inequalities 74
C-43. Solving inequalities (continued) 76
C-44. Solving systems of inequalities 78
S-45. Solving inequalities 81

variable under modulo sign 83
C-47. Degree with integer exponent 87

degree with integer exponent 88
C-49. Standard form of the number 91
S-50. Recording approximate values ​​92
S-51. Elements of statistics 93

(repeat) 95
S-53. Definition of a quadratic function 99
S-54. Function y=ax2 100
S-55. Graph of the function y \u003d ax2 + bx + c 101
S-56. Solving quadratic inequalities 102
S-57. Spacing method 105
Option 2 108
C-1. Converting an Integer Expression
to polynomial (repetition) 108
C-2. Factoring (review) 109
C-3. Integer and fractional software expressions
C-4. Basic property of a fraction.
Fraction reduction 111
C-5. Fraction Reduction (continued) 112
C-6. Addition and subtraction of fractions
with the same denominators 114
C-7. Addition and subtraction of fractions
with different denominators 116
C-8. Addition and subtraction of fractions with different
denominators (continued) 117
C-9. Multiplication of fractions 118
C-10. Division of fractions 119
C-11. All actions with fractions 120
C-12. Feature 121
C-13. Rational and irrational numbers 123
C-14. Arithmetic square root 124
C-15. Solution of equations of the form x2=a 127
C-16. Finding approximate values
square root 129
C-17. Function y=Vx 130
C-18. The square root of the product.
Root product 131
C-19. The square root of a fraction.
Private roots 133
S-20. Square root of 134
C-21. Taking the multiplier out from under the sign of the root
Entering a factor under the sign of the root 137
C-22. Expression conversion,
containing square roots 138
C-23. Equations and their roots 141
S-24. Definition of a quadratic equation.
Incomplete Quadratic Equations 142
S-25. Solving quadratic equations 144
C-26. Solving quadratic equations
(continued) 146
C-27. Vieta's theorem 148
C-28. Solving problems with
quadratic equations 149
C-29. Decomposition of a square trinomial into
factors. Biquadratic Equations 150
S-30. Fractional rational equations 152
C-31. Solving problems with
rational equations 157
S-32. Number comparison (review) 158
C-33. Properties of numerical inequalities 160
S-34. Addition and multiplication of inequalities 161
S-35. Proof of inequalities 162
S-36. Expression Value Evaluation 163
C-37. Approximation Error Estimation 165
S-38. Rounding Numbers 165
S-39. Relative error 166
S-40. Intersection and Union of Sets 166
C-41. Number gaps 167
S-42. Solving inequalities 172
C-43. Solving inequalities (continued) 174
C-44. Solving systems of inequalities 176
S-45. Solving inequalities 179
S-46. Equations and inequalities containing
variable under modulo sign 181
C-47. Degree with integer exponent 185
C-48. Converting expressions containing
degrees with an integer exponent 187
C-49. Standard form of the number 189
S-50. Recording approximate values ​​190
S-51. Elements of statistics 192
S-52. The concept of a function. Function Graph
(repeat) 193
S-53. Definition of a quadratic function 197
S-54. Function y=ax2 199
S-55. Graph of the function y=ax2+txr+c 200
S-56. Solving quadratic inequalities 201
S-57. Spacing method 203
Examinations 206
Option 1 206
K-1 206
K-2 208
K-3 212
K-4 215
K-5 218
K-6 221
K-7 223
K-8 226
K-9 229
K-10 (final) 232
Option 2 236
K-1A 236
K-2A 238
K-ZA 242
K-4A 243
K-5A 246
K-6A 249
K-7A 252
K-8A 255
K-9A (final) 257
Final repetition by topic 263
Autumn Olympics 274
Spring Olympics 275

ALGEBRA
Lessons for grade 9

LESSON #5

Subject. Termwise addition and multiplication of inequalities. Applying Properties of Numeric Inequalities to Evaluate Expression Values

The purpose of the lesson: to achieve students' assimilation of the content of the concept of "add inequalities term by term" and "multiply inequalities term by term", as well as the content of the properties of numerical inequalities, expressed by the theorems on term-by-term addition and term-by-term multiplication of numerical inequalities and consequences from them. Develop the ability to reproduce the named properties of numerical inequalities and use these properties to evaluate the values ​​of expressions, as well as continue working on developing the skills of proving inequalities, comparing expressions using the definition and properties of numerical inequalities

Type of lesson: mastering knowledge, developing primary skills.

Visibility and equipment: reference abstract No. 5.

During the classes

I. Organizational stage

The teacher checks the readiness of students for the lesson, sets them up for work.

II. Checking homework

Students complete test tasks with subsequent verification.

III. Formulation of the purpose and objectives of the lesson.
Motivation of educational activity of students

For the conscious participation of students in the formulation of the goal of the lesson, it is possible to offer them practical problems of geometric content (for example, to estimate the perimeter and area of ​​a rectangle, the lengths of adjacent sides of which are estimated in the form of double inequalities). During the conversation, the teacher should direct the students' thoughts to the fact that although the tasks are similar to those that were solved in the previous lesson (see lesson number 4, evaluate the meaning of expressions), however, unlike those named, they cannot be solved by the same means, since it is necessary to evaluate the values ​​of expressions containing two (and in the future more) letters. Thus, students are aware of the existence of a contradiction between the knowledge they have received up to this point and the need to solve a certain problem.

The result of the work done is the formulation of the goal of the lesson: to study the question of such properties of inequalities that can be applied in cases similar to those described in the proposed task for students; for which it is necessary to clearly formulate the mathematical language and in verbal form, and then bring the corresponding properties of numerical inequalities and learn how to use them in combination with the previously studied properties of numerical inequalities to solve typical problems.

IV. Updating the basic knowledge and skills of students

oral exercises

1. Compare the numbers a and b if:

1) a - b = -0.2;

2) a - b = 0.002;

3) a \u003d b - 3;

4) a - b \u003d m 2;

5) a \u003d b - m 2.

3. Compare the values ​​of the expressions a + b and abif a \u003d 3, b \u003d 2. Justify your answer. The resulting ratio will be fulfilled if:

1) a = -3, b = -2;

2) a = -3, b = 2?

V. Formation of knowledge

Plan for learning new material

1. The property of term-by-term addition of numerical inequalities (with fine-tuning).

2. Property of term-by-term multiplication of numerical inequalities (with fine-tuning).

3. Consequence. Property on term-by-term multiplication of numerical inequalities (with fine-tuning).

4. Examples of application of proven properties.

Reference Note No. 5

Theorem (property) on term-by-term addition of numerical inequalities
If a b and c d , then a + c b + d .
Bringing .
Theorem (property) on term-by-term multiplication of numerical inequalities
If 0 a b and 0 c d , then ac bd . Bringing . Consequence. If 0 a b , then an bn , where n is a natural number.
Bringing
			(by term-by-term multiplication of numerical inequalities).
Example 1. It is known that 3 a 4; 2 b 3. Estimate the value of the expression: 1) a + b; 2) a - b; 3) b; 4) .
		2) a - b \u003d a + (-b) 2 b 31 ∙ (-1) 2 > -b > -3				(0) 2 b 3
Example 2. Let us prove the inequality (m + n )(mn + 1) > 4mn if m > 0, n > 0.
Bringing Using the inequality (where a ≥ 0, b ≥ 0) and the resulting inequality a + b ≥ 2 (a ≥ 0, b ≥ 0), for m ≥ 0 and n ≥ 0 we have:
m + n ≥ 2, (1) mn + 1 ≥ 2. (2)
By the theorem on the term-by-term multiplication of inequalities, we multiply inequalities (1) and (2) term-by-term. Then we have: (m + n )(mn + 1) ≥ 2∙ 2, (m + n )(mn + 1) ≥ 4, hence (m + n )(mn + 1) ≥ 4mn , where m ≥ 0, n ≥ 0.

Methodological comment

For a conscious perception of new material, the teacher can, at the stage of updating the basic knowledge and skills of students, offer solutions to oral exercises with reproduction, respectively, the definition of comparing numbers and the properties of numerical inequalities studied in previous lessons (see above), as well as considering the corresponding properties of numerical inequalities.

Usually, students learn well the content of theorems on term-by-term addition and multiplication of numerical inequalities, however, work experience indicates that students tend to make certain false generalizations. Therefore, in order to prevent errors in the formation of students' knowledge on this issue by demonstrating examples and counterexamples, the teacher should focus on the following points:

Conscious application of the properties of numerical inequalities is impossible without the ability to write these properties both in mathematical language and in verbal form;

· theorems on term-by-term addition and multiplication of numerical inequalities are fulfilled only for irregularities of the same sign;

the property of term-by-term addition of numerical inequalities is fulfilled under a certain condition (see above) for any numbers, and the theorem of term-by-term multiplication (in the form stated in the reference abstract No. 5) only for positive numbers;

theorems on term-by-term subtraction and term-by-term division of numerical inequalities are not studied, therefore, in cases where it is necessary to evaluate the difference or proportion of expressions, these expressions are presented as a sum or product, respectively, and then, under certain conditions, use the properties of term-by-term addition and multiplication of numerical inequalities .

VI. Formation of skills

oral exercises

1. Add term by term inequalities:

1) a > 2, b > 3;

2) s -2, d 4.

Or can the same inequalities be multiplied term by term? Justify the answer.

2. Multiply the inequalities term by term:

1) a > 2, b > 0.3;

2) c > 2, d > 4.

Or can the same irregularities be added? Justify the answer.

3. Determine and justify whether the statement is correct that if 2 a 3, 1 b 2, then:

1) 3 a + b 5;

2) 2 ab 6;

3) 2 - 1 a - b 3 - 2;

Written exercises

To achieve the didactic goal of the lesson, you should solve the exercises of the following content:

1) add and multiply term by term these numerical inequalities;

2) evaluate the value of the sum, difference, product and quotient of two expressions according to the given estimates of each of these numbers;

3) evaluate the meaning of expressions containing these letters, according to the estimates of each of these letters;

4) prove the inequality using term-by-term addition and multiplication theorems for numerical inequalities and using classical inequalities;

5) to repeat the properties of numerical inequalities studied in previous lessons.

Methodological comment

The written exercises that are offered for solving at this stage of the lesson should contribute to the development of stable skills of term-by-term addition and multiplication of inequalities in simple cases. (At the same time, a very important point is worked out: checking the correspondence of the record of inequalities in the condition of the theorem and the correct recording of the sum and product of the left and right parts of the inequalities. Preparatory work is carried out during oral exercises.) For better assimilation of the material, students should be required to reproduce the studied theorems when commenting actions.

After students have successfully worked through theorems in simple cases, they can gradually move on to more complex cases (for evaluating the difference and quotient of two expressions and more complex expressions). At this stage of the work, the teacher should be careful to ensure that the students do not make typical mistakes, trying to make a difference and estimate the share for their own false rules.

Also in the lesson (of course, if time and the level of assimilation of the content of the material by students allows), attention should be paid to exercises on the application of the studied theorems to prove more complex inequalities.

VII. Lesson summary
Control task

It is known that 4 a 5; 6 b 8. Find incorrect inequalities and correct the mistakes. Justify the answer.

1) 10 a + b 13;

2) -4 a - b -1;

3) 24 ab 13;

4) ;

5) ;

7) 100 a2 + b 2 169?

VIII. Homework

1. Study the theorems on term-by-term addition and multiplication of numerical inequalities (with refinement).

2. Perform exercises of a reproductive nature, similar to those in class work.

3. For repetition: exercises for applying the definition of comparing numbers (for bringing irregularities and for comparing expressions).

In this article, we will analyze, firstly, what is meant by evaluation of the values ​​of an expression or function, and, secondly, how the values ​​of expressions and functions are evaluated. First, we introduce the necessary definitions and concepts. After that, we describe in detail the main methods for obtaining estimates. Along the way, we will give solutions to typical examples.

What does it mean to evaluate the value of an expression?

We could not find in school textbooks an explicit answer to the question of what is meant by evaluation of the value of an expression. Let's try to deal with this ourselves, starting from those bits of information on this topic, which are nevertheless contained in textbooks and in collections of tasks for preparing for the Unified State Examination and entering universities.

Let's see what can be found on the topic of interest to us in books. Here are some quotes:

The first two examples involve evaluations of numbers and numerical expressions. There we are dealing with the evaluation of a single value of an expression. The rest of the examples feature evaluations related to expressions with variables. Each value of a variable from the ODZ for an expression or from some set X of interest to us (which, of course, is a subset of the range of acceptable values) has its own value of the expression. That is, if the ODZ (or the set X) does not consist of a single number, then the expression with a variable corresponds to the set of values ​​of the expression. In this case, we have to talk about the evaluation of not one single value, but the evaluation of all values ​​of the expression on the ODZ (or the set X ). Such an estimate takes place for any value of the expression corresponding to some value of the variable from the ODZ (or the set X ).

For reasoning, we are a little distracted from the search for an answer to the question of what it means to evaluate the value of an expression. The above examples advance us in this matter, and allow us to accept the following two definitions:

Definition

Evaluate the value of a numeric expression- this means to specify a numeric set containing the value to be evaluated. In this case, the specified numeric set will be an evaluation of the value of the numeric expression.

Definition

Evaluate the values ​​of an expression with a variable on the ODZ (or on the set X ) - this means specifying a numerical set containing all the values ​​that the expression takes on the ODZ (or on the set X ). In this case, the specified set will be an evaluation of the values ​​of the expression.

It is easy to see that more than one evaluation can be specified for one expression. For example, a numeric expression can evaluate to , or , or , or , etc. The same applies to expressions with variables. For example, the expression on ODZ can be estimated as , or , or , etc. In this regard, it is worth adding a clarification to the recorded definitions regarding the specified numerical set, which is an assessment: the assessment should not be anyhow, it must meet the goals for which it is found. For example, to solve the equation suitable score . But this estimate is no longer suitable for solving the equation , here the values ​​of the expression should be evaluated differently, for example: .

It is worth noting separately that one of the estimates of the values ​​of the expression f(x) is the range of the corresponding function y=f(x).

In conclusion of this paragraph, let us turn our attention to the form of recording estimates. Usually, estimates are written using inequalities. You must have noticed this.

Evaluation of expression values ​​and evaluation of function values

By analogy with the evaluation of the values ​​of an expression, we can talk about the evaluation of the values ​​of a function. This looks quite natural, especially if we mean functions defined by formulas, because the evaluation of the values ​​of the expression f(x) and the evaluation of the values ​​of the function y=f(x) are essentially the same thing, which is obvious. Moreover, it is often convenient to describe the process of obtaining estimates in terms of estimating the values ​​of a function. In particular, in certain cases, obtaining an estimate of the expression is carried out by finding the largest and smallest values ​​of the corresponding function.

On the Accuracy of Estimates

In the first paragraph of this article, we said that for an expression, many evaluations of its values ​​can take place. Are some of them better than others? It depends on the problem being solved. Let's explain with an example.

For example, using the expression value evaluation methods described in the following paragraphs, you can obtain two expression value evaluations : the first one is , the second is . The labor costs for obtaining these estimates differ significantly. The first of them is practically obvious, and obtaining the second estimate involves finding the smallest value of the radical expression and further using the monotonicity property of the square root extraction function. In some cases, any of the estimates can cope with the solution of the problem. For example, any of our estimates allows us to solve the equation . It is clear that in this case we would limit ourselves to finding the first obvious estimate, and, of course, would not strain ourselves in finding the second estimate. But in other cases, it may turn out that one of the estimates is not suitable for solving the problem. For example, our first estimate does not solve the equation , and the estimate allows you to do this. That is, in this case, the first obvious estimate would not be enough for us, and we would have to find a second estimate.

Thus, we approached the question of the accuracy of estimates. It is possible to define in detail what is meant by estimation accuracy. But for our needs, this is not particularly necessary; a simplified idea of ​​​​the accuracy of the estimate will be enough for us. Let's agree to perceive the accuracy of the estimate as some analogue approximation accuracy. That is, let's consider from two estimates of the values ​​of some expression f(x) the one that is "closer" to the range of the function y=f(x) to be more accurate. In this sense, the score is the most accurate of all possible estimates of the values ​​of the expression , since it coincides with the range of the corresponding function . It is clear that the assessment more accurate estimates . In other words, the score rougher estimates .

Does it make sense to always look for the most accurate estimates? No. And the point here is that comparatively rough estimates are often enough to solve problems. And the main advantage of such estimates over exact estimates is that they are often much easier to obtain.

Basic methods for obtaining estimates

Estimates for the values ​​of basic elementary functions

Estimation of function values ​​y=|x|

In addition to the basic elementary functions, a well-studied and useful in terms of obtaining estimates is function y=|x|. We know the range of this function: ; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.

Algebra and the beginning of mathematical analysis. Grade 10: textbook. for general education institutions: basic and profile. levels / [Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; ed. A. B. Zhizhchenko. - 3rd ed. - M.: Enlightenment, 2010.- 368 p.: Ill.-ISBN 978-5-09-022771-1.

Mathematics. Increased level of the USE-2012 (C1, C3). Thematic tests. Equations, inequalities, systems / edited by F. F. Lysenko, S. Yu. Kulabukhov. - Rostov-on-Don: Legion-M, 2011. - 112 p. - (Getting ready for the exam) ISBN 978-5-91724-094-7

Collection tasks in mathematics for applicants to universities (with solutions). In 2 books. Book. 1. Algebra: Proc. allowance / V. K. Egerev, V. V. Zaitsev, B. A. Kordemsky and others; ed. M. I. Skanavi. - 8th ed., Rev. - M.: Higher. school, 1998. - 528 p.: ill. ISBN 5-06-003524-7