Inequalities are called linear the left and right sides of which are linear functions with respect to the unknown quantity. These include, for example, inequalities:

2x-1-x+3; 7x0;

5 >4 - 6x 9- x< x + 5 .

1) Strict inequalities: ax +b>0 or ax+b<0

2) Non-strict inequalities: ax +b≤0 or ax+b≫ 0

Let's analyze this task. One of the sides of the parallelogram is 7cm. What must be the length of the other side so that the perimeter of the parallelogram is greater than 44 cm?

Let the required side be X cm. In this case, the perimeter of the parallelogram will be represented by (14 + 2x) cm. The inequality 14 + 2x > 44 is a mathematical model of the problem of the perimeter of a parallelogram. If we replace the variable in this inequality X on, for example, the number 16, then we obtain the correct numerical inequality 14 + 32 > 44. In this case, they say that the number 16 is a solution to the inequality 14 + 2x > 44.

Solving the inequality name the value of a variable that turns it into a true numerical inequality.

Therefore, each of the numbers is 15.1; 20;73 act as a solution to the inequality 14 + 2x > 44, but the number 10, for example, is not its solution.

Solve inequality means to establish all its solutions or to prove that there are no solutions.

The formulation of the solution to the inequality is similar to the formulation of the root of the equation. And yet it is not customary to designate the “root of inequality.”

The properties of numerical equalities helped us solve equations. Similarly, the properties of numerical inequalities will help solve inequalities.

When solving an equation, we change it to another, more simple equation, but equivalent to the given one. The answer to inequalities is found in a similar way. When changing an equation to an equivalent equation, they use the theorem about transferring terms from one side of the equation to the opposite and about multiplying both sides of the equation by the same non-zero number. When solving an inequality, there is a significant difference between it and an equation, which lies in the fact that any solution to an equation can be verified simply by substitution into the original equation. In inequalities, this method is absent, since it is not possible to substitute countless solutions into the original inequality. Therefore, there is an important concept, these arrows<=>is a sign of equivalent, or equivalent, transformations. The transformation is called equivalent, or equivalent, if they do not change the set of solutions.

Similar rules for solving inequalities.

If we move any term from one part of the inequality to another, replacing its sign with the opposite one, we obtain an inequality equivalent to this one.

If both sides of the inequality are multiplied (divided) by the same positive number, we obtain an inequality equivalent to this one.

If both sides of the inequality are multiplied (divided) by the same negative number, replacing the inequality sign with the opposite one, we obtain an inequality equivalent to the given one.

Using these rules Let us calculate the following inequalities.

1) Let's analyze the inequality 2x - 5 > 9.

This linear inequality, we will find its solution and discuss the basic concepts.

2x - 5 > 9<=>2x>14(5 was moved to the left side with the opposite sign), then we divided everything by 2 and we have x > 7. Let us plot the set of solutions on the axis x

We have obtained a positively directed beam. We note the set of solutions either in the form of inequality x > 7, or in the form of the interval x(7; ∞). What is a particular solution to this inequality? For example, x = 10 is a particular solution to this inequality, x = 12- this is also a particular solution to this inequality.

There are many partial solutions, but our task is to find all the solutions. And there are usually countless solutions.

Let's sort it out example 2:

2) Solve inequality 4a - 11 > a + 13.

Let's solve it: A move it to one side 11 move it to the other side, we get 3a< 24, и в результате после деления обеих частей на 3 the inequality has the form a<8 .

4a - 11 > a + 13<=>3a< 24 <=>a< 8 .

We will also display the set a< 8 , but already on the axis A.

We either write the answer in the form of inequality a< 8, либо A(-∞;8), 8 does not turn on.

It has been necessary to compare quantities and quantities when solving practical problems since ancient times. At the same time, words such as more and less, higher and lower, lighter and heavier, quieter and louder, cheaper and more expensive, etc. appeared, denoting the results of comparing homogeneous quantities.

The concepts of more and less arose in connection with counting objects, measuring and comparing quantities. For example, mathematicians of Ancient Greece knew that the side of any triangle is less than the sum of the other two sides and that the larger side lies opposite the larger angle in a triangle. Archimedes, while calculating the circumference, established that the perimeter of any circle is equal to three times the diameter with an excess that is less than a seventh of the diameter, but more than ten seventy times the diameter.

Symbolically write relationships between numbers and quantities using the signs > and b. Records in which two numbers are connected by one of the signs: > (greater than), You also encountered numerical inequalities in the lower grades. You know that inequalities can be true, or they can be false. For example, \(\frac(1)(2) > \frac(1)(3)\) is a correct numerical inequality, 0.23 > 0.235 is an incorrect numerical inequality.

Inequalities involving unknowns may be true for some values ​​of the unknowns and false for others. For example, the inequality 2x+1>5 is true for x = 3, but false for x = -3. For an inequality with one unknown, you can set the task: solve the inequality. Problems of solving inequalities in practice are posed and solved no less often than problems of solving equations. For example, many economic problems are reduced to the study and solution of systems of linear inequalities. In many branches of mathematics, inequalities are more common than equations.

Some inequalities serve as the only auxiliary means of proving or disproving the existence of a certain object, for example, the root of an equation.

Numerical inequalities

Can you compare integers? decimals. Do you know the rules of comparison? ordinary fractions with the same denominators but different numerators; with the same numerators but different denominators. Here you will learn how to compare any two numbers by finding the sign of their difference.

Comparing numbers is widely used in practice. For example, an economist compares planned indicators with actual ones, a doctor compares a patient’s temperature with normal, a turner compares the dimensions of a machined part with a standard. In all such cases, some numbers are compared. As a result of comparing numbers, numerical inequalities arise.

Definition. Number a more number b, if the difference a-b is positive. Number a less number b, if the difference a-b is negative.

If a is greater than b, then they write: a > b; if a is less than b, then they write: a Thus, the inequality a > b means that the difference a - b is positive, i.e. a - b > 0. Inequality a For any two numbers a and b, from the following three relations a > b, a = b, a To compare the numbers a and b means to find out which of the signs >, = or Theorem. If a > b and b > c, then a > c.

Theorem. If you add the same number to both sides of the inequality, the sign of the inequality will not change.
Consequence. Any term can be moved from one part of the inequality to another by changing the sign of this term to the opposite.

Theorem. If both sides of the inequality are multiplied by the same positive number, then the sign of the inequality does not change. If both sides of the inequality are multiplied by the same negative number, then the sign of the inequality will change to the opposite.
Consequence. If both sides of the inequality are divided by the same positive number, then the sign of the inequality will not change. If both sides of the inequality are divided by the same negative number, then the sign of the inequality will change to the opposite.

You know that numerical equalities can be added and multiplied term by term. Next, you will learn how to perform similar actions with inequalities. The ability to add and multiply inequalities term by term is often used in practice. These actions help solve problems of evaluating and comparing the meanings of expressions.

When deciding various tasks Often you have to add or multiply the left and right sides of inequalities term by term. At the same time, it is sometimes said that inequalities add up or multiply. For example, if a tourist walked more than 20 km on the first day, and more than 25 km on the second, then we can say that in two days he walked more than 45 km. Similarly, if the length of a rectangle is less than 13 cm and the width is less than 5 cm, then we can say that the area of ​​this rectangle is less than 65 cm2.

When considering these examples, the following were used: theorems on addition and multiplication of inequalities:

Theorem. When adding inequalities of the same sign, an inequality of the same sign is obtained: if a > b and c > d, then a + c > b + d.

Theorem. When multiplying inequalities of the same sign, whose left and right sides are positive, an inequality of the same sign is obtained: if a > b, c > d and a, b, c, d are positive numbers, then ac > bd.

Inequalities with the sign > (greater than) and 1/2, 3/4 b, c Along with the signs of strict inequalities > and In the same way, the inequality \(a \geq b \) means that the number a is greater than or equal to b, i.e. .and not less b.

Inequalities containing the \(\geq \) sign or the \(\leq \) sign are called non-strict. For example, \(18 \geq 12 , \; 11 \leq 12 \) are not strict inequalities.

All properties of strict inequalities are also valid for non-strict inequalities. Moreover, if for strict inequalities the signs > were considered opposite and you know that to solve a number of applied problems you have to create a mathematical model in the form of an equation or a system of equations. Next you will find out that mathematical models For solving many problems there are inequalities with unknowns. The concept of solving an inequality will be introduced and how to test whether a given number is a solution to a particular inequality will be shown.

Inequalities of the form
\(ax > b, \quad ax in which a and b are given numbers, and x is an unknown, are called linear inequalities with one unknown.

Definition. The solution to an inequality with one unknown is the value of the unknown at which this inequality becomes a true numerical inequality. Solving an inequality means finding all its solutions or establishing that there are none.

You solved the equations by reducing them to the simplest equations. Similarly, when solving inequalities, one tries to reduce them, using properties, to the form of simple inequalities.

Solving second degree inequalities with one variable

Inequalities of the form
\(ax^2+bx+c >0 \) and \(ax^2+bx+c where x is a variable, a, b and c are some numbers and \(a \neq 0 \), called inequalities of the second degree with one variable.

Solution to inequality
\(ax^2+bx+c >0 \) or \(ax^2+bx+c can be considered as finding intervals in which the function \(y= ax^2+bx+c \) takes positive or negative values To do this, it is enough to analyze how the graph of the function \(y= ax^2+bx+c\) is located in the coordinate plane: where the branches of the parabola are directed - up or down, whether the parabola intersects the x-axis and if it does, then at what points.

Algorithm for solving second degree inequalities with one variable:
1) find the discriminant of the square trinomial \(ax^2+bx+c\) and find out whether the trinomial has roots;
2) if the trinomial has roots, then mark them on the x-axis and through the marked points draw a schematic parabola, the branches of which are directed upward for a > 0 or downward for a 0 or at the bottom for a 3) find intervals on the x-axis for which the points parabolas are located above the x-axis (if they solve the inequality \(ax^2+bx+c >0\)) or below the x-axis (if they solve the inequality
\(ax^2+bx+c Solving inequalities using the interval method

Consider the function
f(x) = (x + 2)(x - 3)(x - 5)

The domain of this function is the set of all numbers. The zeros of the function are the numbers -2, 3, 5. They divide the domain of definition of the function into the intervals \((-\infty; -2), \; (-2; 3), \; (3; 5) \) and \( (5; +\infty)\)

Let us find out what the signs of this function are in each of the indicated intervals.

The expression (x + 2)(x - 3)(x - 5) is the product of three factors. The sign of each of these factors in the intervals under consideration is indicated in the table:

In general, let the function be given by the formula
f(x) = (x-x 1)(x-x 2) ... (x-x n),
where x is a variable, and x 1, x 2, ..., x n are numbers that are not equal to each other. The numbers x 1 , x 2 , ..., x n are the zeros of the function. In each of the intervals into which the domain of definition is divided by zeros of the function, the sign of the function is preserved, and when passing through zero its sign changes.

This property is used to solve inequalities of the form
(x-x 1)(x-x 2) ... (x-x n) > 0,
(x-x 1)(x-x 2) ... (x-x n) where x 1, x 2, ..., x n are numbers not equal to each other

Considered method solving inequalities is called the interval method.

Let us give examples of solving inequalities using the interval method.

Solve inequality:

\(x(0.5-x)(x+4) Obviously, the zeros of the function f(x) = x(0.5-x)(x+4) are the points \(x=0, \; x= \frac(1)(2) , \; x=-4 \)

We plot the zeros of the function on the number axis and calculate the sign on each interval:

We select those intervals at which the function is less than or equal to zero and write down the answer.

Answer:
\(x \in \left(-\infty; \; 1 \right) \cup \left[ 4; \; +\infty \right) \)

The set of all real numbers can be represented as the union of three sets: the set of positive numbers, the set of negative numbers and the set consisting of one number - the number zero. To indicate that the number A positive, use the recording a > 0, to indicate a negative number use another notation a< 0 .

The sum and product of positive numbers are also positive numbers. If the number A negative, then the number -A positive (and vice versa). For any positive number a there is a positive rational number r, What r< а . These facts underlie the theory of inequalities.

By definition, the inequality a > b (or, what is the same, b< a) имеет место в том и только в том случае, если а - b >0, i.e. if the number a - b is positive.

Consider, in particular, the inequality A< 0 . What does this inequality mean? According to the above definition, it means that 0 - a > 0, i.e. -a > 0 or, in other words, what is the number -A positively. But this takes place if and only if the number A negative. So inequality A< 0 means that the number but negative.

The notation is also often used ab(or, what is the same, ba).
Record ab, by definition, means that either a > b, or a = b. If we consider the record ab as an indefinite statement, then in the notation of mathematical logic we can write

(a b) [(a > b) V (a = b)]

Example 1. Are the inequalities 5 0, 0 0 true?

The inequality 5 0 is a complex statement consisting of two simple statements connected by the logical connective “or” (disjunction). Either 5 > 0 or 5 = 0. The first statement 5 > 0 is true, the second statement 5 = 0 is false. By the definition of a disjunction, such a complex statement is true.

The entry 00 is discussed similarly.

Inequalities of the form a > b, a< b we will call them strict, and inequalities of the form ab, ab- not strict.

Inequalities a > b And c > d(or A< b And With< d ) will be called inequalities of the same meaning, and inequalities a > b And c< d - inequalities of opposite meaning. Note that these two terms (inequalities of the same and opposite meaning) refer only to the form of writing the inequalities, and not to the facts themselves expressed by these inequalities. So, in relation to inequality A< b inequality With< d is an inequality of the same meaning, and in the notation d>c(meaning the same thing) - an inequality of the opposite meaning.

Along with inequalities of the form a>b, ab so-called double inequalities are used, i.e., inequalities of the form A< с < b , ac< b , a< cb ,
acb. By definition, a record

A< с < b (1)
means that both inequalities hold:

A< с And With< b.

The inequalities have a similar meaning acb, ac< b, а < сb.

Double inequality (1) can be written as follows:

(a< c < b) [(a < c) & (c < b)]

and double inequality a ≤ c ≤ b can be written in the following form:

(a c b) [(a< c)V(a = c) & (c < b)V(c = b)]

Let us now proceed to the presentation of the basic properties and rules of action on inequalities, having agreed that in this article the letters a, b, c stand for real numbers, and n means natural number.

1) If a > b and b > c, then a > c (transitivity).

Proof.

Since by condition a > b And b > c, then the numbers a - b And b - c are positive, and therefore the number a - c = (a - b) + (b - c), as the sum of positive numbers, is also positive. This means, by definition, that a > c.

2) If a > b, then for any c the inequality a + c > b + c holds.

Proof.

Because a > b, then the number a - b positively. Therefore, the number (a + c) - (b + c) = a + c - b - c = a - b is also positive, i.e.
a + c > b + c.

3) If a + b > c, then a > b - c, that is, any term can be transferred from one part of the inequality to another by changing the sign of this term to the opposite.

The proof follows from property 2) it is sufficient for both sides of the inequality a + b > c add number - b.

4) If a > b and c > d, then a + c > b + d, that is, when adding two inequalities of the same meaning, an inequality of the same meaning is obtained.

Proof.

By virtue of the definition of inequality, it is sufficient to show that the difference
(a + c) - (b + c) positive. This difference can be written as follows:
(a + c) - (b + d) = (a - b) + (c - d).
Since according to the condition of the number a - b And c - d are positive, then (a + c) - (b + d) there is also a positive number.

Consequence. From rules 2) and 4) it follows next Rule subtraction of inequalities: if a > b, c > d, That a - d > b - c(for proof it is enough to apply both sides of the inequality a + c > b + d add number - c - d).

5) If a > b, then for c > 0 we have ac > bc, and for c< 0 имеем ас < bc.

In other words, when multiplying both sides of an inequality with either a positive number, the inequality sign is preserved (i.e., an inequality of the same meaning is obtained), but when multiplied by a negative number, the inequality sign changes to the opposite (i.e., an inequality of the opposite meaning is obtained.

Proof.

If a > b, That a - b is a positive number. Therefore, the sign of the difference ac-bc = c(a - b) matches the sign of the number With: If With is a positive number, then the difference ac - bc is positive and therefore ac > bc, and if With< 0 , then this difference is negative and therefore bc - ac positive, i.e. bc > ac.

6) If a > b > 0 and c > d > 0, then ac > bd, that is, if all terms of two inequalities of the same meaning are positive, then when multiplying these inequalities term by term, an inequality of the same meaning is obtained.

Proof.

We have ac - bd = ac - bc + bc - bd = c(a - b) + b(c - d). Because c > 0, b > 0, a - b > 0, c - d > 0, then ac - bd > 0, i.e. ac > bd.

Comment. From the proof it is clear that the condition d > 0 in the formulation of property 6) is unimportant: for this property to be valid, it is sufficient that the conditions be met a > b > 0, c > d, c > 0. If (if the inequalities are fulfilled a > b, c > d) numbers a, b, c will not all be positive, then the inequality ac > bd may not be fulfilled. For example, when A = 2, b =1, c= -2, d= -3 we have a > b, c > d, but inequality ac > bd(i.e. -4 > -3) failed. Thus, the requirement that the numbers a, b, c be positive in the formulation of property 6) is essential.

7) If a ≥ b > 0 and c > d > 0, then (division of inequalities).

Proof.

We have The numerator of the fraction on the right side is positive (see properties 5), 6)), the denominator is also positive. Hence,. This proves property 7).

Comment. Let's note an important special case rule 7), obtained when a = b = 1: if c > d > 0, then. Thus, if the terms of the inequality are positive, then when passing to the reciprocals we obtain an inequality of the opposite meaning. We invite readers to check that this rule also holds in 7) If ab > 0 and c > d > 0, then (division of inequalities).

Proof. That.

We have proved above several properties of inequalities written using the sign > (more). However, all these properties could be formulated using the sign < (less), since inequality b< а means, by definition, the same as inequality a > b. In addition, as is easy to verify, the properties proved above are also preserved for non-strict inequalities. For example, property 1) for non-strict inequalities will have the following form: if ab and bc, That ac.

Of course, the above does not limit the general properties of inequalities. There is also whole line inequalities general view related to the consideration of power, exponential, logarithmic and trigonometric functions. The general approach for writing this kind of inequalities is as follows. If some function y = f(x) increases monotonically on the segment [a, b], then for x 1 > x 2 (where x 1 and x 2 belong to this segment) we have f (x 1) > f(x 2). Likewise, if the function y = f(x) monotonically decreases on the interval [a, b], then when x 1 > x 2 (where x 1 And X 2 belong to this segment) we have f(x 1)< f(x 2 ). Of course, what has been said is no different from the definition of monotonicity, but this technique is very convenient for memorizing and writing inequalities.

So, for example, for any natural number n the function y = x n is monotonically increasing along the ray }