Irrational equivalence and unevenness 10. Irrational irrationality. Selection of selected personal information

Lesson "Resolution of irrational irregularities",

10th grade

Tsіl : learn about irrational inconsistencies and methods of their perfection

lesson type : development of new material

Ownership: Heading guide “Algebra and analysis on the cob. 10-11 grade”, Sh.A. Alimov, research material on algebra, presentation on this topic.

Lesson plan:

Stage lesson

Stage goal

Hour

Organizational moment

Povodomlennya those lesson; staging a lesson; step by step lesson.

2 min

Usna robot

Propaedeutics of the purpose of irrational equivalence.

4 min

Introduction of new material

Recognize with irrational inconsistencies and in three ways their perfection

20 min

The accomplishment of tasks

Formuvaty vminnya virishuvati irrational irrationality

14 min

Pouch for the lesson

Repeat the designation of irrational inconsistency and ways of virishennya.

3 min

Homework

Instruction for homework.

2 min

Hid lesson

Organizational moment.

Usna robot (Slide 4.5)

How equal are they called irrational?

How are such equals irrational?

Know the destination area

Explain why the number of equals does not make a decision on the impersonal real numbers

Old Greek teachings - a doslіdnik, which has established the foundation of irrational numbers (Slide 6)

Who has done the root image today (Slide 7)

Vyvchennya new material.

Write down the indications of irrational inconsistencies with supporting material: (Slide 8) Inconsistencies that avenge the unknown under the sign of the root are called irrational.

Irrational inconsistencies - the main folding of the school mathematics course. The manifestation of irrational inconsistencies is aggravated by this furnishing, which here, as a rule, the possibility of reverification is turned on, this needs to be done and all transformations are equally strong.

In order to hide the pardons under the hour of the manifestation of irrational inconsistencies, the next is only those meanings of the change, for which all the yaks are included in the unevenness of the function assigned, tobto. to know the UN, that buv grounded zdіysnyuvati equally strong transition throughout the UN chi її parts.

The main method of developing irrational irrationality is to bring irrationality up to an equal-strength system of aggregation of systems of rational irrationality. In conclusion, with supporting material, we will write down the main methods of developing irrational inconsistencies by analogy to methods of developing irrational equivalences. (Slide 9)

When developing irrational irregularities, remember the rule: (Slide 10) 1. when both parts of the unevenness are brought into the unpaired foot, the unevenness will always enter, equally strong this unevenness; 2. if the offending parts of the nervousness are caused at the boy's feet, then the nervousness will be stronger, if it is more powerful than that, if the offending parts of the nervousness are non-negative.

Let's look at the solution of irrational inconsistencies, in which the rights of a part are equal in number. (Slide 11)

We square the insults of some of the unevenness, but we can also square the non-negative numbers. Father, we know the UN, tobto. impersonal such meanings, in which they may feel offended by some part of the nervousness. The rights of the part of the unevenness are assigned for all admissible values ​​of x, and the left for

x-40. This unevenness is equal to the system of irregularities:

Vidpovid.

The right part is negative, and the left part is invisible for all values, for which it is not indicated. Tse means that the last part is more for the right with all the values ​​of x, which pleases the minds of x3.

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Whether it be unevenness, before the warehouse, which function enters, which stands under the root, is called irrational. There are two types of such irregularities:

For the first type, the root is smaller than the function g (x), for the other, it is larger. Yakscho g(x) - constant, the nervousness is sharply expressed. Catch respect: the sounds and inconsistencies are already similar, but the schemes of the stench are fundamentally different.

Today we learn to overcome the irrational inconsistencies of the first type - the stench of the simplest and the most sensible. The sign of nervousness can be suvorim or not suvorim. Їx is more correct:

Theorem. Be irrational to the mind

Equivalently to the system of irregularities:

Not weak? Let's take a look, the stars are taken from such a system:

1. f (x) ≤ g 2 (x) - everything made sense here. Tse vyhіdna nerіvnіst, zvedena at the square;
2. f (x) ≥ 0 - ce ODZ of the root. Guessing: arithmetic square root can't see much numbers;
3. g (x) ≥ 0 – the whole area is the value of the root. Starring unevenness in the square, we burn minus. As a result, the root can be blamed. Unevenness g (x) ≥ 0 vіdsіkaє їх.

A lot of learners are "fixated" on the first unevenness of the system: f (x) ≤ g 2 (x) - and het-purely forget the other two. The result of transfers: wrong decision, wasted balls.

Oskіlki іrrationalnі nerіvnostі - dosit foldable topic, let's take a look at 4 butts. Kind of elementary to fairly foldable. Usі zavdannya is taken from the entrance exams of MDU im. M. V. Lomonosov.

Apply the solution of tasks

Manager. To untie the nervousness:

We have a classic irrational unevenness: f(x) = 2x + 3; g(x) = 2 is a constant. Maemo:

From three irregularities to the end, the decision was left with only two. That's why unevenness 2 ≥ 0 wins forever. Let's move on the nervousness that we have lost:

Also, x ∈ [−1,5; 0.5]. Mustaches zafarbovani, shards unevenness.

Manager. To untie the nervousness:

Let's prove the theorem:

I overcome my nervousness. For whom the cut is the square of the cut. Maemo:

2x 2 − 18x + 16< (x − 4) 2 ;
2x 2 − 18x + 16< x 2 − 8x + 16:
x 2 − 10x< 0;
x (x − 10)< 0;
x ∈ (0; 10).

Now we swear to each other's nervousness. There tezh square trinomial:

2x 2 − 18x + 16 ≥ 0;
x 2 − 9x + 8 ≥ 0;
(x − 8)(x − 1) ≥ 0;
x ∈ (−∞; 1]∪∪∪∪)