Our "Reshebnik" contains answers to all tasks and exercises from " Didactic materials 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.
Per a short time parents can become quite effective home tutors.
Option 1 4
to polynomial (repetition) 4
C-2. Factoring (review) 5
C-3. Whole and fractional expressions 6
C-4. Basic property of a fraction. Fraction reduction. 7
C-5; Fraction Reduction (continued) 9
With 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. Solution 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 quadratic function 99
S-54. Function y=ax2 100
S-55. Graph of the function y \u003d ax2 + bx + c 101
S-56. Solution square 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 square trinomial on the
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
ALGEBRA
Lessons for grade 9
LESSON #5
Topic. 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 perform test tasks followed by verification.
III. Formulation of the purpose and objectives of the lesson.
Motivation learning activities 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; Why should it be clearly defined? 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 |
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If a b and c d , then a + c b + d . |
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Bringing
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Theorem (property) on term-by-term multiplication of numerical inequalities |
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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. |
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Bringing |
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(by term-by-term multiplication of numerical inequalities). |
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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; four) . |
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2) a - b \u003d a + (-b) 2 b 31 ∙ (-1) 2 > -b > -3
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(0) 2 b 3
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Example 2. Let us prove the inequality (m + n )(mn + 1) > 4mn if m > 0, n > 0. |
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Bringing Using the inequality |
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m + n ≥ 2, (1) mn + 1 ≥ 2. (2) |
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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. |
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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 signs;
the term-by-term addition property of numerical inequalities is satisfied under a certain condition (see above) for any numbers, and the term-by-term multiplication theorem (in the form as stated in 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 notation of inequalities in the condition of the theorem and the correct notation of the sum and product of the left and right parts of the inequalities. Preparatory work is carried out during the performance of oral exercises.) For better assimilation of the material, students should be required to reproduce the studied theorems when commenting on actions.
After students have successfully worked through theorems in simple cases, they can gradually move on to more difficult cases(for evaluating the difference and quotient of two expressions and more complex expressions). At this stage of work, the teacher should carefully monitor that students do not allow common mistakes by trying the difference and evaluating the share behind 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).
35 connects the signs of numbers 3 and 5. Three resonates with vibrations of inspiration and joy, enthusiasm and self-expression. This is the trinity of past, present and future; body, mind and spirit. A person under the sign of the three is energetic, talented, honest, proud and independent.
Five adds to the treasury of the general vibration a share of emotionality and free choice. Among the minuses are excessive sensitivity and frequent mood swings, the negative effect of which is compensated by the optimism of the troika. 35 in in general terms personifies creative energy, favorable opportunities, the desire to change places.
Relationship between number and character
What does the number 35 mean in the fate of a person, if it is determined by the date of birth? It gives him a special charisma that attracts friends and followers to him. Such people are always surrounded by fans who choose them for the role. public figure or informal leader.
The negative side of this numerical combination is that a person uses his authority for personal enrichment. The representatives of 35 have a poorly developed spiritual sphere. Infected with pragmatism and vanity, they are able, regardless of the faces, to “go over their heads” towards the intended goal.
magical properties
The mystical meaning of 35 is due to the fact that it predicts a meeting with a deadly temptation. It is possible to avoid the serious mistakes of such a test only by maintaining calmness and prudence.
Sacred comparisons of the number can be found in the Bible, where it is mentioned 5 times. It was on the thirty-fifth day of the wilderness fast that Lucifer approached Jesus to tempt him.
What does the number 35 mean if it occurs often
If the guardian angels make you see 35 all the time, they show that you are not achieving your goals. You are honest and diligent, but luck bypasses you.
You face countless obstacles and are puzzled about your future. This influence is exerted on your life by the ruler of the number 35 - the planet Saturn. Its hidden action is manifested through the number 8, which is obtained by adding 3 and 5. Perhaps you are deviating from your destiny and playing someone else's role. To find your true calling, listen to what your soul is asking for and follow its unspoken call.
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 the variable from the ODZ for the expression or from some set X of interest to us (which, of course, is a subset of the region allowed values) matches its value of the expression. That is, if the ODZ (or the set X ) does not consist of singular, 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 to specify 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, while obtaining the second estimate involves finding the smallest value root expression and further use of the monotonicity property of the square root 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 main elementary functions, 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.
