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 a request on the site, we may collect various information, including your name, phone number, address email etc.
How we use your personal information:
- Collected by us personal information allows us to contact you and inform you about 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 auditing, data analysis and various studies 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, legal proceedings, and/or based on public requests or requests from government agencies on the territory of the Russian Federation - 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.
Any inequality that includes a function under the root is called irrational. There are two types of such inequalities:
In the first case, the root less function g (x), in the second - more. If g(x) - constant, the inequality is greatly simplified. Please note: outwardly these inequalities are very similar, but their solution schemes are fundamentally different.
Today we will learn how to solve irrational inequalities of the first type - they are the simplest and most understandable. The inequality sign can be strict or non-strict. The following statement is true for them:
Theorem. Any irrational inequality of the form
Equivalent to the system of inequalities:
Not weak? Let's look at where this system comes from:
- f (x) ≤ g 2 (x) - everything is clear here. This is the original inequality squared;
- f (x) ≥ 0 is the ODZ of the root. Let me remind you: the arithmetic square root exists only from non-negative numbers;
- g(x) ≥ 0 is the range of the root. By squaring inequality, we burn away the negatives. As a result, extra roots may appear. The inequality g(x) ≥ 0 cuts them off.
Many students “get hung up” on the first inequality of the system: f (x) ≤ g 2 (x) - and completely forget the other two. The result is predictable: wrong decision, lost points.
Since irrational inequalities are enough complex topic, let's look at 4 examples at once. From basic to really complex. All problems are taken from entrance exams Moscow State University named after M. V. Lomonosov.
Examples of problem solving
Task. Solve the inequality:
Before us is a classic irrational inequality: f(x) = 2x + 3; g(x) = 2 is a constant. We have:
Of the three inequalities, only two remained at the end of the solution. Because the inequality 2 ≥ 0 always holds. Let's cross the remaining inequalities:
So, x ∈ [−1.5; 0.5]. All points are shaded because the inequalities are not strict.
Task. Solve the inequality:
We apply the theorem:
Let's solve the first inequality. To do this, we will reveal the square of the difference. We have:
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 let's solve the second inequality. There too quadratic trinomial:
2x 2 − 18x + 16 ≥ 0;
x 2 − 9x + 8 ≥ 0;
(x − 8)(x − 1) ≥ 0;
x ∈ (−∞; 1]∪∪∪∪)
