We already know that the set of real numbers $R$ consists of rational and irrational numbers.

Rational numbers can always be represented as decimal fractions (finite or infinite periodic).

Irrational numbers are written as infinite but non-periodic decimal fractions.

The set of real numbers $R$ also includes the elements $-\infty $ and $+\infty $, for which the inequalities $-\infty hold

Let's look at ways to represent real numbers.

Common fractions

Ordinary fractions are written using two natural numbers and a horizontal decimal line. The fraction bar actually replaces the division sign. The number below the line is the denominator of the fraction (divisor), the number above the line is the numerator (dividend).

Definition

A fraction is called proper if its numerator less than the denominator. Conversely, a fraction is called an improper fraction if its numerator is greater than or equal to the denominator.

For ordinary fractions, there are simple, almost obvious, comparison rules ($m$,$n$,$p$ - natural numbers):

  1. of two fractions with the same denominators, the one with the larger numerator is greater, that is, $\frac(m)(p) >\frac(n)(p) $ for $m>n$;
  2. of two fractions with the same numerators, the one with the smaller denominator is greater, that is, $\frac(p)(m) >\frac(p)(n) $ for $ m
  3. a proper fraction is always less than one; an improper fraction is always greater than one; a fraction in which the numerator is equal to the denominator is equal to one;
  4. Every improper fraction is greater than every proper fraction.

Decimal numbers

The notation of a decimal number (decimal fraction) has the form: integer part, decimal point, fractional part. The decimal notation of a common fraction can be obtained by dividing the numerator by the denominator with the “angle”. This can result in either a finite decimal fraction or an infinite periodic decimal fraction.

Definition

The digits of the fractional part are called decimals. In this case, the first digit after the decimal point is called the tenths digit, the second - the hundredths digit, the third - the thousandths digit, etc.

Example 1

Determine the value of the decimal number 3.74. We get: $3.74=3+\frac(7)(10) +\frac(4)(100) $.

Decimal number can be rounded. In this case, you must indicate the digit to which rounding is performed.

The rounding rule is as follows:

  1. all digits to the right of this digit are replaced with zeros (if these digits are before the decimal point) or discarded (if these digits are after the decimal point);
  2. if the first digit following a given digit is less than 5, then the digit of this digit is not changed;
  3. if the first digit following a given digit is 5 or more, then the digit of this digit is increased by one.

Example 2

  1. Let's round the number 17302 to thousands: 17000.
  2. Let's round the number 17378 to hundreds: 17400.
  3. Let's round the number 17378.45 to tens: 17380.
  4. Let's round the number 378.91434 to the nearest hundredth: 378.91.
  5. Let's round the number 378.91534 to the nearest hundredth: 378.92.

Convert a decimal number to a fraction.

Case 1

A decimal number represents a terminating decimal fraction.

The following example demonstrates the conversion method.

Example 2

We have: $3.74=3+\frac(7)(10) +\frac(4)(100) $.

We bring to common denominator and we get:

The fraction can be reduced: $3.74=\frac(374)(100) =\frac(187)(50) $.

Case 2

A decimal represents an infinite periodic decimal fraction.

The conversion method is based on the fact that the periodic part of a periodic decimal fraction can be considered as the sum of the terms of an infinite decreasing geometric progression.

Example 4

$0,\left(74\right)=\frac(74)(100) +\frac(74)(10000) +\frac(74)(1000000) +\ldots $. The first term of the progression is $a=0.74$, the denominator of the progression is $q=0.01$.

Example 5

$0.5\left(8\right)=\frac(5)(10) +\frac(8)(100) +\frac(8)(1000) +\frac(8)(10000) +\ldots $ . The first term of the progression is $a=0.08$, the denominator of the progression is $q=0.1$.

The sum of the terms of an infinite decreasing geometric progression is calculated by the formula $s=\frac(a)(1-q) $, where $a$ is the first term and $q$ is the denominator of the progression $ \left (0

Example 6

Let's convert the infinite periodic decimal fraction $0,\left(72\right)$ into a regular one.

The first term of the progression is $a=0.72$, the denominator of the progression is $q=0.01$. We get: $s=\frac(a)(1-q) =\frac(0.72)(1-0.01) =\frac(0.72)(0.99) =\frac(72)( 99) =\frac(8)(11) $. Thus, $0,\left(72\right)=\frac(8)(11) $.

Example 7

Let's convert the infinite periodic decimal fraction $0.5\left(3\right)$ into a regular one.

The first term of the progression is $a=0.03$, the denominator of the progression is $q=0.1$. We get: $s=\frac(a)(1-q) =\frac(0.03)(1-0.1) =\frac(0.03)(0.9) =\frac(3)( 90) =\frac(1)(30) $.

Thus, $0.5\left(3\right)=\frac(5)(10) +\frac(1)(30) =\frac(5\cdot 3)(10\cdot 3) +\frac( 1)(30) =\frac(15)(30) +\frac(1)(30) =\frac(16)(30) =\frac(8)(15) $.

Real numbers can be represented by points on the number axis.

In this case, we call the number axis an infinite straight line on which the origin (point $O$), positive direction (indicated by an arrow) and scale (for displaying values) are selected.

There is a one-to-one correspondence between all real numbers and all points on the number axis: each point corresponds to a single number and, conversely, each number corresponds to a single point. Consequently, the set of real numbers is continuous and infinite, just as the number line is continuous and infinite.

Some subsets of the set of real numbers are called numerical intervals. The elements of a numerical interval are numbers $x\in R$ that satisfy a certain inequality. Let $a\in R$, $b\in R$ and $a\le b$. In this case, the types of intervals can be as follows:

  1. Interval $\left(a,\; b\right)$. At the same time $a
  2. Segment $\left$. Moreover, $a\le x\le b$.
  3. Half-segments or half-intervals $\left$. Moreover $ a \le x
  4. Infinite intervals, for example $a

A type of interval called a neighborhood of a point is also important. The neighborhood of a given point $x_(0) \in R$ is an arbitrary interval $\left(a,\; b\right)$ containing this point inside itself, that is, $a 0$ is its radius.

Absolute value of a number

The absolute value (or modulus) of a real number $x$ is a non-negative real number $\left|x\right|$, determined by the formula: $\left|x\right|=\left\(\begin(array)(c) (\; \; x\; \; (\rm at)\; \; x\ge 0) \\ (-x\; \; (\rm at)\; \; x

Geometrically, $\left|x\right|$ means the distance between points $x$ and 0 on the number line.

Properties of absolute values:

  1. from the definition it follows that $\left|x\right|\ge 0$, $\left|x\right|=\left|-x\right|$;
  2. for the modulus of the sum and for the modulus of the difference of two numbers, the following inequalities are valid: $\left|x+y\right|\le \left|x\right|+\left|y\right|$, $\left|x-y\right|\le \left|x\right|+\left|y\right|$, as well as $\left|x+y\right|\ge \left|x\right|-\left|y\right|$,$\ left|x-y\right|\ge \left|x\right|-\left|y\right|$;
  3. for the modulus of the product and the modulus of the quotient of two numbers, the following equalities are true: $\left|x\cdot y\right|=\left|x\right|\cdot \left|y\right|$ and $\left|\frac(x)( y) \right|=\frac(\left|x\right|)(\left|y\right|) $.

Based on the definition of the absolute value for an arbitrary number $a>0$, we can also establish the equivalence of the following pairs of inequalities:

  1. if $\left|x\right|
  2. if $\left|x\right|\le a$, then $-a\le x\le a$;
  3. if $\left|x\right|>a$, then either $xa$;
  4. if $\left|x\right|\ge a$, then either $x\le -a$ or $x\ge a$.

Example 8

Solve the inequality $\left|2\cdot x+1\right|

This inequality is equivalent to the inequalities $-7

From here we get: $-8

The number line, the number axis, is the line on which real numbers are depicted. On the straight line, select the origin - point O (point O represents 0) and point L, representing unity. Point L is usually located to the right of point O. The segment OL is called a unit segment.

The dots to the right of point O represent positive numbers. Points to the left of a point. Oh, they represent negative numbers. If point X represents a positive number x, then distance OX = x. If point X represents a negative number x, then the distance OX = - x.

The number showing the position of a point on a line is called the coordinate of this point.

Point V shown in the figure has a coordinate of 2, and point H has a coordinate of -2.6.

The modulus of a real number is the distance from the origin to the point corresponding to this number. The modulus of a number x is denoted as follows: | x |. It is obvious that | 0 | = 0.

If the number x is greater than 0, then | x | = x, and if x is less than 0, then | x | = - x. The solution of many equations and inequalities with the module is based on these properties of the module.

Example: Solve Equation | x – 3 | = 1.

Solution: Consider two cases - the first case, when x -3 > 0, and the second case, when x - 3 0.

1. x - 3 > 0, x > 3.

In this case | x – 3 | = x – 3.

The equation takes the form x – 3 = 1, x = 4. 4 > 3 – satisfy the first condition.

2. x -3 0, x 3.

In this case | x – 3 | = - x + 3

The equation takes the form x + 3 = 1, x = - 2. -2 3 – satisfy the second condition.

Answer: x = 4, x = -2.

Numeric expressions.

A numerical expression is a collection of one or more numbers and functions connected by arithmetic symbols and parentheses.
Examples of numeric expressions:

The value of a numeric expression is a number.
Operations in numerical expression are performed in the following sequence:

1. Actions in brackets.

2. Calculation of functions.

3. Exponentiation

4. Multiplication and division.

5. Addition and subtraction.

6. Similar operations are performed from left to right.

So the value of the first expression will be the number 12.3 itself
In order to calculate the value of the second expression, we will perform the actions in the following sequence:



1. Let's perform the actions in brackets in the following sequence - first we raise 2 to the third power, then we subtract 11 from the resulting number:

3 4 + (23 - 11) = 3 4 + (8 - 11) = 3 4 + (-3)

2. Multiply 3 by 4:

3 4 + (-3) = 12 + (-3)

3. Perform sequential operations from left to right:

12 + (-3) = 9.
An expression with variables is a collection of one or more numbers, variables and functions connected by arithmetic symbols and parentheses. The values ​​of expressions with variables depend on the values ​​of the variables included in it. The sequence of operations here is the same as for numerical expressions. It is sometimes useful to simplify expressions with variables by performing various actions - putting them out of brackets, opening brackets, grouping, reducing fractions, bringing similar ones, etc. Also, to simplify expressions, various formulas are often used, for example, abbreviated multiplication formulas, properties of various functions, etc.

Algebraic expressions.

An algebraic expression is one or more algebraic quantities (numbers and letters), interconnected by signs of algebraic operations: addition, subtraction, multiplication and division, as well as taking the root and raising to an integer power (and the exponents of the root and the power must necessarily be integers) and signs of the sequence of these actions (usually brackets various types). The number of quantities included in an algebraic expression must be finite.

Example algebraic expression:

“Algebraic expression” is a syntactic concept, that is, something is an algebraic expression if and only if it obeys certain grammatical rules (see Formal grammar). If the letters in an algebraic expression are considered variables, then the algebraic expression takes on the meaning of an algebraic function.

Equations with modules, solution methods. Part 1.

Before you begin directly studying techniques for solving such equations, it is important to understand the essence of the module, its geometric meaning. It is in understanding the definition of the module and its geometric meaning that the main methods for solving such equations are laid. The so-called method of intervals when opening modular brackets is so effective that using it it is possible to solve absolutely any equation or inequality with moduli. In this part, we will study in detail two standard methods: the interval method and the population replacement method.

However, as we will see, these methods are always effective, but not always convenient and can lead to long and even not very convenient calculations, which naturally require more time to solve. Therefore, it is important to know those methods that significantly simplify the solution of certain equation structures. Squaring both sides of an equation, a method for introducing a new variable, graphic method, solving equations containing a modulus under the modulus sign. We will look at these methods in the next part.

Determination of the modulus of a number. Geometric meaning of the module.

First of all, let's get acquainted with the geometric meaning of the module:

Modulus of numbers a (|a|) call the distance on the number line from the origin (point 0) to the point A(a).

Based on this definition, let's look at some examples:

|7| - this is the distance from 0 to point 7, of course it is equal to 7. → | 7 |=7

|-5|- this distance from 0 to point -5 and it is equal to: 5. → |-5| = 5

We all understand that distance cannot be negative! Therefore |x| ≥ 0 always!

Let's solve the equation: |x |=4

This equation can be read like this: the distance from point 0 to point x is 4. Yeah, it turns out that from 0 we can move both to the left and to the right, which means moving to the left at a distance equal to 4 we will end up at the point: -4, and moving to the right we will end up at point: 4. Indeed, |-4 |=4 and |4 |=4.

Hence the answer is x=±4.

If you carefully study the previous equation, you will notice that: the distance to the right along the number line from 0 to the point is equal to the point itself, and the distance to the left from 0 to the number is equal to the opposite number! Understanding that the numbers to the right of 0 are positive, and the numbers to the left of 0 are negative, we formulate definition of the modulus of a number: modulus ( absolute value) numbers X(|x|) is the number itself X, if x ≥0, and number – X, if x<0.

Here we need to find a set of points on the number line, the distance from 0 to which will be less than 3, let's imagine a number line, point 0 on it, go to the left and count one (-1), two (-2) and three (-3), stop. Next will be points that lie further than 3 or the distance to which from 0 is greater than 3, now we go to the right: one, two, three, stop again. Now we select all our points and get the interval x: (-3;3).

It is important that you see this clearly, if you still can’t, draw it on paper and look so that this illustration is completely clear to you, don’t be lazy and try to see the solutions to the following tasks in your mind:

|x |=11, x=? |x|=-5, x=?

|x |<8, х-? |х| <-6, х-?

|x |>2, x-? |x|> -3, x-?

|π-3|=? |-x²-10|=?

|√5-2|=? |2х-х²-3|=?

|x²+2|=? |x²+4|=0

|x²+3x+4|=? |-x²+9| ≤0

Did you notice the strange tasks in the second column? Indeed, the distance cannot be negative therefore: |x|=-5- has no solutions, of course it cannot be less than 0, therefore: |x|<-6 тоже не имеет решений, ну и естественно, что любое расстояние будет больше отрицательного числа, значит решением |x|>-3 are all numbers.

After you learn to quickly see pictures with solutions, read on.

In this article we will analyze in detail the absolute value of a number. We will give various definitions of the modulus of a number, introduce notation and provide graphic illustrations. At the same time, let's look at various examples of finding the modulus of a number by definition. After this, we will list and justify the main properties of the module. At the end of the article, we’ll talk about how the modulus of a complex number is determined and found.

Page navigation.

Number module - definition, notation and examples

First we introduce number modulus designation. We will write the modulus of the number a as , that is, to the left and right of the number we will put vertical dashes to form the modulus sign. Let's give a couple of examples. For example, module −7 can be written as ; module 4.125 is written as , and the module has a notation of the form .

The following definition of modulus refers to , and therefore to , and to integers, and to rational, and to irrational numbers, as constituent parts of the set of real numbers. We will talk about the modulus of a complex number in.

Definition.

Modulus of number a– this is either the number a itself, if a is a positive number, or the number −a, the opposite of the number a, if a is a negative number, or 0, if a=0.

The voiced definition of the modulus of a number is often written in the following form , this entry means that if a>0 , if a=0 , and if a<0 .

The record can be presented in a more compact form . This notation means that if (a is greater than or equal to 0), and if a<0 .

There is also the entry . Here we should separately explain the case when a=0. In this case we have , but −0=0, since zero is considered a number that is opposite to itself.

Let's give examples of finding the modulus of a number using the stated definition. For example, let's find the modules of the numbers 15 and . Let's start by finding . Since the number 15 is positive, its modulus, by definition, is equal to this number itself, that is, . What is the modulus of a number? Since is a negative number, its modulus is equal to the number opposite to the number, that is, the number . Thus, .

To conclude this point, we present one conclusion that is very convenient to use in practice when finding the modulus of a number. From the definition of the modulus of a number it follows that the modulus of a number is equal to the number under the modulus sign without taking into account its sign, and from the examples discussed above this is very clearly visible. The stated statement explains why the module of a number is also called absolute value of the number. So the modulus of a number and the absolute value of a number are one and the same.

Modulus of a number as a distance

Geometrically, the modulus of a number can be interpreted as distance. Let's give determining the modulus of a number through distance.

Definition.

Modulus of number a– this is the distance from the origin on the coordinate line to the point corresponding to the number a.

This definition is consistent with the definition of the modulus of a number given in the first paragraph. Let's clarify this point. The distance from the origin to the point corresponding to a positive number is equal to this number. Zero corresponds to the origin, therefore the distance from the origin to the point with coordinate 0 is equal to zero (you do not need to set aside a single unit segment and not a single segment that makes up any fraction of a unit segment in order to get from point O to a point with coordinate 0). The distance from the origin to a point with a negative coordinate is equal to the number opposite to the coordinate of this point, since it is equal to the distance from the origin to the point whose coordinate is the opposite number.

For example, the modulus of the number 9 is equal to 9, since the distance from the origin to the point with coordinate 9 is equal to nine. Let's give another example. The point with coordinate −3.25 is located at a distance of 3.25 from point O, so .

The stated definition of the modulus of a number is a special case of the definition of the modulus of the difference of two numbers.

Definition.

Modulus of the difference of two numbers a and b is equal to the distance between the points of the coordinate line with coordinates a and b.


That is, if points on the coordinate line A(a) and B(b) are given, then the distance from point A to point B is equal to the modulus of the difference between the numbers a and b. If we take point O (origin) as point B, then we get the definition of the modulus of a number given at the beginning of this paragraph.

Determining the modulus of a number using the arithmetic square root

Occasionally occurs determining modulus via arithmetic square root.

For example, let's calculate the moduli of the numbers −30 and based on this definition. We have. Similarly, we calculate the module of two thirds: .

The definition of the modulus of a number through the arithmetic square root is also consistent with the definition given in the first paragraph of this article. Let's show it. Let a be a positive number, and let −a be a negative number. Then And , if a=0 , then .

Module properties

The module has a number of characteristic results - module properties. Now we will present the main and most frequently used of them. When justifying these properties, we will rely on the definition of the modulus of a number in terms of distance.

    Let's start with the most obvious property of the module - The modulus of a number cannot be a negative number. In literal form, this property has the form for any number a. This property is very easy to justify: the modulus of a number is a distance, and distance cannot be expressed as a negative number.

    Let's move on to the next module property. The modulus of a number is zero if and only if this number is zero. The modulus of zero is zero by definition. Zero corresponds to the origin; no other point on the coordinate line corresponds to zero, since each real number is associated with a single point on the coordinate line. For the same reason, any number other than zero corresponds to a point different from the origin. And the distance from the origin to any point other than point O is not zero, since the distance between two points is zero if and only if these points coincide. The above reasoning proves that only the modulus of zero is equal to zero.

    Go ahead. Opposite numbers have equal modules, that is, for any number a. Indeed, two points on the coordinate line, the coordinates of which are opposite numbers, are at the same distance from the origin, which means the modules of the opposite numbers are equal.

    The following property of the module is: The modulus of the product of two numbers is equal to the product of the moduli of these numbers, that is, . By definition, the modulus of the product of numbers a and b is equal to either a·b if , or −(a·b) if . From the rules of multiplication of real numbers it follows that the product of the moduli of numbers a and b is equal to either a·b, , or −(a·b) if , which proves the property in question.

    The modulus of the quotient of a divided by b is equal to the quotient of the modulus of a number divided by the modulus of b, that is, . Let us justify this property of the module. Since the quotient is equal to the product, then. By virtue of the previous property we have . All that remains is to use the equality , which is valid by virtue of the definition of the modulus of a number.

    The following property of a module is written as an inequality: , a , b and c are arbitrary real numbers. The written inequality is nothing more than triangle inequality. To make this clear, let’s take points A(a), B(b), C(c) on the coordinate line, and consider a degenerate triangle ABC, whose vertices lie on the same line. By definition, the modulus of the difference is equal to the length of the segment AB, - the length of the segment AC, and - the length of the segment CB. Since the length of any side of a triangle does not exceed the sum of the lengths of the other two sides, then the inequality is true , therefore, the inequality is also true.

    The inequality just proved is much more common in the form . The written inequality is usually considered as a separate property of the module with the formulation: “ The modulus of the sum of two numbers does not exceed the sum of the moduli of these numbers" But the inequality follows directly from the inequality if we put −b instead of b and take c=0.

Modulus of a complex number

Let's give definition of the modulus of a complex number. May it be given to us complex number, written in algebraic form, where x and y are some real numbers, representing, respectively, the real and imaginary parts of a given complex number z, and is the imaginary unit.

Definition.

Modulus of a complex number z=x+i·y is the arithmetic square root of the sum of the squares of the real and imaginary parts of a given complex number.

The modulus of a complex number z is denoted as , then the stated definition of the modulus of a complex number can be written as .

This definition allows you to calculate the modulus of any complex number in algebraic notation. For example, let's calculate the modulus of a complex number. In this example, the real part of a complex number is equal to , and the imaginary part is equal to minus four. Then, by the definition of the modulus of a complex number, we have .

The geometric interpretation of the modulus of a complex number can be given through distance, by analogy with the geometric interpretation of the modulus of a real number.

Definition.

Modulus of a complex number z is the distance from the beginning of the complex plane to the point corresponding to the number z in this plane.

According to the Pythagorean theorem, the distance from point O to a point with coordinates (x, y) is found as , therefore, , where . Therefore, the last definition of the modulus of a complex number agrees with the first.

This definition also allows you to immediately indicate what the modulus of a complex number z is equal to, if it is written in trigonometric form as or in demonstrative form. Here . For example, the modulus of a complex number is equal to 5, and the modulus of a complex number is equal to .

You can also notice that the product of a complex number and its complex conjugate number gives the sum of the squares of the real and imaginary parts. Really, . The resulting equality allows us to give another definition of the modulus of a complex number.

Definition.

Modulus of a complex number z is the arithmetic square root of the product of this number and the number complex conjugate of it, that is, .

In conclusion, we note that all the properties of a module formulated in the corresponding paragraph are also valid for complex numbers.

Bibliography.

  • Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
  • Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
  • Luntz G.L., Elsgolts L.E. Functions of a complex variable: a textbook for universities.
  • Privalov I.I. Introduction to the theory of functions of a complex variable.

Video lesson “Geometric meaning of the modulus of a real number” is a visual aid for a mathematics lesson on the relevant topic. The video lesson examines in detail and clearly the geometric meaning of the module, after which examples reveal how to find the module of a real number, and the solution is accompanied by a drawing. The material can be used at the stage of explaining a new topic as a separate part of the lesson or to provide clarity for the teacher’s explanation. Both options contribute to increasing the effectiveness of a mathematics lesson and help the teacher achieve the lesson’s goals.

This video tutorial contains constructions that clearly demonstrate the geometric meaning of the module. To make the demonstration more visual, these constructions are performed using animation effects. To make the educational material easier to remember, important points are highlighted in color. The solution to the examples is discussed in detail, which, due to animation effects, is presented in a structured, consistent, and understandable manner. When compiling the video, tools were used that help make the video lesson an effective modern learning tool.

The video begins by introducing the topic of the lesson. A construction is being made on the screen - a ray is depicted on which points a and b are marked, the distance between which is marked as ρ(a;b). It is recalled that the distance is measured on the coordinate ray by subtracting the smaller number from the larger number, that is, for a given construction, the distance is equal to b-a for b>a and equal to a-b for a>b. Below we demonstrate a construction on which the marked point a lies to the right of b, that is, corresponding to it numeric value more b. Below is another case when the positions of points a and b coincide. In this case, the distance between the points is zero ρ(a;b)=0. All together, these cases are described by one formula ρ(a;b)=|a-b|.

Next, we consider solving problems in which knowledge about the geometric meaning of the module is used. In the first example, you need to solve the equation |x-2|=3. It is noted that this is an analytical form of writing this equation, which we translate into geometric language to find a solution. Geometrically, this problem means that it is necessary to find points x for which the equality ρ(x;2)=3 is true. On the coordinate line, this will mean that the points x are equidistant from the point x=2 at a distance of 3. To demonstrate the solution on the coordinate line, a ray is drawn on which point 2 is marked. At a distance of 3 from the point x=2, points -1 and 5 are marked. Obviously , that these marked points will be the solution to the equation.

To solve the equation |x+3,2|=2, it is proposed to first reduce it to the form |a-b| in order to solve the problem on the coordinate line. After transformation, the equation becomes |x-(-3,2)|=2. This means that the distance between the point -3.2 and the desired points will be equal to 2, that is, ρ(x;-3.2)=2. Point -3,2 is marked on the coordinate line. From it, at a distance of 2, points -1,2 and -5,2 are located. These points are marked on the coordinate line and indicated as the solution to the equation.

The solution to another equation |x|=2.7 considers the case when the required points are located at a distance of 2.7 from point 0. The equation is rewritten as |x-0|=2.7. It is indicated that the distance to the required points is determined as ρ(x;0)=2.7. The origin point 0 is marked on the coordinate line. Points -2.7 and 2.7 are located at a distance of 2.7 from point 0. These points are marked on the constructed line; they are the solutions to the equation.

To solve the following equation |x-√2|=0 no geometric interpretation is required because if the modulus of the expression is zero, it means that this expression is equal to zero, that is, x-√2=0. From the equation it follows that x=√2.

The following example examines solving equations that require transformation before solution. In the first equation |2x-6|=8 there is a numerical coefficient 2 before x. To get rid of the coefficient and translate the equation into geometric language ρ(x;a)=b, we take the common factor out of brackets, getting |2(x-3) |=2|x-3|. After this, the right and left sides of the equation are reduced by 2. We obtain an equation of the form |x-3|=4. This analytical equation is translated into geometric language ρ(x;3)=4. We mark point 3 on the coordinate line. From this point we plot points located at a distance of 4. The solution to the equation will be points -1 and 7, which are marked on the coordinate line. The second equation considered |5-3x|=6 also contains a numerical coefficient in front of the x variable. To solve the equation, coefficient 3 is taken out of brackets. The equation becomes |-3(x-5/3)|=3|x-5/3|. The right and left sides of the equation can be reduced by 3. After this, an equation of the form |x-5/3|=2 is obtained. We move from the analytical form to the geometric interpretation of ρ(x;5/3)=2. To accompany the solution, a drawing is drawn that depicts a coordinate line. Point 5/3 is marked on this line. At a distance of 2 from point 5/3 there are points -1/3 and 11/3. These points are the solutions to the equation.

The last equation considered was |4x+1|=-2. To solve this equation, no transformations or geometric representation are required. The left side of the equation obviously produces a non-negative number, and the right side contains the number -2. Therefore, this equation has no solutions.

The video lesson “Geometric meaning of the modulus of a real number” can be used in a traditional mathematics lesson at school. The material may be useful to a teacher implementing Remote education. A detailed, clear explanation of how to solve tasks that use the module function will help a student who is mastering the topic on their own to master the material.