Actions on inequalities. Linear inequalities

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 a > b and c > d then a + c > b + d; exactly the same if a< b and with< 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 a > b and c< d , then a - c > b - d; if a< 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 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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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 can add the same function defined in the range of acceptable 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 method of intervals, 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.

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 list and justify properties of numerical inequalities, on which all the principles of working with inequalities are based.

We note right away that many properties of numerical inequalities are similar. Therefore, we will present the material according to the same scheme: we formulate the property, give its justification and examples, and then proceed to the next property.

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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 signs not equal ≠, less than<, больше >, less than or equal to ≤ or greater than or equal to ≥. Based on the above definition, it is convenient to define the numerical inequality:

The meeting with numerical inequalities takes place in mathematics lessons in the first grade immediately after getting acquainted with the first natural numbers from 1 to 9, and getting acquainted with 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 extends to other types of numbers (integer, 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 a number of properties of numerical inequalities. They follow from the concept of inequality introduced by us. In relation to numbers, this concept is given by the following statement, which can be considered the definition of the relations "less than" and "greater than" 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 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 recast into a definition of less than or equal to and greater than or equal to. 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 now review.

Basic properties

We begin our review with three basic 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 >, characteristically:

As for the numerical inequalities written using the signs of non-strict inequalities ≤ and ≥, they have the property of reflexivity (rather than 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 with 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 of great practical importance. Methods for evaluating the values ​​of expressions are based on them, the principles of solution of inequalities etc. Therefore, it is advisable to deal well with them.

In this subsection, we will formulate the properties of inequalities only for one sign of strict inequality, but it should be borne in mind that similar properties will also 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

• if a>b , then a+c>b+c ;
• if a≤b , then a+c≤b+c ;
• if a≥b , then a+c≥b+c .

For convenience, we present the properties of numerical inequalities in the form of a list, while giving the corresponding statement, writing it formally using letters, giving a proof, and then showing examples of use. And at the end of the article we will summarize all the properties of numerical inequalities in a table. Go!