Fractional rational equations 9. Fractional rational equations. Solution algorithm. Respecting your privacy at the company level

Novoselitskaya Municipal educational institution secondary school No. 1

"Solving fractional rational equations."

Open lesson in 9A class

Mathematic teacherDemidenko N.Yu.

S. Novoselitskoye 2015

Lesson topic : Solving fractional rational equations .(Slide 1)

Goals and objectives lesson:

Educational:

consolidation of the concept of a fractional rational equation;

continue to develop the skills to solve fractional rational equations;

repeat solving linear equations;

repeat solving quadratic equations.

Educational:

development of students' memory;

development of skills to overcome difficulties in solving mathematical problems;

development of curiosity;

development of logical thinking, attention, skills to analyze, compare and draw conclusions;

develop interest in the subject.

Educational:

formation of such personality qualities as responsibility, organization, discipline, decency, truthfulness;

promote the formation of a system of knowledge, ideas, concepts;

fostering cognitive interest in the subject;

fostering independence in solving educational problems;

nurturing will and perseverance to achieve final results.

Lesson type: consolidation of the studied material.

Form: workshop lesson.

Lesson equipment: PC, projector, fileMSExcelcontaining test tasks, presentation.

Checking homework

ANSWER THE QUESTIONS (Slide 2)

How many modules are there in the OGE test? What modules are these?

How many points do you need to score to successfully pass the exam?

Formulate the topic of our lesson.

"Solving Equations" (Slide 3)

continue the sentence:

the equation is called...

The root of the equation is...

Verbal counting (Slide 4)

1) x+3=0;

2) 3(x-7)=0;

3) x(x-1)(x+3)(x-9)=0;

4) x³-9x=0;

5) 7x²=0;

6) x²-5=0;

7) -7x²=28.

LET'S REPEAT (Slide 5)

1. What is the name of this equation? How many roots does this equation have?

2. Tell me, what degree is this equation? How many roots does this equation have?

3. Tell me, what degree is this equation? How many roots does this equation have?(X 3 – 1) 2 + x 5 - X 6 = 2

4. What is the name of this equation?

5. How to find the degree of an entire equation? (X 3 – 3) 2 + 5x 2 = 0

CONTINUE PHRASE (Slide 6)

A quadratic equation has 2 roots if......

A quadratic equation has 2 equal roots (or one root) if......

A quadratic equation has no roots if......

The range of acceptable values ​​of a fractional rational equation is.....

SPECIFY ODZ OF EQUATIONS (Slide 7)

a) 2(1-x²) +3x -4 =0;

b)x - 3 = x² - x +1 ;

4 2

c) x² -x - 7 = x +8;

G)2x - 4 = 3__;

x² +1 x +1

d)3x + 1 = x;

x -1

Remember the algorithms for solving equations! (Slide 8)

The equationy ( x ) =0 calledfractional rational equation , Ifexpression y ( x ) isfractional

(i.e. contains division into an expression with variables).(Slide 9)

Algorithms for solving fractional rational equations! (Slide 10)

Find acceptable values ​​of the fractions included in the equation.

Find the common denominator of the fractions in the equation.

Multiply both sides of the equation by the common denominator.

Solve the resulting equation.

5. Eliminate roots that are not included in the permissible values ​​of the fractions of the equation

Example #1: (Slide 11,12)

(Slide 13) Example #2: Kim Option No. 6, task No. 21

(x-2)(x 2 +8x+16) = 7(x+4)

(Slide 14) PHYSICAL MINUTE for the eyes

(Slide 15-19) Independent test work

1. Among these equations, choose the one that is not fractional rational:

1) ;

2)

3) .

(3)

2. At what values ​​of the variableX the equation doesn't make sense:

1) -2;

2) -2 and -1;

3) always makes sense.

(Slide 20) Teacher: Check your result (a table with the correct answers is displayed on the screen).

Let's check the answers with the answers on the board. We put “+” or “-” on the pieces of paper, depending on the correctness of execution. Rate yourself:

everything done correctly – “5”;

one error – “4”;

two mistakes made - “3”;

less than 3 tasks completed – “2”.

(Slide 21) Homework

Tests

Option 20-30 No. 4 (equations)

And I would like to end our lesson with the words of the great scientist A. Einstein:“I have to divide my time between politics and equations. However, equations, in my opinion, are much more important, because politics exists only for this moment, and equations will exist forever.”

(Slide 22) Independent work

Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

Below are some examples of the types of personal information we may collect and how we may use such information.

What personal information do we collect:

• When you submit an application on the site, we may collect various information, including your name, telephone number, email address, etc.

How we use your personal information:

• The personal information we collect allows us to contact you with unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send important notices and communications.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.

Disclosure of information to third parties

We do not disclose the information received from you to third parties.

Exceptions:

• If necessary - in accordance with the law, judicial procedure, in legal proceedings, and/or on the basis of public requests or requests from government authorities in the territory of the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
• In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.

Respecting your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.

Class 9.

Lesson topic:"Fractional rational equations"

Lesson type: combined.

Goals:

1. Educational: give a definition of “fractional rational equations”, show ways to solve such equations.

2. Developmental: development of skills and abilities to solve examples with this type of equations, find the roots of fractional rational equations.

3. Educators: cultivate attention, attentiveness, activity, accuracy; respectful attitude towards the mother.

Tasks:to interest students in the subject, to show the importance of the ability to solve various equations and problems.

Material and technical equipment:

Multimedia projector, screen, presentation for the lesson “Fractional rational equations”

Time: 45 minutes

Lesson plan.

DURING THE CLASSES

I. Reflection on the beginning of the lesson(music; presentation about mother).

Checking readiness for the lesson.

II. Communication of a new topic, goal and task:

Teacher: Hello! Please look at each other and smile from your heart.

I would like to start today’s lesson with the words of M. Gorky:

Slide 1
Without the sun, flowers don't bloom,

without love there is no happiness,

without women there is no love,

without a mother there is neither a poet nor a hero.

All the pride in the world comes from mothers.
(M. Gorky)

Teacher:

– What could be more sacred in the world than the name of a mother! ...

A person who has not yet taken a single step on the ground and is just beginning to “burb”, hesitantly and diligently spells out “mama” syllable by syllable and, feeling his luck, laughs, happy...

When does a baby cry for the first time?

And his mother will touch him carefully,

Her love... Oh, how disturbing she is.

Anxious every day and hour.

Guys, Mother's Day is coming soon, so I want to connect today's lesson with this topic. In previous lessons, we learned how to solve, find the roots of various equations, today we will continue to get acquainted with one of the types of equations - these are fractional rational equations, we will find out the importance of equations, and remember how to solve problems using equations. We will try not to let our mother down, we will decide carefully and without distractions to prepare for the State Examination. The mother of each of you wants her child to be the best. So today we have a lesson on learning a new topic. (slide 2).

III. Repetition of the covered topic.

1. Checking homework(slide 3).

No. 925(a, b), No. 935(a, b), No. 936.

2. We repeat orally(slide 3 ,4,5,6 ).

Let's repeat:

What is the name of this equation? How many roots does this equation have?

IV . Learning new material.(slide 7).

Teacher: The equation y (x ) =0 called fractional rational equation if expression y (x ) is fractional(i.e. contains division into an expression with variables).

To solve a rational equation, it is necessary to transform it into a linear or quadratic equation, solve this equation and discard those roots that are not included in the permissible value range of the original rational equation.

Open the textbook on page 78 and read the rule. You already worked on this topic in 8th grade.

Algorithm for solving fractional rational equations: ( slide 8).

(Annex 1)

Teacher: Now, together with me, let's solve the fractional-rational equation using the algorithm (slide 9).

VI . Independent work(slide 10).

… Your letter. Your native lines.

Your last maternal command:

“The laws of life are wise and cruel.

Live. Work hard. Don't ruin your eyes with tears.

My love is always with you. Forever.

You love life. She's really good.

Love people. And remember - in a person

what's important? High soul."

Let us also try to have a “high soul.” And for this you need to respect and love your parents, of course, try to study and pass the state tests well. exams. Let's start preparing for certification.

Independent work. Self-control – 4 options. Testing your integrity. The work is done in notebooks. While completing the work, students determine for themselves an algorithm for solving fractional rational equations. On each desk there is a table - a reminder “Algorithm for solving fractional rational equations.” Annex 1.

ANSWERS:

Option I:
,
(
;
).

Option II:
(
;
)

Option III:
(

)

IV option:
,
(
;
).

VII . Physical education minute(slide 11).

Teacher: Now for the warm-up.

Turn to me. I speak out sentences. If it is fair, you stand up; if not, then you remain sitting.

1) 5x = 7 has a single root.

2) 0x = 0 has no roots.
3) If D 0, then the quadratic equation has two roots.
4) If D
5) The number of roots is not greater than the degree of the equation.

VIII . Reinforcement and repetition of material.(slide 12).

Teacher. Men want to look only courageous, only strong, only unbending in front of their loved ones. Perhaps this is what makes them men. And only in front of their own mother are they not afraid to expose their weaknesses and failures, to admit mistakes and losses, because no matter how far they have gone in their age and development, in front of her they, even gray-haired, are still children. And she understands in her heart that the poor and the offended, first of all, need a mother more than anyone else. Today everyone will have good grades, so I think there will be no offended people.

Solving the problem No. 942 from the textbook. (Algebra – 9th grade / Yu.N. Makarychev) (slide 13).

Solve the example on the board.(slide 14).

No. 289(a)

VII . Summing up the lesson.

What new did you learn in the lesson?

What did you learn in the lesson?

2. Algorithm for solving fractional rational equations:

The teacher evaluates the students' work and assigns grades.

Teacher. Acquiring the features of a symbol and fulfilling a huge social mission, the mother never lost her usual human features, remaining a hospitable hostess and an intelligent interlocutor, a diligent worker and a born singer, open-minded in a feast and courageous in grief, open in joy and restrained in sadness, and always kind, understanding and feminine! I really want your parents' dreams to come true, may you be worthy people (slide 15).

VIII . Homework. No. 943, No. 940 (a, b), No. 290 (slide 16).

Annex 1.

Algorithm for solving fractional rational equations:

Find acceptable values ​​of the fractions included in the equation.

Find the common denominator of the fractions in the equation.

Multiply both sides of the equation by the common denominator.

Solve the resulting equation.

Eliminate roots that are not included in the acceptable values ​​of fractions of the equation .

Greetings, dear schoolchildren. We invite you to watch the video tutorial on how to solve equations with fractions. Andrey Andreevich Andreev will solve problems in Algebra, and using his example you can try to solve your own problems that were assigned to you.

Solving fractional rational equations

An integer expression is a mathematical expression made up of numbers and literal variables using the operations of addition, subtraction and multiplication. Integers also include expressions that involve division by any number other than zero.

The concept of a fractional rational expression

A fractional expression is a mathematical expression that, in addition to the operations of addition, subtraction and multiplication performed with numbers and letter variables, as well as division by a number not equal to zero, also contains division into expressions with letter variables.

Rational expressions are all whole and fractional expressions. Rational equations are equations in which the left and right sides are rational expressions. If in a rational equation the left and right sides are integer expressions, then such a rational equation is called an integer.

If in a rational equation the left or right sides are fractional expressions, then such a rational equation is called fractional.

Examples of fractional rational expressions

1. x-3/x = -6*x+19

2. (x-4)/(2*x+5) = (x+7)/(x-2)

3. (x-3)/(x-5) + 1/x = (x+5)/(x*(x-5))

Scheme for solving a fractional rational equation

1. Find the common denominator of all fractions that are included in the equation.

2. Multiply both sides of the equation by a common denominator.

3. Solve the resulting whole equation.

4. Check the roots and exclude those that make the common denominator vanish.

Since we are solving fractional rational equations, there will be variables in the denominators of the fractions. This means that they will be a common denominator. And in the second point of the algorithm we multiply by a common denominator, then extraneous roots may appear. At which the common denominator will be equal to zero, which means multiplying by it will be meaningless. Therefore, at the end it is necessary to check the obtained roots.

There may also be a solution and other examples for this, it’s worth visiting the site prostoshkola.com. There, select the desired problem, watch the solution, and also possibly a video about how this equation is solved. Now let’s watch a video tutorial with Andrei Andreevich “Solving equations with fractions.”