This article is devoted to techniques for solving various equations and inequalities containing
variable under the module sign.
If you come across an equation or inequality with a module on the exam, you can solve it by
not knowing any special methods and using only the module definition. True,
it can take an hour and a half of precious exam time.
Therefore, we want to tell you about the techniques that simplify the solution of such problems.
Let us first of all recall that
Consider the different types equations with modulus... (We will move on to inequalities later.)
Module on the left, number on the right
This is the simplest case. Let's solve the equation
There are only two numbers whose modules are equal to four. These are 4 and −4. Hence the equation
is equivalent to a combination of two simple ones:
The second equation has no solutions. Solutions to the first: x \u003d 0 and x \u003d 5.
Answer: 0; five.
Variable both under the module and outside the module
Here you have to expand the module by definition. ... ... or think!
The equation falls into two cases, depending on the sign of the expression under the modulus.
In other words, it is tantamount to a combination of two systems:
Solution of the first system:. The second system has no solutions.
Answer: 1.
First case: x ≥ 3. Remove the module:
The number, being negative, does not satisfy the condition x ≥ 3 and therefore is not a root of the original equation.
Let us find out whether the number satisfies this condition. To do this, compose the difference and determine its sign:
Hence, it is more than three and therefore is the root of the original equation
Second case: x< 3. Снимаем модуль:
Number. greater than, and therefore does not satisfy the condition x< 3. Проверим :
Hence,. is the root of the original equation.
Remove the module by definition? It's scary to even think about it, because the discriminant is not a complete square. Let's better use the following consideration: an equation of the form | A | \u003d B is equivalent to the combination of two systems:
The same, but slightly different:
In other words, we solve two equations, A \u003d B and A \u003d −B, and then select roots that satisfy the condition B ≥ 0.
Let's get started. First, we solve the first equation:
Then we solve the second equation:
Now, in each case, we check the sign of the right side:
Therefore, only and are suitable.
Quadratic equations with the replacement | x | \u003d t
Let's solve the equation:
Since, it is convenient to make the replacement | x | \u003d t. We get:
Answer: ± 1.
Module is equal to module
We are talking about equations of the form | A | \u003d | B |. This is a gift of fate. No module disclosures by definition! It's simple:
For example, consider the equation:. It is equivalent to the following aggregate:
It remains to solve each of the equations of the set and write down the answer.
Two or more modules
Let's solve the equation:
Let's not bother with each module separately and expand it by definition - there will be too many options. There is a more rational way - the method of intervals.
The modulus expressions vanish at the points x \u003d 1, x \u003d 2 and x \u003d 3. These points divide the number line into four intervals (intervals). Let us mark these points on the number line and arrange the signs for each of the expressions under the modules at the intervals obtained. (The order of the signs is the same as the order of the corresponding modules in the equation.)
Thus, we need to consider four cases - when x is in each of the intervals.
Case 1: x ≥ 3. All modules are removed "with a plus":
The resulting value x \u003d 5 satisfies the condition x ≥ 3 and therefore is the root of the original equation.
Case 2: 2 ≤ x ≤ 3. The last module is now removed "with a minus":
The resulting value of x is also good - it belongs to the interval under consideration.
Case 3: 1 ≤ x ≤ 2. The second and third modules are removed "with a minus":
We obtained the correct numerical equality for any x from the considered interval serve as solutions to this equation.
Case 4: x ≤ 1 ≤ 1. The second and third modules are removed "with a minus":
Nothing new. We already know that x \u003d 1 is a solution.
Answer: ∪ (5).
Module in module
Let's solve the equation:
We start by expanding the internal module.
1) x ≤ 3. We get:
The expression under the modulus vanishes at. This point belongs to the considered
interval. Therefore, we have to analyze two subcases.
1.1) We get in this case:
This x value is not valid because it does not belong to the interval under consideration.
1.2). Then:
This x value is also not valid.
So, for x ≤ 3 there are no solutions. We pass to the second case.
2) x ≥ 3. We have:
Here we are in luck: the expression x + 2 is positive in the interval under consideration! Therefore, there will be no more sub-cases: the module is removed "with a plus":
This value of x is in the considered interval and therefore is the root of the original equation.
This is how all tasks of this type are solved - we open the nested modules one by one, starting with the internal one.
The modulus is the absolute value of the expression. To at least somehow denote the module, it is customary to use straight brackets. The value that is enclosed in straight brackets is the value that is taken modulo. The process of solving any module consists in expanding the very right brackets, which are called modular brackets in mathematical language. Their disclosure takes place according to a certain number of rules. Also, in the order of solving modules, there are also the sets of values \u200b\u200bof those expressions that were in the module brackets. In most of all cases, a module is expanded in such a way that an expression that was submodular gets both positive and negative values, including the value zero. Based on the set properties of the module, then in the process, various equations or inequalities from the original expression, which then need to be solved. Let's figure out how to solve modules.
Solution process
The solution to the module begins by writing the original equation with the module. To answer the question of how to solve equations with a module, you need to expand it completely. To solve such an equation, the module is expanded. All modular expressions must be considered. It is necessary to determine at what values \u200b\u200bof the unknown quantities included in its composition, the modular expression in parentheses turns to zero. To do this, it is enough to equate the expression in modular brackets to zero, and then calculate the solution of the resulting equation. The found values \u200b\u200bmust be recorded. In the same way, it is also necessary to determine the value of all unknown variables for all modules in this equation. Next, you need to deal with the definition and consideration of all cases of existence of variables in expressions when they are different from the value zero. To do this, you need to write down some system of inequalities according to all modules in the original inequality. Inequalities should be designed so that they cover all available and possible values \u200b\u200bfor a variable that are found on the number line. Then you need to draw this very numerical line for visualization, on which in the future you will postpone all the obtained values.
Almost everything can now be done on the Internet. The module is no exception to the rule. You can solve it online on one of the many modern resources. All those values \u200b\u200bof the variable that are in the zero module will be a special constraint that will be used in the process of solving the modular equation. In the original equation, it is required to expand all available modular brackets, while changing the sign of the expression so that the values \u200b\u200bof the desired variable coincide with those values \u200b\u200bthat can be seen on the number line. The resulting equation must be solved. The value of the variable that will be obtained during the solution of the equation must be checked against the constraint set by the module itself. If the value of the variable fully satisfies the condition, then it is correct. All roots that will be obtained during the solution of the equation, but will not fit the constraints, should be discarded.
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. In this case, we will consider various examples of finding the modulus of a number by definition. After that, we will list and justify the main properties of the module. At the end of the article, let's talk about how the module is defined and located. complex number.
Page navigation.
Number module - definition, notation and examples
First we introduce modulus of number... The modulus of the number a will be written as, that is, to the left and right of the number we will put vertical dashes forming the modulus sign. Here are a couple of examples. For example, the module −7 can be written as; module 4.125 is written as, and the module is written as.
The following definition of a module 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'll talk about the complex number module in.
Definition.
Modulus of number a Is either the number a itself, if a is a positive number, or the number −a, opposite to the number a, if a is a negative number, or 0, if a \u003d 0.
The sounded definition of the module of a number is often written in the following form 
, this notation means that if a\u003e 0, if a \u003d 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 a record 
... Here, the case where a \u003d 0 should be clarified separately. In this case, we have, but −0 \u003d 0, since zero is considered a number that is opposite to itself.
Let us give examples of finding the modulus of a number using the sounded definition. For example, let's find the modules of 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,. And what is the absolute value of a number? Since is a negative number, its modulus is equal to the opposite number, that is, the number 
... In this way, .
In conclusion of this paragraph, we present one conclusion, which is very convenient to apply in practice when finding the modulus of a number. It follows from the definition of the modulus of a number that modulus of a number is equal to the number under the modulus sign without regard to its sign, and from the examples considered above, this is very clearly visible. The above 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 number as distance
Geometrically, the module of a number can be interpreted as distance... Let us give determination of the modulus of a number through distance.
Definition.
Modulus of number a 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 us explain this point. The distance from the origin to the point to which the positive number corresponds is equal to this number. The origin corresponds to zero, so the distance from the origin to the point with coordinate 0 is equal to zero (you do not need to postpone 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 the 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 absolute value of 9 is 9, since the distance from the origin to the point with coordinate 9 is nine. Let's give another example. The point with the coordinate −3.25 is from the point O at a distance of 3.25, so 
.
The sounded definition of the modulus of a number is a special case of determining the modulus of the difference of two numbers.
Definition.
Difference modulus 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 are given on the coordinate line A (a) and B (b), 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 absolute value of a number given at the beginning of this paragraph.
Determining the modulus of a number through arithmetic square root
Sometimes occurs definition of modulus in terms of arithmetic square root.
For example, let's calculate the absolute values \u200b\u200bof the numbers −30 and based on this definition. We have. Similarly, we calculate the module of the two thirds: 
.
The definition of the modulus of a number through the arithmetic square root also agrees with the definition given in the first paragraph of this article. Let's show it. Let a be a positive number, and the number −a is negative. Then 
and 
, if a \u003d 0, then 
.
Module properties
The module has a number of characteristic results - module properties... Now we will present the main and most frequently used ones. When substantiating 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 a module - modulus of a number cannot be negative... In literal form, this property has the form for any number a. This property is very easy to substantiate: the modulus of a number is distance, and distance cannot be expressed as a negative number.
Let's move on to the next property of the module. The absolute value 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 other than 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.
Move on. 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 that the absolute values \u200b\u200bof opposite numbers are equal.
The next property of the module is as follows: the modulus of the product of two numbers is equal to the product of the moduli of these numbers, i.e, . By definition, the modulus of the product of the numbers a and b is either a b if, or - (a b) if. It follows from the rules for multiplying real numbers that the product of the absolute values \u200b\u200bof the numbers a and b is equal to either a b, or - (a b), if, which proves the property under consideration.
The modulus of the quotient of dividing a by b is equal to the quotient of dividing the modulus of the number a by the modulus of the number b, i.e, . Let's justify this property of the module. Since the quotient is equal to the product, then. By virtue of the previous property, we have 
... It remains only 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 inequality recorded is nothing more than triangle inequality... To make this clear, take the points A (a), B (b), C (c) on the coordinate line, and consider the degenerate triangle ABC, whose vertices lie on one straight line. By definition, the modulus of the difference is equal to the length of the segment AB, is the length of the segment AC, and is 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, the inequality 
hence 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 absolute value of the sum of two numbers does not exceed the sum of the absolute values \u200b\u200bof these numbers". But the inequality follows directly from the inequality if we put −b instead of b and take c \u003d 0.
Complex number module
Let's give determining 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, which are, respectively, the real and imaginary parts of a given complex number z, and is an imaginary unit.
Definition.
By the modulus of a complex number z \u003d x + i · y is called 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 sounded 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 the complex number is and the imaginary part is 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 in terms of distance, by analogy with the geometric interpretation of the modulus of a real number.
Definition.
Complex number module z is the distance from the origin of the complex plane to the point corresponding to the number z in that plane.
According to the Pythagorean theorem, the distance from point O to the 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 exemplary form. Here . For example, the modulus of a complex number 
is 5, and the modulus of a complex number is.
You can also notice that the product of a complex number by a complex conjugate number gives the sum of the squares of the real and imaginary parts. Indeed,. The resulting equality allows us to give one more definition of the modulus of a complex number.
Definition.
Complex number module z is the arithmetic square root of the product of this number and the complex conjugate of it, that is,.
In conclusion, we note that all the properties of the module formulated in the corresponding subsection are also valid for complex numbers.
Bibliography.
- Vilenkin N. Ya. and other Mathematics. Grade 6: textbook for educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for grade 8. educational institutions.
- Lunts 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.