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Numerical segments, intervals, half-intervals and rays are called numerical intervals. Numerical segments, intervals, half-intervals and rays are called numerical intervals Types of numerical intervals

Among sets of numbers, there are sets where the objects are numerical intervals. When indicating a set, it is easier to determine by the interval. Therefore, we write down sets of solutions using numerical intervals.

This article provides answers to questions about numerical intervals, names, notations, images of intervals on a coordinate line, and correspondence of inequalities. Finally, the gap table will be discussed.

Definition 1

Each numerical interval is characterized by:

  • name;
  • the presence of ordinary or double inequality;
  • designation;
  • geometric image on a straight line coordinate.

The numerical interval is specified using any 3 methods from the list above. That is, when using inequality, notation, image on the coordinate line. This method is the most applicable.

Let us describe the numerical intervals with the above-mentioned sides:

Definition 2

  • Open number beam. The name comes from the fact that it is omitted, leaving it open.

This interval has the corresponding inequalities x< a или x >a , where a is some real number. That is, on such a ray there are all real numbers that are less than a - (x< a) или больше a - (x >a) .

The set of numbers that will satisfy an inequality of the form x< a обозначается виде промежутка (− ∞ , a) , а для x >a as (a , + ∞) .

The geometric meaning of an open ray considers the presence of a numerical interval. There is a correspondence between the points of a coordinate line and its numbers, due to which the line is called a coordinate line. If you need to compare numbers, then on the coordinate line the larger number is to the right. Then an inequality of the form x< a включает в себя точки, которые расположены левее, а для x >a – points that are to the right. The number itself is not suitable for the solution, so it is indicated in the drawing by a punctured dot. The gap that is required is highlighted using shading. Consider the figure below.

From the above figure it is clear that the numerical intervals correspond to parts of the line, that is, rays with a beginning at a. In other words, they are called rays without a beginning. That's why it got the name open number beam.

Let's look at a few examples.

Example 1

For a given strict inequality x > − 3, an open beam is specified. This entry can be represented in the form of coordinates (− 3, ∞). That is, these are all points lying to the right than - 3.

Example 2

If we have an inequality of the form x< 2 , 3 , то запись (− ∞ , 2 , 3) является аналогичной при задании открытого числового луча.

Definition 3

  • Number beam. The geometric meaning is that the beginning is not discarded, in other words, the ray retains its usefulness.

Its task is carried out using non-strict inequalities of the form x ≤ a or x ≥ a. For this type, special notations of the form (− ∞, a ] and [ a , + ∞) are accepted, and the presence of a square bracket means that the point is included in the solution or in the set. Consider the figure below.

For a clear example, let's define a numerical ray.

Example 3

An inequality of the form x ≥ 5 corresponds to the notation [ 5 , + ∞), then we obtain a ray of the following form:

Definition 4

  • Interval. A statement using intervals is written using double inequalities a< x < b , где а и b являются некоторыми действительными числами, где a меньше b , а x является переменной. На таком интервале имеется множество точек и чисел, которые больше a , но меньше b . Обозначение такого интервала принято записывать в виде (a , b) . Наличие круглых скобок говорит о том, что число a и b не включены в это множество. Координатная прямая при изображении получает 2 выколотые точки.

Consider the figure below.

Example 4

Interval example − 1< x < 3 , 5 говорит о том, что его можно записать в виде интервала (− 1 , 3 , 5) . Изобразим на координатной прямой и рассмотрим.

Definition 5

  • Numerical segment. This interval differs in that it includes boundary points, then it has the form a ≤ x ≤ b. Such a non-strict inequality suggests that when writing in the form of a numerical segment, square brackets [a, b] are used, which means that the points are included in the set and are depicted as shaded.

Example 5

Having examined the segment, we find that its definition is possible using the double inequality 2 ≤ x ≤ 3, which we represent in the form 2, 3. On the coordinate line, the given points will be included in the solution and shaded.

Definition 6 Example 6

If there is a half-interval (1, 3], then its designation can be in the form of the double inequality 1< x ≤ 3 , при чем на координатной прямой изобразится с точками 1 и 3 , где 1 будет исключена, то есть выколота на прямой.

Definition 7

Intervals can be depicted as:

  • open number beam;
  • number beam;
  • interval;
  • number line;
  • half-interval

To simplify the calculation process, you need to use a special table that contains designations for all types of numerical intervals of a line.

Name Inequality Designation Image
Open number beam x< a - ∞ , a
x>a a , + ∞
Number beam x ≤ a (- ∞ , a ]
x ≥ a [a, + ∞)
Interval a< x < b a, b
Numerical segment a ≤ x ≤ b a, b

Half-interval

“Grade 7 Algebra Tables” - Difference of squares. Expressions. Content. Algebra worksheets.

“Numerical functions” - The set X is called the domain of assignment or the domain of definition of the function f and is denoted D (f). Function graph. However, not every line is a graph of some function. Example 1. A paratrooper jumps from a hovering helicopter. Just one number. Piecewise specification of functions. Natural phenomena are closely related to each other.

“Number sequences” - Lesson-conference. "Number Sequences". Geometric progression. Methods of assignment. Arithmetic progression. Number sequences.

“Limit of a numerical sequence” - Solution: Methods for specifying sequences. Limited number sequence. The quantity уn is called the common term of the sequence. Limit of number sequence. Continuity of a function at a point. Example: 1, 4, 9, 16, ..., n2, ... - limited from below by 1. By specifying an analytical formula. Properties of limits.

“Number sequence” - Number sequence (number series): numbers written out in a certain order. 2. Methods for specifying sequences. 1. Definition. Sequence designation. Sequences. 1. Formula for the nth member of a sequence: - allows you to find any member of the sequence. 3. Number sequence graph.

"Tables" - Oil and gas production. Table 2. Table 5. Tabular information models. The order of constructing the OS type table. Table 4. Annual estimates. Table number. Tables of the “Objects – objects” type. Pupils of 10 "B" class. Table structure. Tables of the object-property type. Pairs of objects are described; There is only one property.

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); at 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,


Among numerical sets, that is sets, the objects of which are numbers, there are so-called numerical intervals. 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.

In this article we will look at all types of numerical intervals. Here we will give their names, introduce notations, depict numerical intervals on the coordinate line, and also show what simple inequalities correspond to them. In conclusion, let us visually present all the information in the form of a table of numerical intervals.

Page navigation.

Types of numerical intervals

Each numerical interval has four inextricably linked things:

  • name of the number interval,
  • corresponding inequality or double inequality,
  • designation,
  • and its geometric image in the form of an image on a coordinate line.

Any numerical interval can be specified by any of the last three methods in the list: either an inequality, or a notation, or its image on a coordinate line. Moreover, using this method of specifying, for example, by inequality, others can be easily restored (in our case, the designation and the geometric image).

Let's get down to specifics. Let us describe all numerical intervals from the four sides indicated above.

Table of numerical intervals

So, in the previous paragraph we defined and described the following numerical intervals:

  • open number beam;
  • number beam;
  • interval;
  • half-interval

For convenience, we summarize all the data on numerical intervals in a table. Let us enter into it the name of the numerical interval, the corresponding inequality, designation and image on the coordinate line. We get the following table of numerical intervals:


Bibliography.

  • 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. 9th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., erased. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.

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.

Heading: Karamzin N.M.
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