Volume 04 Issue 12-2024
312
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
04
ISSUE
12
Pages:
312-317
OCLC
–
1368736135
A
BSTRACT
This article describes the ways in which equations involving integers and fractions are given in school
mathematics classes and their solution methods.
K
EYWORDS
Equations and inequalities involving whole number, fractional part of number, whole number and
fractional part of number are covered.
I
NTRODUCTION
In the plan of measures indicated in the
Resolution of the President of the Republic of
Uzbekistan dated May 7, 2020 "On measures to
increase the quality of education in the field of
mathematics and develop scientific research" No.
PD-4708 tasks such as organization of research in
cooperation with world scientific centers of
development, organization of popularization of
modern science achievements, strengthening of
relations between scientific research activities
and the educational process have been defined.
Journal
Website:
http://sciencebring.co
m/index.php/ijasr
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Research Article
METHODOLOGY FOR SOLVING EQUATIONS WITH
MULTIPLICATIONS OF INTEGER AND FRACTIONAL PARTS IN
SCHOOL MATHEMATICS LESSONS
Submission Date:
December 15,
2024,
Accepted Date:
December 20, 2024,
Published Date:
December 30, 2024
Crossref doi:
https://doi.org/10.37547/ijasr-04-12-48
Zulfikharov Ilhom Mahmudovich
Republic Of Uzbekistan, Andijan Institute Of Mechanical Engineering, Senior Lecturer Of The Department Of
Information Technology, Uzbekistan
Mamasidikov Bokhodir Kobilzhonovich
Republic Of Uzbekistan, Andijan Institute Of Mechanical Engineering, Assistant Of The Department Of
Information Technology, Uzbekistan
Volume 04 Issue 12-2024
313
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
04
ISSUE
12
Pages:
312-317
OCLC
–
1368736135
Today, our community of pedagogues requires
urgent tasks such as developing new methods of
education,
strengthening
inter-disciplinary
connections (integration), educating young
people who can think creatively and
independently in every way.
General secondary education is compulsory
education consisting of I-XI grades. This type of
education covers the primary class (grades I-IV)
and provides students with regular knowledge of
thinking, academic and general cultural
knowledge, spiritual and moral qualities based on
universal national values, work forms skills and
career choice.
Current textbooks, educational and methodical
manuals provide enough information about the
equation and its types. Many methods of solving
them have been developed. However, the method
of solving equations involving whole and
fractional numbers in the school mathematics
course and explaining them to students is not
sufficiently developed. That's why we set
ourselves the goal of "developing a methodology
for solving various equations involving whole and
fractional parts and explaining them to
students"[5].
To achieve this goal, it is necessary to perform the
following tasks:
to determine the current methodical
conditions of solving equations involving the
whole and fractional part of the number among
students of the school;
to determine the content and structure of
the equations in which the whole and fractional
parts of the number are involved;
use and improve the methods of
experienced advanced pedagogues for solving
equations involving whole and fractional
numbers in educational materials for teaching
mathematics in schools;
to determine the content and structure of
the equations in which the whole and fractional
parts of the number are involved;
show the method of solving equations
involving the whole and fractional part of the
number in the secondary school.
We are trying to improve the scientific methodical
solution of equations of different forms (rational
and transcendental) involving whole and
fractional numbers, based on the purpose,
content, form, method and tools of mathematical
education:
we determine the structure and content of
the educational material for school students to
form the methodology of solving equations
involving whole and fractional numbers;
we improve students' methods of solving
rational equations involving whole and fractional
numbers.
If we look at the history of mathematics, the
famous Greek mathematician Euclid in his work
"Fundamentals" explained algebraic expressions
and the operations between them by
intersections, that is, he used geometric algebra.
Volume 04 Issue 12-2024
314
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
04
ISSUE
12
Pages:
312-317
OCLC
–
1368736135
Russian mathematicians I.F. Sharigin, N.B.
Alfutova, G.Z. Genkin, V.L. Kryukova, Ukrainian
mathematician I.A. Kushnir, Tajik mathematician
A. Sufiyev conducted research on the solutions of
problems involving whole and fractional
numbers.
Before we teach students about equations
involving whole and fractional numbers, we
should give them information about the equation
and its solution.
The concept of an equation is presented in a
school mathematics course, and students learn to
find the unknown component when two of the
components involved in addition, subtraction,
and division are known.
An equation with an unknown number is called an
equation. The value of the unknown that makes
the equation a true equation is called the solution
or root of the equation.
We have studied the equations in which whole
and fractional parts of the number are involved,
divided into several groups [2,3,4].
1.
𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒐𝒓𝒎 [𝒇(𝒙)] =
𝒂 𝒂𝒏𝒅 {𝒈(𝒙)} = 𝒃.
From the equation
[𝑓(𝑥)] = 𝑎, 𝑎 ∈ 𝑍
we form the
relation
𝑎 ≤ 𝑓(𝑥) < 𝑎 + 1
, then
[𝑓(𝑥)] = 𝑓(𝑥) −
{𝑓(𝑥)}
and put into the given equation:
𝑓(𝑥) − {𝑓(𝑥)} = 𝑎
→
{𝑓(𝑥)} = 𝑓(𝑥) − 𝑎
Since
0 ≤ {𝑓(𝑥)} < 1
,
0 ≤ 𝑓(𝑥) − 𝑎 < 1
.
Example 1
. We solve the equation
[𝑥
2
+
2𝑥 − 2] = 1
The solution
. We reduce this equation to
a double inequality based on the above.
1 ≤ 𝑥
2
+ 2𝑥 − 2 < 2
1)
𝑥
2
+ 2𝑥 − 2 ≥ 1 → 𝑥
2
+ 2𝑥 − 3 ≥
0 →
→ (𝑥 + 3)(𝑥 − 1) ≥ 0 → 𝑥 ≤ −3; 𝑥 ≥ 1
2)
𝑥
2
+ 2𝑥 − 2 < 2 → 𝑥
2
+ 2𝑥 − 4 < 0 →
→ −1 − √5 < 𝑥 < −1 + √5
−1 − √5 < 𝑥 ≤ −3; ∪ 1 ≤ 𝑥 < −1 + √5
Answer.
−1 − √5 < 𝑥 ≤ −3; ∪ 1 ≤ 𝑥 < −1 +
√5
Example 2
. We solve the equation
{𝑐𝑜𝑠𝑥} =
1
3
, 𝑥 ∈ [0 ; 𝜋]
The solution
. In this equation
𝑐𝑜𝑠𝑥 =
1
3
+ 𝑛,
−1 ≤
1
3
+ 𝑛 ≤ 1, 𝑛 ∈ 𝑍
,
is divided into two simple trigonometric
equations
n
=0 and
n=-1 (
n has no meaning for
other values of the trigonometric equation).
𝑐𝑜𝑠𝑥 =
1
3
and
𝑐𝑜𝑠𝑥 = −
2
3
,
𝑥 ∈ [0 ; 𝜋]
𝑥 = 𝑎𝑟𝑐𝑐𝑜𝑠
1
3
and
𝑥 = 𝜋 − 𝑎𝑟𝑐𝑐𝑜𝑠
2
3
Answer.
𝑎𝑟𝑐𝑐𝑜𝑠
1
3
, 𝜋 − 𝑎𝑟𝑐𝑐𝑜𝑠
2
3
.
2. Equations of the type
𝒌[𝒙] = 𝒙
and
𝒌{𝒙} = 𝒙
To solve equations of this type, first consider it for
the number
𝑘 ∈ 𝑅/0
Let's consider solving the equations
𝑘[𝑥] = 𝑥
and
𝑘{𝑥} = 𝑥
This process is carried out based on the following
algorithm [1,2].
1) Since
[𝑥] =
𝑥
𝑘
from the equation
𝑘[𝑥] = 𝑥
,
𝑥
𝑘
∈
𝑍,
, that is,
𝑥 = 𝑘𝑛, 𝑛 ∈ 𝑍
2) Since
𝑥 − {𝑥} =
𝑥
𝑘
or
{𝑥} = 𝑥 (1 −
1
𝑘
)
,
0 ≤
𝑥(1 −
1
𝑘
) < 1
.
So this is a solution to the equation
Volume 04 Issue 12-2024
315
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
04
ISSUE
12
Pages:
312-317
OCLC
–
1368736135
{
𝑥 = 5𝑛, 𝑛 ∈ 𝑍
0 ≤ 𝑥(1 −
1
𝑘
) < 1
determined from relationships.
Example 3.
We solve the equation
3[𝑥] =
𝑥
The solution.
Since
𝑘 = 3
{
𝑥 = 3𝑛, 𝑛 ∈ 𝑍
0 ≤ 𝑥(1 −
1
3
) < 1
relationship is enough. Since
0 ≤ 𝑥 <
3
2
is equal to
0 ≤ 3𝑛 <
3
2
→ 0 ≤ 𝑛 <
1
2
and
𝑛 = 0
is the
unique integer solution
𝑥 = 0
.
Since
𝑘{𝑥} = 𝑥
from the equation
{𝑥} =
𝑥
𝑘
∈ [0,1)
𝑥 ∈ [0, 𝑘) 𝑎𝑔𝑎𝑟 𝑘 > 0 𝑏𝑜
′
𝑙𝑠𝑎
𝑥 ∈ (𝑘, 0], 𝑎𝑔𝑎𝑟 𝑘 < 0 𝑏𝑜
′
𝑙𝑠𝑎
we form relationships.
Since
{𝑥} = 𝑥 − [𝑥]
the equation under
consideration
𝑘(𝑥 − [𝑥]) = 𝑥 → [𝑥] = (1 −
1
𝑘
) 𝑥
if we take
[𝑥] = 𝑛
since
[𝑥] ∈ 𝑍
to the equation,
𝑘 − 1
𝑘
𝑥 = 𝑛; 𝑥 =
𝑛𝑘
𝑘 − 1
will be.
But taking into account that
𝑥 ∈ [0, 𝑘),
𝑘 > 0
and
𝑥 ∈ (𝑘, 0], 𝑘 < 0
, let
𝑛
be
0 ≤
𝑛𝑘
𝑘−1
< 𝑘,
if
аgаr
𝑘 > 0
and
vа
𝑘 <
𝑛𝑘
𝑘−1
≤ 0
if
𝑘 < 0
, we can
define relations and find the desired solution or
set of solutions.
Example 4
. We solve the equation
3{𝑥} =
𝑥
The solution.
Since
𝑘 = 3 > 0
, the
solution of the equation is in the form
𝑥 =
𝑛𝑘
𝑘−1
=
3𝑛
2
, and the inequality
0 ≤
3𝑛
2
< 3
must be
satisfied. Since
0 ≤ 𝑛 < 2
the corresponding
roots of the unknown
𝑥
consist of
(0 ;
3
2
)
numbers.
3. Equations of the form
[𝒇(𝒙)] = 𝒈(𝒙)
.
When solving equations of this form, we
use the definition of the integral part of the
number, taking into account that the expression
on the left side of the equation is a whole number
and the right part is also a whole number [20].
[𝑓(𝑥)] = 𝑔(𝑥)
from which
𝑔(𝑥) ≤ 𝑓(𝑥) <
𝑔(𝑥) + 1
. If we solve this double inequality, we
get the result.
Example 5.
We solve the equation
[3𝑥 +
4] = 4𝑥 + 5
.
The solution.
Solving this equation using
the property
[𝑥 + 𝑎] = [𝑥] + 𝑎, 𝑥 ∈ 𝑅, 𝑎 ∈ 𝑍
[3𝑥] + 4 = 4𝑥 + 5
→
[3𝑥] = 4𝑥 + 1
we form the equation.
We write this equation in the form of a
double inequality as follows:
4𝑥 + 1 ≤ 3𝑥 < 4𝑥 + 2
we have this system of inequalities:
{
4𝑥 + 1 ≤ 3𝑥
4𝑥 + 2 > 3𝑥
→
{
𝑥 ≤ −1
𝑥 > −2
An integer satisfying this system of
inequalities is -1.
Example 6.
We solve the equation
𝑥
𝑥+4
=
5[𝑥]−7
7[𝑥]−5
The solution
. To solve this equation, we
first introduce the notation
[𝑥] = 𝑛, 𝑛 ∈ 𝑍
:
𝑥
𝑥 + 4
=
5𝑛 − 7
7𝑛 − 5
we find
𝑥
from this equation:
7𝑥𝑛 − 5𝑥 = 5𝑛𝑥 − 7𝑥 + 20𝑛 − 28
Volume 04 Issue 12-2024
316
International Journal of Advance Scientific Research
(ISSN
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2750-1396)
VOLUME
04
ISSUE
12
Pages:
312-317
OCLC
–
1368736135
𝑥 =
10𝑛 − 14
𝑛 + 1
Since the inequality
𝑛 ≤ 𝑥 < 𝑛 + 1
holds
using the integer property
[𝑥] = 𝑛
𝑛 ≤
10𝑛 − 14
𝑛 + 1
< 𝑛 + 1
we get this double inequality.
The solution of this inequality is
𝑛 = 2, 6, 7
.
Now we find the value of
𝑥
. remains equal to
𝑥 =
(2 ;
46
7
; 7)
.
4. Equations involving whole and
fractional parts.
If we are given an equation that contains a
whole and a fractional part in one equation, it is
much more convenient to convert the whole part
into a fractional part or convert the fractional part
into a whole part in order to solve it.
Example 7
. We solve the equation
2
𝑥
=
3
[𝑥]
∙ 4
{𝑥}
The solution.
We logarithmize this
equation according to two bases:
𝑥 = [𝑥]𝑙𝑜𝑔
2
3 + 2{𝑥}
→
{
[𝑥] = 𝑛, 𝑛 ∈ 𝑍
𝑛 ≤ 𝑥 < 𝑛 + 1
𝑥 = 𝑛𝑙𝑜𝑔
2
3 + 2(𝑥 − 𝑛)
let's simplify
{
𝑛 ≤ 𝑥 < 𝑛 + 1
𝑥 = 2𝑛 − 𝑛𝑙𝑜𝑔
2
3
𝑛 ≤ 2𝑛 − 𝑛𝑙𝑜𝑔
2
3 < 𝑛 + 1
0 ≤ 𝑛 − 𝑛𝑙𝑜𝑔
2
3 < 1
0 ≤ 𝑛(1 − 𝑙𝑜𝑔
2
3) < 1
1
1 − 𝑙𝑜𝑔
2
3
< 𝑛 ≤ 0
The solution of this double inequality is
𝑛
1
= −1, 𝑛
2
= 0
.
If we take these values, the
result will be
natija
𝑥 = 𝑙𝑜𝑔
2
3 − 2
and
𝑥 = 0
.
Example 8.
We solve the equation
[𝑥] +
[3𝑥] + [5𝑥] = 4.
The solution
. This equation has no
solution at
𝑥 < 0, 𝑥 > 1
We consider the following cases:
1)
[𝑥] = 0
0 ≤ 𝑥 < 1
,
[3𝑥] = 2
2
3
≤ 𝑥 < 1
,
[5𝑥] = 2
2
5
≤ 𝑥 <
3
5
intersect
2
5
≤ 𝑥 <
3
5
,
[𝑥] = 0
[3𝑥] = 1
[5𝑥] = 2
0 + 1 + 2 = 3 ≠ 4 ∅
,
2)
[𝑥] = 0
0 ≤ 𝑥 < 1
,
[3𝑥] = 1
1
3
≤ 𝑥 <
2
3
,
[5𝑥] = 3
3
5
≤ 𝑥 <
4
5
intersect
3
5
≤ 𝑥 <
4
5
, 3)
[𝑥] =
0
,
[3𝑥] = 1
[5𝑥] = 3
0+1+3=4.
Answer
3
5
≤ 𝑥 <
4
5
.
Example 8.
We solve the equation
[𝑠𝑖𝑛𝑥]{𝑠𝑖𝑛𝑥} = 𝑠𝑖𝑛𝑥
.
The
solution
.
[𝑥] + {𝑥} = 𝑥
,
[𝑠𝑖𝑛𝑥](𝑠𝑖𝑛𝑥 − [𝑠𝑖𝑛𝑥]) = 𝑠𝑖𝑛𝑥
1)
−1 < 𝑠𝑖𝑛𝑥 < 0
,
[𝑠𝑖𝑛𝑥] = −1
𝑠𝑖𝑛𝑥 =
1
2
𝑥 =
±
𝜋
6
+ 𝜋𝑛
.
2)
0 < 𝑠𝑖𝑛𝑥 < 1
,
[𝑠𝑖𝑛𝑥] = 0
𝑠𝑖𝑛𝑥 = 0
𝑥 =
𝜋𝑛
, 3)
𝑠𝑖𝑛𝑥 − 1 = 𝑠𝑖𝑛𝑥
∅
.
One of the indicators of an effectively organized
educational process is the development of
students' mathematical abilities. The student's
research activity is manifested in the ability to
find solutions that differ from standard and
generally accepted solutions, and to apply his
knowledge to various conditions in performing
actions. In order to develop these qualities,
Volume 04 Issue 12-2024
317
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
04
ISSUE
12
Pages:
312-317
OCLC
–
1368736135
providing students with dynamic exercises and
tasks that require research knowledge will
greatly help to stimulate student behavior.
In the teaching of mathematics, students' ability
to solve equations is studied based on the basic
laws, rules, and methodological conditions of the
knowledge system. In this, the issues of creating a
system of mathematical knowledge and its
implementation, in-depth study of the laws and
rules of mathematics, and cases of thorough
mastering are determined.
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EFERENCES
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Decision No. PQ-4708 of the President of the
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quality of education and develop research in
the field of mathematics".
2.
Mathematical Olympiads, Problems and
solutions from around the world, 1998-1999.
Edited by Andreescu T. and Feng Z.
Washington. 2000. p. 216.
3.
Galperin P. Ya., Talizina N. F. Formirovanie
znaniy i umeniy na osnove teorii poetapnogo
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Yunusov.A. S, Afonina.S. I, Berdikulov.M.A,
Yunusova.D. I. Interesting math and Olympiad
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Zulfikharov I.M. Organization of mathematics
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Monograph, 106 pages. "Andijan publishing
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Rakhimov.N.N,
Mamasidikov.B
"Solving
equations involving whole and fractional
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magazine EURASIAN RESEARCH BULLETIN,
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Zulfikharov.I.M.,
Akbarov.S.A.
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