Presentation on the topic of the main properties of numerical inequalities. Mathematics presentation "Numeric inequalities and their properties
Algebra class 8 important role The theme is "Inequality". Therefore, its deep study is extremely important. On the basis of this theory, a number of the most difficult problems are solved, and not only in the course of algebra, but also in other sciences.
This presentation is intended to study the properties of numerical inequalities. Moreover, before the lesson at which this presentation will be considered, a lesson should be held where the properties themselves will be given. To do this, you can also take the presentation “Properties of numerical inequalities. Part 1 ”, where the whole theory on this topic is given. Here you can find many different examples, where the studied properties are applicable. So, in more detail.
slides 1-2 (Presentation topic "Properties of numerical inequalities. Part 2", property)
The first example shows how to prove an inequality using the definition of inequality and some operations on fractions.
The next example also shows the proof of the inequality, which is a bit more complicated. To prove inequality, you need to apply the knowledge and skills of how fractions are added to numbers. That is, you need to be able to bring fractions to a common denominator and add them. And again, a definition is used that says that if the right side of the inequality is subtracted from the left side when the sign is greater, then a positive value should be obtained, which the author comes to as a result. So, the inequality is proven.
slides 3-4 (properties)
In the third example, it is required to find estimates of numbers, which are given seven pieces, if some certain conditions are given. If you go in order, you will notice that when solving these examples, several properties are applied at once. This is the property of multiplying an inequality by a positive and negative number, adding and subtracting two inequalities, raising to a power. The author considers each example in some detail, which allows you to thoroughly assimilate the proposed material and consolidate it with examples.
slides 5-6 (properties)
The next, fourth example is already more complicated than the previous ones. Present here Square root. In the proof, the author again uses the definition of inequalities. In other words, it finds the difference between the left and right sides of the inequality and determines the sign. During the proof, when found common denominator, in the numerator an expression is obtained that can be collapsed by the formula of the square of the difference of two expressions.
The result is a positive expression, which confirms the inequality sign. But here the sign is not strict, so the author checks the equality condition. As a result, it turns out that in order for the expressions to be equal, both numbers given in the condition must be equal, but this is not specified by the condition. Therefore, the inequality has a strictly greater sign for different meanings numbers a and b.
slides 7-8 (properties)
Further, the author demonstrates this example clearly. That is, the left side of this inequality is the arithmetic mean of the given numbers, and the right side is the geometric mean of the same numbers. It follows that the arithmetic mean of two non-negative numbers is not less than their geometric mean. And this is the Cauchy inequality. Here the author draws attention to the remark, which is shown in the figure.
In the last, fifth example, the author suggests comparing numbers. But these numbers are not prime. Here there is a sum, where one of the terms is the square root of the number. Therefore, there is no way to do without properties in order to complete the task. AT this example two cases. In the first case, the author proposes to square both numbers, which is allowed by the properties studied earlier. As a result, new numbers are obtained, which differ in that 9 is added to the same number different number. It remains to compare these two numbers. In the second case, the author proposes to compare the terms in pairs from both parts of the inequality. It turns out that the first and second terms of the first number are less than the first and second terms of the second number, respectively. So the sign is clear.
slide 9 (properties)
The presentation can be used in the lesson of learning new material as an example where the learned properties can be applied. Also, the presentation is suitable for a lesson in consolidating the material studied in the last lesson. It is also suitable for optional or extracurricular activities. At the request of the teacher, the presentation can be supplemented.
"Inequalities"
Presentation of a mathematics teacher of the 1st category
MOU GOOSh, Kalyazina, Tver region
b , or a or a ≥ b , or a ≤ b , set between numbers, then we say that a numerical inequality is specified." width="640"
Numerical inequality
- Inequality is one of the fundamental concepts of mathematics.
- If two real numbers a and b connected by an inequality sign ≠ or one of the order relations a b, or
a or a ≥ b, or a ≤ b, established between numbers, then they say that given numerical inequality .
- If a a b- it means that a-b – positive number ;
- If a a - it means that a-b – a negative number ;
b, c d (or a Inequalities of the form a d and c" width="640"
Equal and Opposite Inequalities
inequalities
Inequalities of opposite meaning
Inequalities of the same meaning
Inequalities of the form
a b, c d (or a
Inequalities of the form a d and
, Relationship inequalities ≥ , ≤ are called non-strict are called strict" width="640"
Strict and non-strict inequalities
inequalities
Non-strict
Strict
Relationship inequalities ,
Relationship inequalities ≥ , ≤ called non-strict
called strict
b and b c , then a c Proof. 1) a b - by condition, i.e. a - b is a positive number. 2) b c - by condition, i.e. b - c is a positive number. 3) Adding positive numbers a - b and b - c, we get a positive number. 4) Therefore, (a - b) + (b - c) = a - c . So a - c is a positive number, i.e. a c " width="640"
Properties of numerical inequalities
- Property 1 .
If a b and b c , then a c
- Proof.
1) a b - by condition, i.e. a-b is a positive number.
2) b c - by condition, i.e. b-c is a positive number.
3) Adding positive numbers a - b and b - c, we get a positive number.
4) Therefore, (a - b) + (b - c) = a - c . So a - c is a positive number, i.e. a c , which was to be proved.
b means that point a is located to the right of point b on the number line, and inequality b c means that point b is located to the right of point c . But then the point a is located on the straight line to the right of the point c, i.e. a c . This property is called the property of transitivity (Figuratively speaking, from point a we get to point c as if in transit, with an intermediate stop at point b) x c b a" width="640"
Justification of property 1, using a number line
Inequality a b means that on the number line the point a located to the right of the point b, and the inequality bc- what's the point b located to the right of the point c. But then the point a located on a straight line to the right of the point c, i.e. a c . This property is called property of transitivity(Figuratively speaking, from point a we get to point c as if in transit, with an intermediate stop at point b)
b , then a + c b + c That is, if the same number is added to both parts of the inequality, then the inequality sign will not change. Example: 6 4 , if 2 is added to both parts of the inequality, then the inequality sign will not change. The following expression will be obtained: 8 6. Based on the first property, we can conclude that any term can be transferred from one part to another by changing its sign to the opposite. Example: 5
Property 2.
- If a a b, then a + c b + c
That is, If the same number is added to both sides of the inequality, then the sign of the inequality does not change.
Example:
6 4 , if 2 is added to both parts of the inequality, then the sign of the inequality will not change. The following expression will be obtained: 8 6. Based on the first property, we can conclude that any term can be transferred from one part to another by changing its sign to the opposite. Example :
5
b and m 0 , then a b m m That is, if both parts of the inequality are divided by the same positive number, then the inequality sign should be preserved; Example: a b, then a b If a b and m 0, then am bm That is, if both parts of the inequality are multiplied by the same positive number, then the inequality sign should be preserved; Example: a b , then 8a 8b If a b and m 0 , then am . That is, if both parts of the inequality are multiplied by the same negative number, then the inequality sign should be changed (, to Example: a, then -9a -9b; If a b, then -a; That is, if you change the signs of both parts inequality, then the inequality sign must also be changed. 8 8" width="640"
Property 3.
- If a a b and m 0 , then a b
That is, if both parts of the inequality are divided by the same positive number, then the inequality sign should be preserved;
Example: a b , then a b
- If a a b and m 0 , then am bm
That is, if both parts of the inequality are multiplied by the same positive number, then the inequality sign should be preserved;
Example: a b , then 8a 8b
- If a a b and m 0 , then am .
That is, if both parts of the inequality are multiplied by the same negative number, then the inequality sign should be changed (, to
Example: a , then -9a -9b ;
- If a a b, then -a ;
That is, If you change the signs of both parts of the inequality, then you must also change the sign of the inequality.
b and c d , then a + c b + d. Proof. I way. 1. a b and c d - by condition, so a - b and c - d are positive numbers. 2. Then their sum, i.e. (a - b) + (c - d) is a positive number. 3. Since (a-b) + (c-d) \u003d (a + c) - (b + d) , then (a + c) - (b + d) is a positive number. Therefore, a + c b + d , which was to be proved. II way. 1. Since a b, then a + c b + c - by property 2. 2. Similarly, since c d , then c + b d + b . 3. So, a + c b + c, b + c b + d. Then, due to the property of transitivity, we obtain that a + c b + d , which was required to prove." width="640"
Property 4.
- If a a b and c d, then a + c b + d.
Proof.
- I way.
1. a b and with d- by condition, that is, a - b and c - d - positive numbers .
2. Then so is their sum, i.e. (a - b) + (c - d) - positive number .
3. Since (a-b) + (c-d) = (a + c)-(b + d), then and (a + c) - (b + d) - positive number. That's why a + c b + d, which was to be proved.
- II way.
1.Because a b, then a + c b + c – by property 2 .
2. Similarly, since with d, then c + b d + b .
3.So, a + c b + c, b + c b + d . Then, due to the property of transitivity, we get that a + c b + d, which was to be proved.
b , c d , then ac bd . That is, when multiplying inequalities of the same meaning, in which the left and right parts are positive numbers, we get an inequality of the same meaning. Proof. 1. Since a b and c 0 , then ac bc - by property 3. 2. Since with d and b 0 , then cb db - by property 3. 3. So, ac bc , bc bd . Then ac bd - by the property of transitivity, which was required to prove." width="640"
Property 5.
If a a, b, c, d – positive numbers and a b , c d, then ac bd .
That is, when multiplying inequalities of the same meaning, in which the left and right parts are positive numbers, we get an inequality of the same meaning.
Proof.
1.Because a b and c 0, then ac bc - by property 3.
2.Because with d and b 0, then cb db - by property 3.
3. So ac bc , bc bd. Then ac bd - by the property of transitivity, which was to be proved.
b , then a n b n , where n is any natural number. That is, if both parts of the inequality are non-negative numbers, then they can be raised to the same natural degree, keeping the inequality sign. Addition: If n - odd number, then for any numbers a and b from the inequality a b follows the inequality of the same meaning a n b n ." width="640"
Property 6.
- If a a and b - non-negative numbers and a b, then a n b n, where n- any natural number .
That is, if both sides of the inequality are non-negative numbers , then they can be raised to the same natural power, preserving the inequality sign.
- Addition:
If a n - odd number, then for any numbers a and b from inequality a b follows an inequality of the same meaning a n b n .
b. Prove that Solution. Consider the difference We have: By condition, a, b, a - b are positive numbers. Hence, is a negative number, i.e. which implies that "width="640"
- Let a and b - positive numbers and a b .
Prove that
- Solution.
Consider the difference
We have:
By the condition, a, b, a - b are positive numbers. Means,
- a negative number, those.
whence it follows that
- Let a - positive number .
Prove that
- Solution.
We got a non-negative number, which means
notice, that
- Let a and b non-negative numbers. Prove that
- Solution.
Compose the difference between the left and right parts of the inequality. We have
In this case, the number
called arithmetic mean numbers a and b ;
The number is called geometric mean numbers a and b .
In this way , the arithmetic mean of two non-negative numbers is not less than their geometric mean. The proved inequality is sometimes called Cauchy inequality in honor of the 19th century French mathematician August Cauchy.
Comment . Cauchy's inequality has an interesting geometric interpretation. Let a right triangle be given and let the height h drawn from the vertex right angle, divides the hypotenuse into segments a and b (Fig. 116). It has been proved in geometry that
(so it is no coincidence that the term “geometric mean” was introduced for this expression). What is it?
This is the length of half of the hypotenuse. But it is known from geometry that the median m right triangle, drawn from the vertex of the right angle, is exactly equal to half the hypotenuse. Thus, the Cauchy inequality means that the median drawn to the hypotenuse, i.e.,
not less than the height drawn to the hypotenuse (i.e.),
An obvious geometric fact (see Fig. 116).
Augustin Louis Cauchy
- Textbook "Algebra" A.G. Mordkovich Grade 8
- http://en.wikipedia.org/wiki
- Yandex pictures
Independent work Option 1 1. Give a definition that the number a is greater than the number b 2. Compare: a) b) a and 8 a 3. Prove the inequality (a - 3) (a + 9)
Theorem 1 If a>b, then b b, then b b, then bb, then bb, then b
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