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1 . If a>b, That b< a ; on the contrary, if A< b , That b > a.
Example. If 5x – 1 > 2x + 1, That 2x +1< 5x — 1 .
2 . If a>b And b > c, That a > c. Similar, A< b And b< с , That a< с .
Example. From inequalities x > 2у, 2y > 10 follows that x >10.
3
. If a > b, That a + c > b + c And a – c > b – c. If A< b
, That a + c
Example 1. Given inequality x + 8>3. Subtracting the number 8 from both sides of the inequality, we find x > - 5.
Example 2. Given inequality x – 6< — 2 . Adding 6 to both sides, we find X< 4 .
4 . If a>b And c > d, That a + c >b + d; exactly the same if A< b And With< d , That a + c< b + d , i.e., two inequalities of the same meaning) can be added term by term. This is true for any number of inequalities, for example if a1 > b1, a2 > b2, a3 > b3, That a1 + a2 + a3 > b1+b2 +b3.
Example 1. Inequalities — 8 > — 10 And 5 > 2 are true. Adding them term by term, we find the true inequality — 3 > — 8 .
Example 2. Given a system of inequalities ( 1/2)x + (1/2)y< 18 ; (1/2)x - (1/2)y< 4 . Adding them up term by term, we find x< 22 .
Comment. Two inequalities of the same meaning cannot be subtracted from each other term by term, since the result may be true, but it may also be incorrect. For example, if from the inequality 10 > 8 2 > 1 , then we obtain the correct inequality 8 > 7 but if from the same inequality 10 > 8 subtract inequality term by term 6 > 1 , then we get absurdity. Compare next point.
5 . If a>b And c< d , That a – c > b – d; If A< b And c - d, That a - c< b — d , that is, from one inequality one can subtract, term by term, another inequality of the opposite meaning), leaving the sign of the inequality from which the other was subtracted.
Example 1. Inequalities 12 < 20 And 15 > 7 are true. Subtracting the second term by term from the first and leaving the sign of the first, we obtain the correct inequality — 3 < 13 . Subtracting the first from the second term by term and leaving the sign of the second, we find the correct inequality 3 > — 13 .
Example 2. Given a system of inequalities (1/2)x + (1/2)y< 18; (1/2)х — (1/2)у > 8 . Subtracting the second from the first inequality, we find y< 10 .
6 . If a > b And m is a positive number, then ma > mb And a/n > b/n, i.e. both sides of the inequality can be divided or multiplied by the same positive number (the sign of the inequality remains the same). If a>b And n is a negative number, then na< nb And a/n< b/n , that is, both sides of the inequality can be multiplied or divided by the same negative number, but the sign of the inequality must be changed to the opposite.
Example 1. Dividing both sides of the true inequality 25 > 20 on 5 , we obtain the correct inequality 5 > 4 . If we divide both sides of the inequality 25 > 20 on — 5 , then you need to change the sign > on < , and then we get the correct inequality — 5 < — 4 .
Example 2. From inequality 2x< 12 follows that X< 6 .
Example 3. From inequality -(1/3)х — (1/3)х > 4 follows that x< — 12 .
Example 4. Given inequality x/k > y/l; it follows from it that lx > ky, if the signs of the numbers l And k are the same, so what lx< ky , if the signs of the numbers l And k opposite.
Inequalities play a prominent role in mathematics. 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 principles of working with inequalities are based.
Let us immediately note that many 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, after which we 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 we called inequalities meaningful algebraic expressions containing the signs not equal to ≠, less than<, 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:
The meeting with numerical inequalities occurs in mathematics lessons in the first grade immediately after getting acquainted with the first natural numbers from 1 to 9, and becoming familiar with the comparison operation. True, there they are simply called inequalities, omitting the definition of “numerical”. For clarity, it wouldn’t 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 extends to other types of numbers (integer, rational, real numbers), the rules for their comparison are studied, and this significantly expands the variety of types of numerical inequalities: −5>−72, 3>−0.275 (7−5, 6) , .
Properties of numerical inequalities
In practice, working with inequalities allows a number of properties of numerical inequalities. They follow from the concept of inequality we introduced. In relation to numbers, this concept is given by the following statement, which can be considered a definition of the relations “less than” and “more than” on a 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 zero.
This definition can be reworked into the definition of the relations “less than or equal to” and “greater than or equal to.” Here is his wording:
Definition.
- number a is greater than or equal to b if and only if a−b is a non-negative number;
- a is less than or equal to b if and only if a−b is a non-positive number.
We will use these definitions when proving the properties of numerical inequalities, to a review of which we proceed.
Basic properties
We begin the review with three main properties of inequalities. Why are they basic? Because they are a reflection of the properties of inequalities in the most general sense, and not only in relation to numerical inequalities.
Numerical inequalities written using signs< и >, characteristic:
As for numerical inequalities written using the weak inequality signs ≤ 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 proof is 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 estimating the values of expressions are based on them; principles are based on them solutions to inequalities and so on. Therefore, it is advisable to understand them well.
In this paragraph, we will formulate the properties of inequalities only for one sign of strict inequality, but it is worth keeping in mind that similar properties will be valid for the opposite sign, as well as for signs of non-strict inequalities. Let's 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, while we will give the corresponding statement, write it 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. Termwise multiplication of identical true inequalities of the form a
Adding (or subtracting) any number to both sides of a true numerical inequality produces a true numerical inequality. In other words, if the numbers a and b are such that a
To prove it, let’s make up 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 subtracting a number c, since on the set of real numbers subtraction can be replaced by adding −c.
For example, if you add the number 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 valid numerical inequality are multiplied (or divided) by the same positive number c, you get a valid 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 inequality will be true. In literal form: if the numbers a and b satisfy the inequality a b·c.
Proof. Let's start with the case when c>0. Let's make up 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's show an example of using the analyzed property on specific numbers. For example, you can have both sides of the correct 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
. From the just discussed property of multiplying both sides of a numerical equality by a number, two practically valuable results follow. So we formulate them in the form of consequences. All the properties discussed above in this paragraph are united by the fact that first a correct numerical inequality is given, and from it, through some manipulations with the parts of the inequality and the sign, another correct numerical inequality is obtained. Now we will present a block of properties in which not one, but several correct numerical inequalities are initially given, and a 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 for the numbers a 1, a 2, …, a n and b 1, b 2, …, b n the following inequalities are true: 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 numerical inequalities of the same sign term by term, both sides of which are represented by positive numbers. In particular, for two inequalities a
To prove it, you can multiply both sides of the inequality a
This property is also true 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 notation for numerical inequalities contains non-positive numbers, then their term-by-term multiplication can lead to incorrect numerical inequalities. For example, 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.
At the end of the article, as promised, we will collect all the studied properties in table of properties of numerical inequalities:
Bibliography.
- Moro M. I.. Mathematics. Textbook for 1 class. beginning school In 2 hours. Part 1. (First half of the year) / M. I. Moro, S. I. Volkova, S. V. Stepanova. - 6th ed. - M.: Education, 2006. - 112 p.: ill.+Add. (2 separate l. ill.). - ISBN 5-09-014951-8.
- Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Mordkovich A. G. Algebra. 8th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
Inequality is a record in which numbers, variables or expressions are connected by a sign<, >, or . That is, 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 can be seen 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 whose parts are connected by a sign< или >. Non-strict inequalities- inequalities in which the parts are connected by the sign or.
Let's consider the basic rules of comparison in algebra:
- Any positive number greater than zero.
- Any negative number is less than zero.
- Of two negative numbers, the one whose absolute value is smaller is greater. For example, -1 > -7.
- a And b positive:
a - b > 0,
That a more b (a > b).
- If the difference of two unequal numbers a And b negative:
a - b < 0,
That a less b (a < b).
- If the number is greater than zero, then it is positive:
a> 0, which means a- positive number.
- If the number is less than zero, then it is negative:
a < 0, значит a- a negative number.
Equivalent inequalities- inequalities that are a consequence of other inequalities. For example, if a less b, That b more a:
a < b And b > a- equivalent inequalities
Properties of inequalities
- If you add the same number to both sides of an inequality or subtract the same number from both sides, you get an equivalent inequality, that is,
If a > b, That a + c > b + c And a - c > b - c
It follows from this that it is possible to transfer terms of inequality from one part to another with the opposite sign. For example, adding to both sides of the inequality a - b > c - d By 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 an equivalent inequality is obtained, that is,
- If both sides of the inequality are multiplied or divided by the same negative number, then the inequality opposite to the given one will be obtained, that is, Therefore, when multiplying or dividing both parts 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 signs of all terms of an inequality by multiplying both sides by -1 and changing the sign of the inequality to the opposite:
-a + b > -c
(-a + b) · -1< (-c) · -1
a - b < c
Inequality -a + b > -c tantamount to inequality a - b < c
