Numerical inequalities and their properties. Linear inequalities. Detailed theory with examples Collection and use of personal information
The field of real numbers has the property of order (item 6, p. 35): for any numbers a, b, one and only one of the three relations holds: or . In this case, the notation a > b means that the difference is positive, and the notation difference is negative. Unlike the field of real numbers, the field of complex numbers is not ordered: for complex numbers, the concepts "greater than" and "less than" are not defined; therefore, this chapter deals only with real numbers.
We call the relations inequalities, the numbers a and b are members (or parts) of the inequality, the signs > (greater than) and Inequalities a > b and c > d are called inequalities of the same (or the same) meaning; inequalities a > b and c It immediately follows from the definition of the inequality that
1) any positive number greater than zero;
2) any negative number less than zero;
3) any positive number is greater than any negative number;
4) of two negative numbers, the one whose absolute value is smaller is greater.
All these statements admit a simple geometric interpretation. Let the positive direction of the number axis go to the right of the starting point; then, whatever the signs of the numbers, the larger of them is represented by a point lying to the right of the point representing the smaller number.
Inequalities have the following main properties.
1. Asymmetry (irreversibility): if , then , and vice versa.
Indeed, if the difference is positive, then the difference is negative. They say that when the terms of the inequality are rearranged, the meaning of the inequality must be changed to the opposite.
2. Transitivity: if , then . Indeed, the positivity of the differences implies the positivity
In addition to inequality signs, inequality signs and are also used. They are defined as follows: a record means that either or Therefore, for example, you can write and also. Usually, inequalities written with signs are called strict inequalities, and those written with signs are called non-strict inequalities. Accordingly, the signs themselves are called signs of strict or non-strict inequality. Properties 1 and 2 discussed above are also true for non-strict inequalities.
Consider now the operations that can be performed on one or more inequalities.
3. From the addition of the same number to the members of the inequality, the meaning of the inequality does not change.
Proof. Let an inequality and an arbitrary number be given. By definition, the difference is positive. We add to this number two opposite numbers from which it will not change, i.e.
This equality can be rewritten like this:
It follows from this that the difference is positive, that is, that
and this was to be proved.
This is the basis for the possibility of skew any term of the inequality from one of its parts to another with the opposite sign. For example, from the inequality
follows that
4. When multiplying the terms of the inequality by the same positive number, the meaning of the inequality does not change; when the terms of the inequality are multiplied by the same negative number, the meaning of the inequality changes to the opposite.
Proof. Let then If then since the product of positive numbers is positive. Expanding the brackets on the left side of the last inequality, we obtain , i.e. . The case is considered in a similar way.
Exactly the same conclusion can be drawn regarding the division of the parts of the inequality by some non-zero number, since division by a number is equivalent to multiplying by a number and the numbers have the same signs.
5. Let the terms of the inequality be positive. Then, when its members are raised to the same positive power, the meaning of the inequality does not change.
Proof. Let in this case, by the property of transitivity, and . Then, due to the monotonic increase of the power function at and positive, we have
In particular, if where is a natural number, then we get
i.e., when extracting the root from both parts of the inequality with positive terms, the meaning of the inequality does not change.
Let the terms of the inequality be negative. Then it is easy to prove that when its members are raised to an odd natural power, the meaning of the inequality does not change, and when it is raised to an even natural power, it changes to the opposite. From inequalities with negative terms, you can also extract the root of an odd degree.
Let, further, the terms of the inequality have different signs. Then, when it is raised to an odd power, the meaning of the inequality does not change, and when it is raised to an even power, nothing definite can be said in the general case about the meaning of the resulting inequality. Indeed, when a number is raised to an odd power, the sign of the number is preserved and therefore the meaning of the inequality does not change. When raising the inequality to an even power, an inequality with positive terms is formed, and its meaning will depend on the absolute values of the terms of the original inequality, an inequality of the same meaning as the original one, an inequality of the opposite meaning, and even equality!
It is useful to check everything that has been said about raising inequalities to a power using the following example.
Example 1. Raise the following inequalities to the indicated power, changing the inequality sign to the opposite or to the equal sign, if necessary.
a) 3 > 2 to the power of 4; b) to the power of 3;
c) to the power of 3; d) to the power of 2;
e) to the power of 5; e) to the power of 4;
g) 2 > -3 to the power of 2; h) to the power of 2,
6. From inequality, you can go to the inequality between if the terms of the inequality are both positive or both negative, then between their reciprocals there is an inequality of the opposite meaning:
Proof. If a and b are of the same sign, then their product is positive. Divide by inequality
i.e., which was required to obtain.
If the terms of the inequality have opposite signs, then the inequality between their reciprocals has the same meaning, since the signs of the reciprocals are the same as the signs of the quantities themselves.
Example 2. Check the last property 6 on the following inequalities:
7. The logarithm of inequalities can be performed only in the case when the terms of the inequalities are positive (negative numbers and zero do not have logarithms).
Let be . Then when will
and when will
The correctness of these statements is based on the monotonicity of the logarithmic function, which increases if the base and decreases if
So, when taking the logarithm of an inequality consisting of positive terms, with a base greater than one, an inequality of the same meaning as the given one is formed, and when taking its logarithm with a positive base less than one, an inequality of the opposite meaning is formed.
8. If , then if , but , then .
This immediately follows from the monotonicity properties of the exponential function (Sec. 42), which increases in the case and decreases if
When adding inequalities of the same meaning term by term, an inequality of the same meaning as the data is formed.
Proof. Let us prove this statement for two inequalities, although it is true for any number of summed inequalities. Let the inequalities
By definition, numbers will be positive; then their sum also turns out to be positive, i.e.
Grouping the terms differently, we get
and hence
and this was to be proved.
Nothing definite can be said in the general case about the meaning of an inequality resulting from the addition of two or more inequalities of different meanings.
10. If another inequality of opposite meaning is subtracted term by term from one inequality, then an inequality of the same meaning as the first is formed.
Proof. Let two inequalities of different meanings be given. The second of them, by the property of irreversibility, can be rewritten as follows: d > c. Let us now add two inequalities of the same meaning and obtain the inequality
the same meaning. From the latter we find
and this was to be proved.
Nothing definite can be said in the general case about the meaning of an inequality obtained by subtracting another inequality of the same meaning from one inequality.
It is customary to call a system of inequalities a record of several inequalities under the sign of a curly bracket (in this case, the number and type of inequalities included in the system can be arbitrary).
To solve the system, it is necessary to find the intersection of the solutions of all the inequalities included in it. A solution to an inequality in mathematics is any value of a variable for which the given inequality is true. In other words, it is required to find the set of all its solutions - it will be called the answer. As an example, let's try to learn how to solve a system of inequalities using the interval method.
Properties of inequalities
To solve the problem, it is important to know the basic properties inherent in inequalities, which can be formulated as follows:
- To both parts of the inequality, one and the same function can be added, defined in the area of admissible values (ODV) of this inequality;
- If f(x) > g(x) and h(x) is any function defined in the DDE of the inequality, then f(x) + h(x) > g(x) + h(x);
- If both parts of the inequality are multiplied by a positive function defined in the ODZ of the given inequality (or by a positive number), then we obtain an inequality equivalent to the original one;
- If both parts of the inequality are multiplied by the negative function defined in the ODZ of the given inequality (or by a negative number) and the sign of the inequality is reversed, then the resulting inequality is equivalent to the given inequality;
- Inequalities of the same meaning can be added term by term, and inequalities of the opposite meaning can be subtracted term by term;
- Inequalities of the same meaning with positive parts can be multiplied term by term, and inequalities formed by non-negative functions can be raised term by term to a positive power.
To solve a system of inequalities, you need to solve each inequality separately, and then compare them. As a result, a positive or negative answer will be received, which means whether the system has a solution or not.
Spacing method
When solving a system of inequalities, mathematicians often resort to the interval method, as one of the most effective. It allows us to reduce the solution of the inequality f(x) > 0 (<, <, >) to the solution of the equation f(x) = 0.
The essence of the method is as follows:
- Find the range of acceptable values of inequality;
- Reduce the inequality to the form f(x) > 0(<, <, >), that is, move the right side to the left and simplify;
- Solve the equation f(x) = 0;
- Draw a diagram of a function on a number line. All points marked on the ODZ and limiting it divide this set into so-called intervals of constant sign. On each such interval, the sign of the function f(x) is determined;
- Write the answer as a union of separate sets on which f(x) has the corresponding sign. ODZ points that are boundary are included (or not included) in the answer after additional checking.
Inequality is a notation 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 ⩾ 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 whose parts are connected by the sign or .
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 with the smaller absolute value 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 means 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 that are a consequence of another inequality. For example, if a less b, then b more a:
a < b And b > a- equivalent inequalities
Properties of inequalities
- If the same number is added to both parts of the inequality or the same number is subtracted from both parts, then an equivalent inequality will be obtained, 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 parts of the inequality are multiplied or divided by the same positive number, then an equivalent inequality will be obtained, that is,
- If both parts 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 inequality sign 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 reversing the sign of the inequality:
-a + b > -c
(-a + b) · -one< (-c) · -one
a - b < c
Inequality -a + b > -c is equivalent to the inequality a - b < c
1 . If a > b, then b< a ; vice versa if but< b , then b > a.
Example. If 5x - 1 > 2x + 1, then 2x +1< 5x — 1 .
2 . If a > b And b > c, then a > c. Similar, but< b And b< с , then a< с .
Example. From the inequalities x > 2y, 2y > 10 follows that x>10.
3
. If a > b then a + c > b + c And a - c > b - c. If but< b
, then a + c
Example 1. Given the inequality x + 8>3. Subtracting the number 8 from both parts of the inequality, we find x > - 5.
Example 2. Given the inequality x - 6< — 2 . Adding 6 to both parts, we find X< 4 .
4 . If a > b And c > d then a + c > b + d; exactly the same if but< b And from< d , then 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, then a1 + a2 + a3 > b1+b2 +b3.
Example 1. inequalities — 8 > — 10 And 5 > 2 are true. Adding them term by term, we find the correct 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 term by term, we find x< 22 .
Comment. Two inequalities of the same meaning cannot be subtracted term by term from each other, since the result may be true, but it may also be wrong. For example, if from the inequality 10 > 8 2 > 1 , then we get the correct inequality 8 > 7 but if from the same inequality 10 > 8 subtract inequality term by term 6 > 1 , then we get an absurdity. Compare next item.
5 . If a > b And c< d , then a - c > b - d; if but< b And c - d, then a - c< b — d , i.e., one inequality can be subtracted 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 term by term the second from the first and leaving the sign of the first, we obtain the correct inequality — 3 < 13 . Subtracting term by term the first from the second 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 parts of the inequality can be divided or multiplied by the same positive number (the inequality sign remains the same). If a > b And n is a negative number, then na< nb And a/n< b/n , i.e. both parts of the inequality can be multiplied or divided by the same negative number, but the inequality sign must be reversed.
Example 1. Dividing both sides of the true inequality 25 > 20 on the 5 , we obtain the correct inequality 5 > 4 . If we divide both sides of the inequality 25 > 20 on the — 5 , then you need to change the sign > on the < , 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)x - (1/3)x > 4 follows that x< — 12 .
Example 4. Given the inequality x/k > y/l; it follows that lx > ky if signs of numbers l And k are the same and that lx< ky if signs of numbers l And k are opposite.
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