Numerical segments, intervals, half-intervals and rays are called numerical intervals. Numeric intervals Function.Function graph

B) Number line

Consider the number line (Fig. 6):

Consider the set of rational numbers

Each rational number is represented by some point on the number line. 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 the expression instead of it, we find: , whence it follows that is an even number. We have obtained a contradiction, which proves the assertion.

So, not all points of the number axis represent rational numbers. Those dots that do not represent rational numbers represent numbers called irrational.

Any number of the form , , is either integer or irrational.

Numeric spans

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

Let's represent on the coordinate line the numbers a And b, as well as the number x between them.

The set of all numbers that meet the condition a ≤ x ≤ b, is called numerical segment or just a cut. It is marked like this: a; b]-It reads like this: a segment from a to b.

The set of numbers that meet the condition a< x < b , is called interval. It's marked like this: a; b)

It reads like this: the 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), is read like this: a half-interval from a before b, including a.

A bunch of a<x ≤ b marked like this: a; b], reads like this: a half-interval from a before b, including b.

Now imagine Ray with a dot a, to the right and left of which is a set of numbers.

a, satisfying the condition x ≥ a, is called number beam.

It is marked like this: a; +∞) - It reads like this: a numerical beam from a up to plus infinity.

Lots of numbers to the right of the dot a corresponding to the inequality x > a, is called open number beam.

It's marked like this: a; +∞) - It reads like this: an open numerical beam from a up to plus infinity.

a, satisfying the condition x ≤ a, is called number line from minus infinity toa .

It's labeled like this: -∞; a]-It reads like this: a numerical ray from minus infinity to a.

Set of numbers to the left of the dot a corresponding to the inequality x< a , is called open numerical beam from minus infinity toa .

It's marked like this: -∞; a) - It reads like this: an open numerical ray from minus infinity to a.

The set of real numbers is represented by the entire coordinate line. He is called number line. It's labeled like this: - ∞; + ∞ )

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 the 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 roots of the first equation are roots of the second equation and vice versa, all 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 transfer the term from one part to another in the equation, changing its sign, you will get an equation that is equivalent to this one.

If both sides of the equation are multiplied or divided by the same non-zero number, then an equation is obtained that is 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 any value of x satisfies the equation.

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 in both parts of the equation, move all the terms with x to the left side of the equation, and the terms that do not contain x to the right side, we get:

16x-15x=88-40-12

Example 2. Solve 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; .

Factoring 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 of this equation are the numbers x1=2, x2=3, x3=-3.

c) Let's represent 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.

Recall the definition of the modulus of a number:

For example: ½3½=3, ½0½=0, ½-4½= 4.

In this equation, under the module sign are the numbers x-1 and x + 1. If x is less than -1, then x+1 is negative, then ½x+1½=-x-1. And if x>-1, then ½x+1½=x+1. For x=-1 ½x+1½=0.

Thus,

Similarly

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's show a brief record of the solution of the equation, expanding the sign of the modulus "by intervals".

x £-2, -(x + 2) -3x \u003d -2 (x-1), - 4x \u003d 4, x \u003d -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=-2W(1; +¥)

Answer: [-2; 0]
Example 5. Solve the equation: (a-1) (a + 1) x \u003d (a-1) (a + 2), for all values ​​of the parameter a.

This equation actually has two variables, but considers x to be the unknown and a to be the parameter. It is required to solve the equation with respect to 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 \u003d -1, then the equation has the form 0 × x \u003d -2, this equation does not satisfy any number.

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 some numerical value, then we get a numerical inequality expressing either a true or a false statement. Let, for example, the inequality 5x-1>3x+2 be given. With x=2 we get 5 2-1> 3 2+2 - true statement (true numerical statement); for x=0 we get 5·0-1>3·0+2 – a false statement. Any value of a variable for 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 one variable x are said to be equivalent if the solution sets of these inequalities are the same.

The main idea of ​​solving the inequality is as follows: we replace the given inequality with another one, simpler, but equivalent to the given one; the resulting inequality is again replaced by a simpler equivalent inequality, and so on.

Such replacements are carried out on the basis of the following assertions.

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 inequality sign unchanged, then an inequality equivalent to the given one will be obtained.

Theorem 2. If both parts of an inequality with one variable are multiplied or divided by the same positive number, while leaving the inequality sign unchanged, then an inequality equivalent to the given one will be obtained.

Theorem 3. If both parts 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 (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,

"Tables in algebra grade 7" - The difference of squares. Expressions. Content. Algebra tables.

"Numerical functions" - The set X is called the task area or the area of ​​\u200b\u200bdefinition of the function f and is denoted by D (f). Function graph. However, not every line is a graph of some function. Example 1. A skydiver jumps from a hovering helicopter. Just one number. Piecewise specification of functions. Natural phenomena are closely related to each other.

"Numeric sequences" - Lesson-conference. "Number Sequences". Geometric progression. Task methods. Arithmetic progression. Numeric sequences.

"Limit of a numerical sequence" - Solution: Methods for specifying sequences. Limited number sequence. The value уn is called the common member of the sequence. The limit of the numerical sequence. Continuity of a function at a point. Example: 1, 4, 9, 16, ..., n2, ... - limited from below 1. By setting an analytical formula. Limit properties.

"Number sequence" - Numerical sequence (number series): numbers written out in a certain order. 2. Methods for setting sequences. 1. Definition. Sequence notation. Sequences. 1. Formula of the n-th member of the sequence: - allows you to find any member of the sequence. 3. Graph of the numerical sequence.

"Tables" - Oil and gas production. Table 2. Table 5. Tabular information models. Order of construction of OS type table. Table 4. Annual estimates. Table number. Tables of the type "Objects - objects". Pupils of 10 "B" class. Table structure. Tables of type objects-properties. Pairs of objects are described; There is only one property.

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

This article gives answers to questions about numerical gaps, names, notation, images of gaps on the coordinate line, correspondence of inequalities. In conclusion, the table of gaps will be considered.

Each number span is characterized by:

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

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

Let's make a description of the numerical intervals with the above indicated sides:

Definition 2

• Open number beam. The name is due to 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 , like (a , + ∞) .

The geometric meaning of an open beam considers the presence of a numerical gap. There is a correspondence between the points of the coordinate line and its numbers, due to which the line is called the coordinate line. If it is necessary 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 solving, therefore, in the drawing it is indicated by a punched out dot. The gap that is needed is highlighted by hatching. Consider the figure below.

From the above figure, it can be seen that the numerical gaps correspond to a part of the line, that is, rays starting at a. In other words, they are called rays without a beginning. Therefore, it was called the open number ray.

Let's look at a few examples.

Example 1

For a given strict inequality x > − 3, an open ray is given. This entry can be represented as 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 leaves behind its usefulness.

Its assignment goes with the help of non-strict inequalities of the form x ≤ a or x ≥ a . For this type, special notation of the form (− ∞ , a ] and [ a , + ∞) is 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 an illustrative example, let's set a numerical ray.

Example 3

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

Definition 4

• Interval. Setting 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

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

Example 5

Having considered the segment, we get that its specification is possible using the double inequality 2 ≤ x ≤ 3 , which is represented as 2 , 3 . On the coordinate line, the data 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 a 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.

Gaps can be shown as:

• open number beam;
• number beam;
• interval;
• numerical segment;
• half-interval.

To simplify the calculation process, it is necessary to use a special table, where there are designations for all types of numerical intervals of a straight 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

Number intervals include rays, segments, intervals and half-intervals.

Types of numerical intervals

Table a And b are the boundary points, and x- a variable that can take the coordinate of any point belonging to the numerical interval.

boundary point is a point that defines the boundary of the numerical interval. The boundary point may or may not belong to the numerical interval. In the drawings, boundary points that do not belong to the numerical interval under consideration are indicated by an unfilled circle, and those belonging to a filled circle.

Open and closed beam

open beam is the set of points on a straight line that lie on one side of a boundary point that is not included in the given set. A ray is called open precisely because of the boundary point, which does not belong to it.

Consider the 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 beams are denoted with parentheses - (2; +∞), this entry reads as follows: an open numerical beam from two to plus infinity.

The set corresponding to the inequality x < 2, можно обозначить (-∞; 2) или изобразить в виде луча, все точки которого лежат с левой стороны от точки 2:

closed beam is the set of points on a line that lie on the same side of a boundary point belonging to the given set. In the drawings, the boundary points belonging to the set under consideration are indicated by a filled circle.

Closed numerical rays are defined by non-strict inequalities. For example, the inequalities x 2 and x 2 can be shown 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 gap.

Line segment

Line segment is the set of points on a line that lie between two boundary points belonging to the given set. Such sets are given by double non-strict inequalities.

Consider a segment of the 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 denote [-2; 3], such an entry reads as follows: a segment from minus two to three.

Interval and half-interval

Interval is the set of points on a line that lie between two boundary points that do not belong to the given set. Such sets are defined by double strict inequalities.

Consider a segment of the coordinate line with ends at points -2 and 3:

The set of points that make up this 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 that lie between two boundary points, one of which belongs to the set and the other does not. Such sets are given by double inequalities:

These half-intervals are designated as follows: (-2; 3] and [-2; 3). It reads like this: a half-interval from minus two to three, including 3, and a half-interval from minus two to three, including minus two.