The formula 1 and the properties of inequality are the basis for proving and solving inequality.
The basic properties of inequality are:
(1) symmetry: A >；; bb & lta；
(2) Transitivity: If a>b, b>c, then a & gtc；;
(3) additivity: a & gtba+c & gt；; b+ c；
(4) Availability: a>b, when c>0, ac> BC; When c<0, alternating current
Properties of inequality operation:
(1) addition in the same direction: if a >；; B, c>d, then A+C > b+ d；
(2) Anisotropic subtraction.
(3) multiply positive numbers in the same direction: if a >；; B>0, c>d>0, then ac & gtbd.
(4) power law: if A >；; b & gt0，n？ N+, then;
(5) Prescription rule: If a>b>0, N? N+, then;
(6) Reciprocity rule: If ab>0, a>, then B.
2. Basic inequality
Theorem: If, then (if and only if a=b? =? Number)
Inference: If, then (if and only if a=b? =? Number)
Arithmetic average; Geometric average;
Promotion: If, then
If and only if a=b? =? Number;
3. Absolute inequality
The solution set of |x|0) is: {x |-a.
| x | >； a(a & gt； The solution set of 0) is: {x | x >；; A or x < -a}.
Attachment: Summary of inequality proof knowledge
The problem of inequality proof is often not solved by one method, and there is no fixed law to follow because of its variety of questions, diverse methods and strong skills. It is the flexible use of various methods and the concentrated expression of various thinking methods, so it is difficult. The solution to this problem lies in mastering the nature of inequality and some basic inequalities, and using commonly used proof methods flexibly.
First of all, the main points are analyzed.
1. comparison method is one of the most basic and important methods to prove inequality. It is a direct application of the size order and operational properties of two real numbers. Comparison method can be divided into difference comparison method (referred to as difference method) and business comparison method (referred to as business method).
The theoretical basis of (1) difference comparison method is the basic property of inequality:? a-b？ Is it? b； a-b？ Is it? b？ . The general steps are as follows: ① making difference: investigating the difference formula formed by the left and right sides of inequality and treating it as a whole; (2) Deformation: the difference between two sides of inequality is deformed into a constant, the product of several factors, the sum of one or several squares, etc. Among them, deformation is the key of difference method, and formula and factorization are commonly used deformation means; ③ Judgment: According to the known conditions and the above deformation results, judge the sign of the difference between the two sides of the inequality, and finally affirm the conclusion that the inequality is established. Application: When both ends of inequality are polynomials, fractions or logarithms, the difference comparison method is generally used.
(2) The theoretical basis of the quotient comparison method is:? If a, b R+, a/b? 1a？ b； a/b？ 1a？ b？ . The general steps are as follows: ① Quotient: left and right quotient; ② Deformation: Simplify the quotient to the simplest form; ③ The relationship between judgment quotient and 1 means that the quotient is greater than 1 or less than 1. Scope of application: When both ends of the proved inequality contain power sum exponential expressions, quotient comparison method is generally adopted.
2. The synthesis method is based on the known facts (known conditions, important inequalities or proven inequalities), with the help of the properties of inequalities and related theorems, and through gradual logical reasoning, the inequalities to be proved are finally derived. What are its characteristics and ideas? Cause leads to result? From where? Known? Do you see it? Need to know? , gradually introduced? Conclusion? . The logical relationship is AB 1.
B2·B3？ BnB, that is, starting from the known A, gradually deduces the necessary conditions for the inequality to be established, and draws the conclusion B.
3. The analytic method is to analyze the sufficient condition for the inequality to be established from the inequality to be proved, and then judge whether that condition is satisfied. What are its characteristics and ideas? Fruit-holding business? In other words, from? Unknown? See? Need to know? , getting closer? Known? . The analysis proves that the logical relationship of AB is BB 1B 1.
B3？
BnA, written as: in order to prove that proposition B is true, as long as proposition B 1 is proved to be true, then what? This only needs to prove that B2 is true, so there is? This only needs to prove that A is true. Since A is known to be true, then B must be true. This model tells us that analytical method is a sufficient condition for seeking the last step.
4. The proof of some inequalities by reduction to absurdity is not clear from the positive proof, but it can be considered from the positive difficulty and negative point of view, that is, to prove the inequality A>b, assuming a? B, from the topic and other properties, deduce the contradiction, so as to confirm that A> B. Is the proof inequality involved here a negative proposition or contains a unique proposition? Most? 、? At least? 、? Does not exist? 、? Impossible? When waiting for the word, you can consider reducing to absurdity.
5. method of substitution and method of substitution introduced one or more variables to replace some inequalities with complex structure, many variables and unclear relations among variables, thus simplifying the original structure or realizing some transformation and adaptation, and bringing new enlightenment and methods to the proof. There are two main forms of substitution. (1) Triangular substitution method: it is often used to prove conditional inequalities. When the given conditions are complicated and one variable cannot be easily expressed by another variable, we can consider triangle substitution and use the same parameter to express two variables. If this method is used properly, it can communicate the relationship between trigonometry and algebra and transform complex algebraic problems into trigonometric problems. According to specific problems, the triangle substitution method is as follows: ① If x2+y2= 1, let x=cos? Y = sin? ; ② If x2+y2? 1，x=rcos？ ，y=rsin？ (0? r？ 1); (3) For inequalities, because |x|? 1，x=cos？ ; (4) If x+y+z=xyz, it is known by tanA+tanB+tanC=tanAtan-BtanC, x=taaA, y=tanB, z=tanC, where A+B+C=? . (2) Incremental substitution method: in the symmetrical formula (two letters can be exchanged arbitrarily, and the algebraic formula remains unchanged) and the given alphabetical order (such as A >；; B>c etc. ), consider changing elements by incremental method, with the purpose of reducing elements by changing elements, making problems difficult and easy, and simplifying the complex. For example, a+b= 1 can be substituted by a= 1-t, b=t or a= 1/2+t, b =1/2-t.
6. scaling method scaling method is to prove inequality a.
Second, a difficult breakthrough
1. When using quotient comparison method to prove inequality, we should pay attention to the sign of denominator to determine the direction of inequality.
2. Analytical method and synthetic method are two aspects of the unity of opposites. The former is conducive to thinking because of its clear direction, natural thinking and easy mastery. The latter is caused by cause and is suitable for expression, because the organization is clear and the form is concise, which is suitable for people's thinking habits. However, it is only an important way to explore and prove inequality by analytical method, which is not a good writing form because it is more complicated to describe. If you put? Just prove it? It will be a mistake if you don't write. In the form of comprehensive writing, the process of analysis and exploration is hidden. So when proving inequality, analysis and synthesis are often inseparable. If inequality is proved by comprehensive method, it is difficult to explore the way to prove the problem by analytical method, and then write its proof process in the form of comprehensive method to adapt to people's accustomed thinking law. Some inequalities are difficult to prove, and need to be analyzed and synthesized at the same time, so as to achieve the purpose of proving the problem by tilting both ends to the middle. This fully shows the dialectical unity of mutual premise, mutual penetration and mutual transformation between analysis and synthesis. The end of analysis is the starting point of synthesis, and the end of synthesis becomes the starting point of further analysis.
3. Analysis proves that every step in the process is not necessarily? Step by step reversibility? , there is no need to ask? Step by step reversibility? Because at this time we only need to find sufficient conditions, not necessary and sufficient conditions. If I have to? Step by step reversibility? It limits the scope of analytical methods to solve problems, so that analytical methods can only be used to prove equivalent propositions. When using analytical method to prove problems, we must use it correctly, right? Want a certificate? 、? Just a certificate? 、? Direct evidence? 、? Which is the certificate? Words like that.
4. When proving inequality by reduction to absurdity, it is necessary to deduce contradictions from various situations in which the propositional conclusion is opposite.
5. In triangle substitution, due to the limitation of known conditions, there may be some restrictions on the angle introduced, which should be paid great attention to, otherwise wrong results may occur. This is the key and difficult point of method of substitution, so we should pay attention to the application of the overall thought.