Inequality is a record in which numbers, variables or expressions are connected by a sign<, >, or . That is, an inequality can be called a comparison of numbers, variables, or expressions. Signs < , > , ⩽ and ⩾ are called inequality signs.
Types of inequalities and how they are read:
As you can see from the examples, all inequalities consist of two parts: left and right, connected by one of the inequality signs. Depending on the sign connecting the parts of the inequalities, they are divided into strict and non-strict.
Strict inequalities- inequalities in which parts are connected by a sign< или >. Lax inequalities- inequalities in which parts are connected by the sign or.
Let's consider the basic rules of comparison in algebra:
- Any positive number is greater than zero.
- Any negative number is less than zero.
- Of the two negative numbers, the larger is the one with the lower absolute value. For example, -1> -7.
- a and b positive:
a - b > 0,
That a more b (a > b).
- If the difference between two unequal numbers a and b negative:
a - b < 0,
That a smaller b (a < b).
- If the number is greater than zero, then it is positive:
a> 0, hence a is a positive number.
- If the number is less than zero, then it is negative:
a < 0, значит a- a negative number.
Equivalent inequalities- inequalities resulting from other inequalities. For example, if a smaller b, then b more a:
a < b and b > a- equivalent inequalities
Properties of inequalities
- If you add the same number to both sides of the inequality or subtract the same number from both sides, you get an equivalent inequality, that is,
if a > b, then a + c > b + c and a - c > b - c
It follows from this that it is possible to transfer the terms of the inequality from one part to another with the opposite sign. For example, adding to both sides of the inequality a - b > c - d on d, we get:
a - b > c - d
a - b + d > c - d + d
a - b + d > c
- If both sides of the inequality are multiplied or divided by the same positive number, then we get an equivalent inequality, that is,
- If both sides of the inequality are multiplied or divided by the same negative number, then the inequality is opposite to the given one, that is, Therefore, when multiplying or dividing both sides of the inequality by a negative number, the sign of the inequality must be changed to the opposite.
This property can be used to change the sign of all members of an inequality by multiplying both sides by -1 and reversing the sign of the inequality:
-a + b > -c
(-a + b) · -1< (-c) · -1
a - b < c
Inequality -a + b > -c tantamount to inequality a - b < c
Municipal budgetary educational institution "Kachalinskaya secondary school No. 2"
Ilovlinsky district of the Volgograd region
Developing a lesson using an interactive whiteboard
in algebra for grade 8 students
on this topic"Numerical inequalities"
Mathematic teacher
Postoeva Zh.V.
Stanitsa Kachalinskaya
2009 r.
The lesson on the topic "Numerical inequalities" was developed for pupils of the 8th grade on the basis of the textbook "Algebra" by Yu.N. Makarychev.
Goals:
Continue to improve the skills of applying the reduced multiplication formulas. Print a way to compare numbers and literal expressions. To achieve from students the ability to apply knowledge to perform tasks of the standard type (training exercises), reconstructive-variable type, creative type;
Development of skills in applying knowledge in a specific situation; development of logical thinking, the ability to compare, generalize, correctly formulate tasks and express thoughts; development of students' independent activities.
Fostering interest in the subject through the content of the educational material, fostering such character traits as communicative when working in a group, persistence in achieving goals.
Lesson type: learning new material.
The form: lesson - research.
Equipment:
Interactive whiteboard and multimedia equipment
Lesson structure
Lesson stage
Screenshot of the program window Notebook
To work in the lesson, students are seated in groups of 3-4 people.
Lesson topic message
Communication of the goals and objectives of the lesson.
Activation of the knowledge and skills of students necessary for the perception of new knowledge.
Examples of abbreviated multiplication and comparison of various numbers are repeated:
Decimal fractions,
Ordinary fractions with the same numerators,
Ordinary fractions with different denominators,
Regular and improper fractions.
Natural
Decimal fractions
Ordinary fractions
first the number was smaller second, and the difference was obtained negative .
Oral work on comparing different numbers:
Natural
Decimal fractions
Ordinary fractions
and comparing the resulting differences with zero.
For comparison, numbers are taken such that first the number was more second, and the difference was obtained positive .
Hidden behind the curtain is a conclusion that students must come to on their own.
Oral work on comparing different numbers:
Decimal fractions
Ordinary fractions
and comparing the resulting differences with zero.
For comparison, numbers are taken such that first the number was equals the second, and the difference was obtained equal to zero .
Hidden behind the curtain is a conclusion that students must come to on their own.
The teacher suggests performing an oral comparison exercise if the difference is known.
If students find it difficult to answer, there is a hint behind the screen that you can use.
This exercise is also done orally. Students must give reasons for their answer.
Teacher: who can formulate: when one number is greater than another;
when one number is less than the other
when two numbers are equal.
Who can tell what to do to compare two numbers?
Hidden behind the curtain is the formulation of the method of comparing numbers, which is revealed after the students' answers.
An example for proof is offered - comparison of two letter expressions. The proof is carried out with the students, while the teacher gradually opens the curtain.
The teacher once again returns to the formulation of the method of comparing numbers.
Exercise number 728 is given for the application of knowledge Tasks a) and b) the exercises are performed by students in notebooks and on the board with comments on the solution. Tasks c) and d) are performed independently in groups.
The teacher reviews the solutions in groups, answers the students' questions.
Task a) students solve on the blackboard and in notebooks, b) it is proposed to solve it orally with comments, c) - independently.
Students perform tasks a) and b) in groups. The teacher reviews the solutions, with one of the group explaining the solution.
Task d) is performed on a board with comments.
To consolidate the new material, students are offered questions, after answering which, from behind the screen, rules are drawn for repeated visual perception.
Lesson outcome: comments on the work of students in the lesson, grading, writing homework in diaries.
Inequalities in mathematics play a prominent role. At school, we mainly deal with numerical inequalities, with the definition of which we will begin this article. And then we will list and justify properties of numerical inequalities, on which all the principles of working with inequalities are based.
We note right away that many of the properties of numerical inequalities are similar. Therefore, we will present the material according to the same scheme: we formulate a property, give its justification and examples, and then move on to the next property.
Page navigation.
Numerical inequalities: definition, examples
When we introduced the concept of inequality, we noticed that inequalities are often defined by the way they are written. So inequalities we called meaningful algebraic expressions containing signs not equal to ≠, less<, больше >, less than or equal to ≤ or greater than or equal to ≥. Based on the above definition, it is convenient to give a definition of a numerical inequality:
A meeting with numerical inequalities occurs in mathematics lessons in the first grade immediately after meeting the first natural numbers from 1 to 9, and getting to know the comparison operation. True, there they are simply called inequalities, omitting the definition of "numerical". For clarity, it does not hurt to give a couple of examples of the simplest numerical inequalities from that stage of their study: 1<2 , 5+2>3 .
And further from natural numbers, knowledge is extended to other types of numbers (integers, rational, real numbers), the rules for their comparison are studied, and this significantly expands the species diversity of numerical inequalities: −5> −72, 3> −0.275 (7−5, 6),.
Properties of numerical inequalities
In practice, working with inequalities allows the series properties of numerical inequalities... They follow from the concept of inequality introduced by us. In relation to numbers, this concept is defined by the following statement, which can be considered a definition of the relationship "less" and "more" on the set of numbers (it is often called the difference definition of inequality):
Definition.
- number a is greater than b if and only if the difference a - b is a positive number;
- the number a is less than the number b if and only if the difference a - b is a negative number;
- the number a is equal to the number b if and only if the difference a - b is equal to zero.
This definition can be rewritten to define the less than or equal to and greater than or equal relationship. Here is its wording:
Definition.
- number a is greater than or equal to b if and only if a - b is a non-negative number;
- the number a is less than or equal to the number b if and only if a - b is a non-positive number.
We will use these definitions in proving the properties of numerical inequalities, which we will now review.
Basic properties
We begin our survey with three main properties of inequalities. Why are they essential? Because they are a reflection of the properties of inequalities in the most general sense, and not just in relation to numerical inequalities.
Numerical inequalities written using signs< и >, typically:
As for the numerical inequalities written using the signs of non-strict inequalities ≤ and ≥, they have the property of reflexivity (and not anti-reflexivity), since the inequalities a≤a and a≥a include the case of equality a = a. They are also characterized by antisymmetry and transitivity.
So, numerical inequalities written using the signs ≤ and ≥ have the following properties:
- reflexivity a≥a and a≤a are true inequalities;
- antisymmetry, if a≤b, then b≥a, and if a≥b, then b≤a.
- transitivity, if a≤b and b≤c, then a≤c, and also, if a≥b and b≥c, then a≥c.
Their proofs are very similar to those already given, so we will not dwell on them, but move on to other important properties of numerical inequalities.
Other important properties of numerical inequalities
Let us supplement the basic properties of numerical inequalities with a series of results that are of great practical importance. Methods for evaluating the values of expressions are based on them, the principles are based on them solutions to inequalities etc. Therefore, it is advisable to deal with them well.
In this subsection, the properties of inequalities will be formulated only for one sign of a strict inequality, but it should be borne in mind that similar properties will be valid for the opposite sign, as well as for the signs of non-strict inequalities. Let us explain this with an example. Below we formulate and prove the following property of inequalities: if a
For convenience, we will present the properties of numerical inequalities in the form of a list, in this case we will give the corresponding statement, write it down formally using letters, give a proof, and then show examples of use. And at the end of the article, we will summarize all the properties of numerical inequalities in a table. Go!
Consequence. Term-by-term multiplication of the same true inequalities of the form a
Adding (or subtracting) any number to both sides of a valid numeric inequality produces a valid numeric inequality. In other words, if the numbers a and b are such that a
For the proof, compose the difference between the left and right sides of the last numerical inequality, and show that it is negative under the condition a (a + c) - (b + c) = a + c − b − c = a − b... Since by condition a
We do not dwell on the proof of this property of numerical inequalities for the subtraction of the number c, since the subtraction on the set of real numbers can be replaced by the addition of −c.
For example, if you add 15 to both sides of the correct numerical inequality 7> 3, you get the correct numerical inequality 7 + 15> 3 + 15, which is the same thing, 22> 18.
If both sides of a true numerical inequality are multiplied (or divided) by the same positive number c, then you get the correct numerical inequality. If both sides of the inequality are multiplied (or divided) by a negative number c, and the sign of the inequality is reversed, then the correct inequality is obtained. In literal form: if for numbers a and b the inequality a b c.
Proof. Let's start with the case when c> 0. Let us compose the difference between the left and right sides of the numerical inequality being proved: a c - b c = (a - b) c. Since by condition a 0, then the product (a - b) · c will be a negative number as the product of a negative number a - b and a positive number c (which follows from). Therefore, a c - b c<0
, откуда a·c We do not dwell on the proof of the considered property for dividing both sides of a true numerical inequality by the same number c, since division can always be replaced by multiplication by 1 / c. Let us show an example of applying the analyzed property to concrete numbers. For example, you can both sides of the true numerical inequality 4<6
умножить на положительное число 0,5
, что дает верное числовое неравенство −4·0,5<6·0,5
, откуда −2<3
. А если обе части верного числового неравенства −8≤12
разделить на отрицательное число −4
, и изменить знак неравенства ≤ на противоположный ≥, то получится верное числовое неравенство −8:(−4)≥12:(−4)
, откуда 2≥−3
. Two practically valuable results follow from the property just examined of multiplying both sides of a numerical equality by a number. So we will formulate them in the form of consequences. All the properties discussed above in this subsection are united by the fact that first the correct numerical inequality is given, and from it, by means of some manipulations with the parts of the inequality and the sign, another correct numerical inequality is obtained. Now we will give a block of properties in which not one, but several correct numerical inequalities are initially given, and the new result is obtained from their joint use after adding or multiplying their parts. If the numbers a, b, c, and d satisfy the inequalities a
Let us prove that (a + c) - (b + d) is a negative number, this will prove that a + c By induction, this property extends to term-by-term addition of three, four, and, in general, any finite number of numerical inequalities. So, if the numbers a 1, a 2,…, a n and b 1, b 2,…, b n satisfy the inequalities a 1 a 1 + a 2 +… + a n . For example, we are given three correct numerical inequalities of the same sign −5<−2
, −1<12
и 3<4
. Рассмотренное свойство числовых неравенств позволяет нам констатировать, что неравенство −5+(−1)+3<−2+12+4
– тоже верное. You can multiply term-by-term numerical inequalities of the same sign, both sides of which are represented by positive numbers. In particular, for two inequalities a
For the proof, we can multiply both sides of the inequality a
The indicated property is also valid for the multiplication of any finite number of true numerical inequalities with positive parts. That is, if a 1, a 2, ..., a n and b 1, b 2, ..., b n are positive numbers, and a 1 a 1 · a 2 ·… · a n . Separately, it is worth noting that if the record of numerical inequalities contains non-positive numbers, then their term-by-term multiplication can lead to incorrect numerical inequalities. For example, the numerical inequalities 1<3
и −5<−4
– верные и одного знака, почленное умножение этих неравенств дает 1·(−5)<3·(−4)
, что то же самое, −5<−12
, а это неверное неравенство.
In conclusion of the article, as promised, we will collect all the studied properties in numerical inequality property table:
Bibliography.
- Moro M.I.... Maths. Textbook. for 1 cl. early shk. At 2 o'clock, Part 1. (First half of the year) / M. I. Moro, S. I. Volkova, S. V. Stepanova. - 6th ed. - M .: Education, 2006 .-- 112 p .: ill. + App. (2 separate l. Ill.). - ISBN 5-09-014951-8.
- Maths: textbook. for 5 cl. general education. institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., Erased. - M .: Mnemosina, 2007 .-- 280 p .: ill. ISBN 5-346-00699-0.
- Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.
- A. G. Mordkovich Algebra. 8th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., Erased. - M .: Mnemozina, 2009 .-- 215 p .: ill. ISBN 978-5-346-01155-2.
Lesson 8th grade on the topic "Numerical inequalities"
Goals:
Educational: introduce a definition of the concepts "more" and "less", numerical inequality, teach them to apply them to the proof of inequalities;
Developing: to develop the ability to use theoretical knowledge in solving practical problems, the ability to analyze and generalize the data obtained; develop a cognitive interest in mathematics, broaden horizons;
Educational: to form a positive motivation for learning.
During the classes:
1. Preparation and motivation.
Today we begin to study the important and relevant topic "Numerical Inequalities". If we slightly change the words of the great Chinese teacher Confucius (he lived more than 2400 years ago), we can formulate the task of our lesson: “I hear and forget. I see and remember. I do and understand. "Let's formulate together the purpose of the lesson. (Students formulate a goal, the teacher complements).
Explore numerical inequalities and their definitions, and learn how to apply them in practice.
In practice, we often have to compare values. For example, the area of the territory of Russia (17 098 242 ) and the area of the territory of France (547 030 ) , the length of the Oka river (1500 km) and the length of the Don river (1870 km).
2. Updating basic knowledge .
Guys, let's take a look at everything we know about inequality.
Guys, look at the board, compare:
3.6748 and 3.675
36.5810 and 36.581
and 0.45
5.5 and
15 and -23
115 and -127
What is inequality?
Inequality -the ratio between numbers (or any mathematical expression capable of taking a numerical value), indicating which of them is greater or less than the other.
Inequality signs (›;‹) first appeared in 1631, but the concept of inequality, like the concept of equality, arose in ancient times. In the development of mathematical thought, without comparing quantities, without the concepts of "more" and "less", it was impossible to reach the concept of equality, identity, equation.
What rules were used to compare numbers?
a) of two positive numbers, the greater is the one whose modulus is greater;
b) of two negative numbers, the greater is the one whose modulus is less;
c) any negative number is less than positive;
d) any positive number is greater than zero;
e) any negative number is less than zero.
What rule do we use to compare numbers located on the coordinate line?
(On the coordinate line, the larger number is represented by the point to the right, and the smaller, by the point to the left.)
Note that, depending on the specific type of numbers, we used one or another comparison method. It is not comfortable. It would be easier for us to have a universal way of comparing numbers that would cover all cases.
3. Learning new material.
Arrange in ascending order of numbers: 8; 0; -3; -1.5.
What is the smallest number? What is the largest number?
What numbers can be substituted foraandb?
a - b = 8
a - b = -3
a - b = -8
a - b = 1.5
a - b = 0
Note that subtracting a smaller number from a larger number results in a positive number; subtracting the larger from a smaller number results in a negative number.
The universal way to compare numbers is based on the definition of numerical inequalities: Numberamore numbersbif the differencea – b- positive number; number a is less than numberbif the differencea – b- a negative number. Note that if the differencea – b= 0, then the numbers a andbare equal.
4. Securing new material.
Compare the numbers a andb, if:
A) a -b= - 0.8 (and lessbsince difference - negative number)
B) a -b= 0 (a =b)
B) a -b= 5, 903 (and morebsince difference - positive number).
Solve with an explanation at the board number 724, 725 (orally), 727 (if time permits), 728 (a, d), 729 (c, d), 730, 732.
5. Lesson summary. D / z. learn. def. No. 726, 728 (a, d), 729 (c, d), 731.
Guys, today in the lesson we repeated the previously studied material on inequalities and learned a lot about inequalities.
1) What is “inequality”?
2) How do you compare two numbers?
3) Guys, raise your hand, who had difficulties in the lesson?