Numerical intervals include rays, segments, intervals and half-intervals.
Types of numerical intervals
| Name | Image | Inequality | Designation |
|---|---|---|---|
| Open beam | x > a | (a; +∞) | |
| x < a | (-∞; a) | ||
| Closed beam | x ⩾ a | [a; +∞) | |
| x ⩽ a | (-∞; a] | ||
| Line segment | a ⩽ x ⩽ b | [a; b] | |
| Interval | a < x < b | (a; b) | |
| Half-interval | a < x ⩽ b | (a; b] | |
| a ⩽ x < b | [a; b) |
In the table a And b are boundary points, and x- a variable that can take the coordinate of any point belonging to a numerical interval.
Boundary point- this is the point that defines the boundary of the numerical interval. A boundary point may or may not belong to a numerical interval. In the drawings, boundary points that do not belong to the numerical interval under consideration are indicated by an open circle, and those that belong to them are indicated by a filled circle.
Open and closed beam
Open beam is a set of points on a line lying on one side of a boundary point that is not included in this set. The ray is called open precisely because of the boundary point that does not belong to it.
Let's consider a set of points on the coordinate line that have a coordinate greater than 2, and therefore located to the right of point 2:
Such a set can be defined by the inequality x> 2. Open rays are denoted using parentheses - (2; +∞), this entry reads like this: open numeric ray from two to plus infinity.
The set to which the inequality corresponds x < 2, можно обозначить (-∞; 2) или изобразить в виде луча, все точки которого лежат с левой стороны от точки 2:
Closed beam is a set of points on a line lying on one side of a boundary point belonging to a given set. In the drawings, boundary points belonging to the set under consideration are indicated by a filled circle.
Closed number rays are defined by non-strict inequalities. For example, inequalities x 2 and x 2 can be depicted like this:
These closed rays are designated as follows: , it is read like this: a numerical ray from two to plus infinity and a numerical ray from minus infinity to two. The square bracket in the notation indicates that point 2 belongs to the numerical interval.
Line segment
Line segment is the set of points on a line lying between two boundary points belonging to a given set. Such sets are defined by double non-strict inequalities.
Consider a segment of a coordinate line with ends at points -2 and 3:
The set of points that make up a given segment can be specified by the double inequality -2 x 3 or designate [-2; 3], such a record reads like this: a segment from minus two to three.
Interval and half-interval
Interval- this is the set of points on a line lying between two boundary points that do not belong to this set. Such sets are defined by double strict inequalities.
Consider a segment of a coordinate line with ends at points -2 and 3:
The set of points that make up a given interval can be specified by the double inequality -2< x < 3 или обозначить (-2; 3), такая запись читается так: интервал от минус двух до трёх.
Half-interval is the set of points on a line lying between two boundary points, one of which belongs to the set and the other does not. Such sets are defined by double inequalities:
These half-intervals are designated as follows: (-2; 3] and [-2; 3), it is read like this: half-interval from minus two to three, including 3, and half-interval from minus two to three, including minus two.
Numeric interval
Interval, open span, interval- the set of points on the number line between two given numbers a And b, that is, a set of numbers x, satisfying the condition: a < x < b . The interval does not include ends and is denoted by ( a,b) (Sometimes ] a,b[ ), in contrast to the segment [ a,b] (closed interval), including the ends, that is, consisting of points.
In recording ( a,b), numbers a And b are called the ends of the interval. The interval includes all real numbers, the interval includes all numbers smaller a and the interval - all numbers are large a .
Term interval used in complex terms:
- upon integration - integration interval,
- when clarifying the roots of the equation - insulation span
- when determining the convergence of power series - interval of convergence of power series.
By the way, in English the word interval called a segment. And to denote the concept of interval, the term is used open interval.
Literature
- Vygodsky M. Ya. Handbook of higher mathematics. M.: “Astrel”, “AST”, 2002
see also
Links
Wikimedia Foundation. 2010.
See what “Numerical interval” is in other dictionaries:
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Answer - The set (-∞;+∞) is called a number line, and any number is a point on this line. Let a be an arbitrary point on the number line and δ
Positive number. The interval (a-δ; a+δ) is called the δ-neighborhood of point a.
A set X is bounded from above (from below) if there is a number c such that for any x ∈ X the inequality x≤с (x≥c) holds. The number c in this case is called the upper (lower) bound of the set X. A set that is bounded both above and below is called bounded. The smallest (largest) of the upper (lower) bounds of a set is called the exact upper (lower) bound of this set.
A numerical interval is a connected set of real numbers, that is, such that if 2 numbers belong to this set, then all the numbers between them also belong to this set. There are several somewhat different types of non-empty number intervals: Line, open ray, closed ray, segment, half-interval, interval
Number line
The set of all real numbers is also called the number line. They write.
In practice, there is no need to distinguish between the concept of a coordinate or number line in a geometric sense and the concept of a number line introduced by this definition. Therefore, these different concepts are denoted by the same term.
Open beam
The set of numbers such that is called an open number ray. They write 
or accordingly: 
.
Closed beam
The set of numbers such that is called a closed number line. They write 
or accordingly:.
A set of numbers is called a number segment.
Comment. The definition does not stipulate that . It is assumed that the case is possible. Then the numerical interval turns into a point.
Interval
A set of numbers such that is called a numerical interval.
Comment. The coincidence of the designations of an open beam, a straight line and an interval is not accidental. An open ray can be understood as an interval, one of whose ends is removed to infinity, and a number line - as an interval, both ends of which are removed to infinity.
Half-interval
A set of numbers such as this is called a numerical half-interval.
They write or, respectively,
3.Function.Graph of the function. Methods for specifying a function.
Answer - If two variables x and y are given, then the variable y is said to be a function of the variable x if such a relationship is given between these variables that allows for each value to uniquely determine the value of y.
The notation F = y(x) means that a function is being considered that allows for any value of the independent variable x (from among those that the argument x can generally take) to find the corresponding value of the dependent variable y.
Methods for specifying a function.
The function can be specified by a formula, for example:
y = 3x2 – 2.
The function can be specified by a graph. Using a graph, you can determine which function value corresponds to a specified argument value. This is usually an approximate value of the function.
4.Main characteristics of the function: monotonicity, parity, periodicity.
Answer - Periodicity Definition. A function f is called periodic if there is such a number 
, that f(x+ 
)=f(x), for all x 
D(f). Naturally, there are countless numbers of such numbers. The smallest positive number ^ T is called the period of the function. Examples. A. y = cos x, T = 2 
. V. y = tg x, T = 
. S. y = (x), T = 1. D. y = 
, this function is not periodic. Parity Definition. A function f is called even if the property f(-x) = f(x) holds for all x in D(f). If f(-x) = -f(x), then the function is called odd. If none of the indicated relations are satisfied, then the function is called a general function. Examples. A. y = cos (x) - even; V. y = tg (x) - odd; S. y = (x); y=sin(x+1) – functions of general form. Monotony Definition. A function f: X -> R is called increasing (decreasing) if for any 
the condition is met: 
Definition. A function X -> R is called monotonic on X if it is increasing or decreasing on X. If f is monotone on some subsets of X, then it is called piecewise monotone. Example. y = cos x - piecewise monotonic function.
B) Number line
Consider the number line (Fig. 6):
Consider the set of rational numbers
Each rational number is represented by a certain point on the number axis. So, the numbers are marked in the figure.
Let's prove that .
Proof. Let there be a fraction: . We have the right to consider this fraction irreducible. Since , then - the number is even: - odd. Substituting its expression, we find: , which implies that is an even number. We have obtained a contradiction that proves the statement.
So, not all points on the number axis represent rational numbers. Those points that do not represent rational numbers represent numbers called irrational.
Any number of the form , , is either an integer or an irrational number.
Numeric intervals
Numerical segments, intervals, half-intervals and rays are called numerical intervals.
| Inequality specifying a numerical interval | Designation of a numerical interval | Name of the number interval | It reads like this: |
| a ≤ x ≤ b | [a; b] | Numerical segment | Segment from a to b |
| a< x < b | (a; b) | Interval | Interval from a to b |
| a ≤ x< b | [a; b) | Half-interval | Half-interval from a before b, including a. |
| a< x ≤ b | (a; b] | Half-interval | Half-interval from a before b, including b. |
| x ≥ a | [a; +∞) | Number beam | Number beam from a up to plus infinity |
| x>a | (a; +∞) | Open number beam | Open numerical beam from a up to plus infinity |
| x ≤ a | (- ∞; a] | Number beam | Number ray from minus infinity to a |
| x< a | (- ∞; a) | Open number beam | Open number ray from minus infinity to a |
Let us represent the numbers on the coordinate line a And b, as well as the number x between them.
The set of all numbers that meet the condition a ≤ x ≤ b, called numerical segment or just a segment. It is designated as follows: [ a; b] - It reads like this: a segment from a to b.
The set of numbers that meet the condition a< x < b , called interval. It is designated as follows: ( a; b)
It reads like this: interval from a to b.
Sets of numbers satisfying the conditions a ≤ x< b или a<x ≤ b, are called half-intervals. Designations:
Set a ≤ x< b обозначается так:[a; b), reads like this: half-interval from a before b, including a.
A bunch of a<x ≤ b is indicated as follows:( a; b], reads like this: half-interval from a before b, including b.
Now let's imagine Ray with a dot a, to the right and left of which there is a set of numbers.
a, meeting the condition x ≥ a, called numerical beam.
It is designated as follows: [ a; +∞)-Reads like this: a numerical ray from a to plus infinity.
Set of numbers to the right of a point a, corresponding to the inequality x>a, called open number beam.
It is designated as follows: ( a; +∞)-Reads like this: an open numerical ray from a to plus infinity.
a, meeting the condition x ≤ a, called numerical ray from minus infinity toa .
It is designated as follows:( - ∞; a]-Reads like this: a numerical ray from minus infinity to a.
Set of numbers to the left of the point a, corresponding to the inequality x< a , called open number ray from minus infinity toa .
It is designated as follows: ( - ∞; a)-Reads like this: an open number ray from minus infinity to a.
The set of real numbers is represented by the entire coordinate line. He is called number line. It is designated as follows: ( - ∞; + ∞ )
3) Linear equations and inequalities with one variable, their solutions:
An equation containing a variable is called an equation with one variable, or an equation with one unknown. For example, an equation with one variable is 3(2x+7)=4x-1.
The root or solution of an equation is the value of a variable at which the equation becomes a true numerical equality. For example, the number 1 is a solution to the equation 2x+5=8x-1. The equation x2+1=0 has no solution, because the left side of the equation is always greater than zero. The equation (x+3)(x-4) =0 has two roots: x1= -3, x2=4.
Solving an equation means finding all its roots or proving that there are no roots.
Equations are called equivalent if all the roots of the first equation are roots of the second equation and vice versa, all the roots of the second equation are roots of the first equation or if both equations have no roots. For example, the equations x-8=2 and x+10=20 are equivalent, because the root of the first equation x=10 is also the root of the second equation, and both equations have the same root.
When solving equations, the following properties are used:
If you move a term in an equation from one part to another, changing its sign, you will get an equation equivalent to the given one.
If both sides of an equation are multiplied or divided by the same non-zero number, you get an equation equivalent to the given one.
The equation ax=b, where x is a variable and a and b are some numbers, is called a linear equation with one variable.
If a¹0, then the equation has a unique solution.
If a=0, b=0, then the equation is satisfied by any value of x.
If a=0, b¹0, then the equation has no solutions, because 0x=b is not executed for any value of the variable.
Example 1. Solve the equation: -8(11-2x)+40=3(5x-4)
Let's open the brackets on both sides of the equation, move all terms with x to the left side of the equation, and terms that do not contain x to the right side, we get:
16x-15x=88-40-12
Example 2. Solve the equations:
x3-2x2-98x+18=0;
These equations are not linear, but we will show how such equations can be solved.
3x2-5x=0; x(3x-5)=0. The product is equal to zero, if one of the factors is equal to zero, we get x1=0; x2= .
Answer: 0; .
Factor the left side of the equation:
x2(x-2)-9(x-2)=(x-2)(x2-9)=(x-2)(x-3)(x-3), i.e. (x-2)(x-3)(x+3)=0. This shows that the solutions to this equation are the numbers x1=2, x2=3, x3=-3.
c) Imagine 7x as 3x+4x, then we have: x2+3x+4x+12=0, x(x+3)+4(x+3)=0, (x+3)(x+4)= 0, hence x1=-3, x2=- 4.
Answer: -3; - 4.
Example 3. Solve the equation: ½x+1ç+½x-1ç=3.
Let us recall the definition of the modulus of a number: 
For example: ½3½=3, ½0½=0, ½- 4½= 4.
In this equation, under the modulus sign are the numbers x-1 and x+1. If x is less than –1, then the number x+1 is negative, then ½x+1½=-x-1. And if x>-1, then ½x+1½=x+1. At x=-1 ½x+1½=0.
Thus, 
Likewise
a) Consider this equation½x+1½+½x-1½=3 for x £-1, it is equivalent to the equation -x-1-x+1=3, -2x=3, x=, this number belongs to the set x £-1.
b) Let -1< х £ 1, тогда данное уравнение равносильно уравнению х+1-х+1=3, 2¹3 уравнение не имеет решения на данном множестве.
c) Consider the case x>1.
x+1+x-1=3, 2x=3, x= . This number belongs to the set x>1.
Answer: x1=-1.5; x2=1.5.
Example 4. Solve the equation:½x+2½+3½x½=2½x-1½.
Let us show a short record of the solution to the equation, revealing the sign of the modulus “over intervals”.
x £-2, -(x+2)-3x=-2(x-1), - 4x=4, x=-2О(-¥; -2]
–2<х£0, х+2-3х=-2(х-1), 0=0, хÎ(-2; 0]
0<х£1, х+2+3х=-2(х-1), 6х=0, х=0Ï(0; 1]
x>1, x+2+3x=2(x-1), 2x=- 4, x=-2П(1; +¥)
Answer: [-2; 0]
Example 5. Solve the equation: (a-1)(a+1)x=(a-1)(a+2), for all values of parameter a.
There are actually two variables in this equation, but consider x to be the unknown and a to be the parameter. It is required to solve the equation for the variable x for any value of the parameter a.
If a=1, then the equation has the form 0×x=0; any number satisfies this equation.
If a=-1, then the equation looks like 0×x=-2; not a single number satisfies this equation.
If a¹1, a¹-1, then the equation has a unique solution.
Answer: if a=1, then x is any number;
if a=-1, then there are no solutions;
if a¹±1, then .
B) Linear inequalities with one variable.
If the variable x is given any numerical value, then we get a numerical inequality expressing either a true or false statement. Let, for example, the inequality 5x-1>3x+2 be given. For x=2 we get 5·2-1>3·2+2 – a true statement (true numerical statement); for x=0 we get 5·0-1>3·0+2 – a false statement. Any value of a variable at which a given inequality with a variable turns into a true numerical inequality is called a solution to the inequality. Solving an inequality with a variable means finding the set of all its solutions.
Two inequalities with the same variable x are said to be equivalent if the sets of solutions to these inequalities coincide.
The main idea of solving the inequality is as follows: we replace the given inequality with another, simpler, but equivalent to the given one; we again replace the resulting inequality with a simpler inequality equivalent to it, etc.
Such replacements are made on the basis of the following statements.
Theorem 1. If any term of an inequality with one variable is transferred from one part of the inequality to another with the opposite sign, while leaving the sign of the inequality unchanged, then an inequality equivalent to the given one will be obtained.
Theorem 2. If both sides of an inequality with one variable are multiplied or divided by the same positive number, leaving the sign of the inequality unchanged, then an inequality equivalent to the given one will be obtained.
Theorem 3. If both sides of an inequality with one variable are multiplied or divided by the same negative number, while changing the sign of the inequality to the opposite, then an inequality equivalent to the given one will be obtained.
An inequality of the form ax+b>0 is called linear (respectively, ax+b<0, ax+b³0, ax+b£0), где а и b – действительные числа, причем а¹0. Решение этих неравенств основано на трех теоремах равносильности изложенных выше.
Example 1. Solve the inequality: 2(x-3)+5(1-x)³3(2x-5).
Opening the brackets, we get 2x-6+5-5x³6x-15,
Numerical intervals. Context. Definition
An equality (equation) has one point on the number line (although this point depends on the transformations done and the root chosen). The solution to the equation itself will be a numerical set (sometimes consisting of a single number). However, all this on the number line (visualization of a set of real numbers) will be displayed only pointwise, but there are also more generalized types of relationships between two numbers - inequalities. In them, the number line is divided by a certain number and a certain part is cut off from it - the values of an expression or a numerical interval.
It is logical to discuss the topic of numerical intervals together with inequalities, but this does not mean that it is connected only with them. Numerical intervals (intervals, segments, rays) are a set of variable values that satisfy a certain inequality. That is, in essence, this is the set of all points on the number line, limited by some kind of framework. Therefore, the topic of numerical intervals is most closely related to the concept variable. Where there is a variable, or an arbitrary point x on the number line, and it is used, there are also numerical intervals, intervals - x values. Often the value can be anything, but this is also a numerical interval covering the entire number line.
Let's introduce the concept numerical interval. Among numerical sets, that is, sets whose objects are numbers, so-called numerical intervals are distinguished. Their value is that it is very easy to imagine a set corresponding to a specified numerical interval, and vice versa. Therefore, with their help it is convenient to write down many solutions to an inequality. Whereas the set of solutions to the equation will not be a numerical interval, but simply several numbers on the number line, with inequalities, in other words, any restrictions on the value of a variable, numerical intervals appear.
A number interval is the set of all points on the number line, limited by a given number or numbers (points on the number line).
A numerical interval of any kind (a set of x values enclosed between certain numbers) can always be represented by three types of mathematical notation: special notation for intervals, chains of inequalities (single inequality or double inequality) or geometrically on a number line. Essentially, all these designations have the same meaning. They provide a constraint(s) on the values of some mathematical object, variable (some variable, any expression with a variable, function, etc.).
From the above it can be understood that since the area of the number line can be limited in different ways (there are different types of inequalities), then the types of numerical intervals are different.
Types of numerical intervals
Each type of numerical interval has its own name, a special designation. To indicate numerical intervals, round and square brackets are used. A parenthesis means that the final, boundary-defining point on the number line (end) of this parenthesis is not included in the set of points of this interval. The square bracket means the end fits into the gap. With infinity (on this side the interval is not limited) use a parenthesis. Sometimes, instead of parentheses, you can write square brackets turned in the opposite direction: (a;b) ⇔]a;b[
| Type of gap (name) | Geometric image (on a number line) | Designation | Writing using inequalities (always chained for brevity) |
|---|---|---|---|
| Interval (open) | (a;b) | a< x < b | |
| Segment (section) | a ≤ x ≤ b | ||
| Half-interval (half-segment) | a< x ≤ b | ||
| Ray | x ≤ b | ||
| Open beam | (a;+∞) | x>a | |
| Open beam | (-∞;b) | x< b | |
| The set of all numbers (on a coordinate line) | (-∞;+∞) , although here it is necessary to indicate the specific set-carrier of the algebra with which the work is performed; example: | x ∈(they usually talk about the set of real numbers; to represent complex numbers they use the complex plane, not the straight line) | |
| Equality | or x=a | x = a(a special case of a non-strict inequality: a ≤ x ≤ a- an interval of length 1, where both ends coincide - a segment consisting of one point) | |
| Empty set | ∅ | The empty set is also an interval - the variable x has no values (the empty set). Designation: x∈∅⇔x∈( ). |
The names of the intervals can be confusing: there are a huge number of options. Therefore, it is always better to indicate them accurately. In English literature only the term is used interval ("interval") - open, closed, half-open (half-closed). There are many variations.
Intervals in mathematics are used to denote a very large number of things: there are intervals of isolation when solving equations, intervals of integration, intervals of convergence of series. When studying a function, intervals are always used to denote its range of values and domain of definition. Gaps are very important, for example, there are Bolzano-Cauchy theorem(you can find out more on Wikipedia).
Systems and sets of inequalities
System of inequalities
So, a variable x or the value of some expression can be compared with some constant value - this is an inequality, but this expression can be compared with several quantities - a double inequality, a chain of inequalities, etc. This is exactly what was shown above - as an interval and a segment. Both is system of inequalities.
So, if the task is to find a set of common solutions to two or more inequalities, then we can talk about solving a system of inequalities (just like with equations - although we can say that equations are a special case).
Then it is obvious that the value of the variable used in the inequalities, at which each of them becomes true, is called the solution to the system of inequalities.
All inequalities included in the system are combined with a curly brace - "(". Sometimes they are written in the form double inequality(as shown above) or even chain of inequalities. Example of a typical entry: f x ≤ 30 g x 5 .
The solution to systems of linear inequalities with one variable in the general case comes down to these 4 types: x > a x > b (1) x > a x< b (2) x < a x >b(3)x< a x < b (4) . Здесь предполагается, что b > a.
Any system can be solved graphically using the number line. Where the solutions of the inequalities that make up the system intersect, there will be a solution to the system itself.
Let us present a graphical solution for each case.
(1) x>b (2) aFurther, systems of inequalities can be classified as equivalent if they have a common set of solutions. From here (as can be seen above) it follows that more complex systems can be simplified (for example, using a geometric solution).
The curly brace can be roughly speaking, roughly speaking, called the equivalent of the conjunction " AND" for inequalities
Set of inequalities
However, there are other cases. So, in addition to the intersection of sets of solutions, there is their union: if the task is to find the set of all such values of a variable, each of which is a solution to at least one of the given inequalities, then they say that it is necessary to solve the set of inequalities.
So, all inequalities in the aggregate are united by the aggregate bracket "[". If the value of a variable satisfies at least one inequality from the population, then it belongs to the set of solutions of the entire population. The same goes for equations (again, they can be called a special case).
If the curly brace is And, then the aggregate bracket is, conditionally, in simple terms, the equivalent of the union " OR" for inequalities (although this will of course be a logical or, including the case satisfying both conditions).
So, the solution to a set of inequalities is the value of the variable at which at least one inequality becomes true.
The set of solutions, both collections and systems of inequalities, can be defined through two basic binary operations for working with sets - intersection and union. The set of solutions to a system of inequalities is intersection sets of solutions to inequalities that constitute it. The set of solutions to a set of inequalities is Union sets of solutions to inequalities that constitute it. This can also be illustrated. Let's say we have a system and a set of two inequalities. We denote the set of solutions of the first A, and denote the set of solutions of the second B. An excellent illustration would be the Euler-Venn diagram.
A ∪ B - solution to a system of inequalities A ∩ B - solution to a set of inequalities