Basic number gaps. Numerical segments, intervals, half-intervals and rays are called numerical intervals. Open and closed beam

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.

Inequality defining a numerical gap Number gap notation The name of the number range 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 number 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'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.

Lots 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 in the equation we transfer the term from one part to another by changing its sign, then we will get an equation equivalent to the given 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; - four.
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.

In this way,

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 the 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 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 a negative number, while changing the sign of inequality to the opposite, then we get an inequality equivalent to the given one.

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,


Among the number sets, that is sets, whose objects are numbers, distinguish the so-called number gaps. Their value is that it is very easy to imagine a set corresponding to a specified numerical range, and vice versa. Therefore, with their help it is convenient to write down the set of solutions of the inequality.

In this article, we will analyze all types of numerical intervals. Here we give their names, introduce notation, draw numerical intervals on the coordinate line, and also show which simplest inequalities correspond to them. In conclusion, we will 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:

  • the name of the number range,
  • 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 in any of the last three ways in the list: either by an inequality, or by a designation, or by its image on a coordinate line. Moreover, according to this method of assignment, for example, by inequality, others are easily restored (in our case, the designation and the geometric image).

Let's get down to specifics. Let us describe all the numerical intervals on 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's put in it the name of the numerical interval, the inequality corresponding to it, the notation and the image on the coordinate line. We get the following range table:


Bibliography.

  • Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
  • Mordkovich A. G. Algebra. Grade 9 At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., Sr. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.

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.

Definition 1

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 a straight 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 given point 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.

Definition 7

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

Answer - The set (-∞;+∞) is called a number line, and any number is called a point of this line. Let a be an arbitrary point on the real line and δ

Positive number. The interval (a-δ; a+δ) is called the δ-neighborhood of the point a.

The set X is bounded from above (from below) if there is such a number c that for any x ∈ X the inequality x≤с (x≥c) is satisfied. The number c in this case is called the upper (lower) bound of the set X. A set bounded both above and below is called bounded. The smallest (largest) of the upper (lower) faces 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 enclosed between them also belong to this set. There are several, in a sense, different types of non-empty numerical intervals: Line, open ray, closed ray, line 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 the geometric sense and the concept of a number line introduced by this definition. Therefore these different concepts denoted by the same term.

open beam

The set of numbers such that or is called an open number ray. Write or respectively: .

closed beam

The set of numbers such that or is called a closed number ray. Write or respectively:

The set of numbers such that is called a number segment.

Comment. The definition does not state that . It is assumed that the case is possible. Then the numerical interval turns into a point.

Interval

A set of numbers such as 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 the ends of which is removed to infinity, and a number line - as an interval, both ends of which are removed to infinity.

Half interval

The set of numbers such that or is called a numerical half-interval.

Write or, respectively,

3.Function.Function graph. Ways to set a function.

Answer - If two variables x and y are given, then they say that the variable y is a function of the variable x, if such a relationship between these variables is given that allows for each value to uniquely determine the value of y.

The notation F = y(x) means that we are considering a function that allows for any value of the independent variable x (out of those that the argument x can take at all) to find the corresponding value of the dependent variable y.

Ways to set a function.

A function can be defined by a formula, for example:

y \u003d 3x2 - 2.

The function can be given by a graph. Using the graph, you can determine which value of the function corresponds to the specified value of the argument. Usually this is an approximate value of the function.

4. The main characteristics of the function: monotonicity, parity, periodicity.

Answer - Periodicity Definition. A function f is called periodic if there exists such a number
, that f(x+
)=f(x), for all x D(f). Naturally, there are an infinite number of such numbers. The smallest positive number ^ T is called the period of the function. Examples. A. y \u003d cos x, T \u003d 2 . B. y \u003d tg x, T \u003d . S. y = (x), T = 1. D. y = , this function is not periodic. Parity Definition. A function f is called even if for all x from D(f) the property f(-x) = f(x) is satisfied. If f (-x) = -f (x), then the function is called odd. If none of these relations is satisfied, then the function is called a function of general form. Examples. A. y \u003d cos (x) - even; B. y \u003d tg (x) - odd; S. y \u003d (x); y=sin(x+1) – general functions. Monotony Definition. A function f: X -> R is called increasing (decreasing) if for any
condition is met:
Definition. A function X -> R is said to be 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 \u003d cos x is a piecewise monotone function.

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