1. The core problem of solving inequality is the homosolution deformation of inequality, and the nature of inequality is the theoretical basis of inequality deformation. The roots, functions and images of the equation are closely related to the solution of inequality, so we should be good at connecting them organically. Mutual transformation. In solving inequalities, method of substitution and graphic method are one of the commonly used skills. By changing elements, more complex inequalities can be reduced to simpler or basic inequalities. By combining constructors with numbers and shapes, inequalities can be reduced to intuitive and vivid graphic relations. For inequalities with parameters, The graphic method can make the classification standard clear. < P > 2. The solution of algebraic expression inequality (mainly linear and quadratic inequality) is the basis of solving inequality, and the basic idea of solving inequality is to classify fractional inequality and absolute inequality into algebraic expression inequality (group) by using the properties and monotonicity of inequality. Classification, substitution and combination of numbers and shapes are common methods to solve inequality. The roots of equations, the properties and images of functions are all the same as inequalities. We should be good at connecting them organically, transforming and changing each other. < P > 3. In solving inequalities, method of substitution and graphic method are one of the commonly used skills. By changing elements, more complex inequalities can be reduced to simpler or basic inequalities, and by constructing functions, the solutions of inequalities can be reduced to intuitive and vivid image relationships. For inequalities with parameters, the graphic method can make the classification criteria clearer. Through review, It is realized that the core problem of inequality is isomorphic deformation of inequality, and the theory of isomorphic deformation of inequality plays an important role. < P > 4. Comparative method is the most basic and commonly used method in inequality proof, and the general steps of comparative method are: making difference (quotient) → deformation → judging symbol (value). < P > 5. The methods of proving inequality are flexible and diverse, rich in content. This will play a good role in promoting the development of comprehensive analytical ability and positive and negative thinking. Before proving the inequality, we should choose an appropriate proof method according to the structural characteristics and internal relations of the question and the inequality to be proved. Through the operation of equality or inequality, the inequality to be proved will be turned into an obvious and well-known inequality, so that the original inequality can be proved. On the other hand, we can also start with the obvious and well-known inequality, and derive the inequality to be proved through a series of operations. The former is "the cause of the fruit" and the latter is "the cause leads to the fruit". As a way of communication, analytical synthesis is often used in the proof, and the two sides attack each other and complement each other to achieve the purpose of proof. < P > 6. The methods of proving inequality are flexible and diverse. However, comparison, synthesis, analysis and mathematical induction are still the basic methods to prove inequality. We should choose an appropriate proof method according to the structural characteristics and internal relations of the questions and questions, be familiar with the reasoning thinking in various proofs, and master the corresponding steps, skills and language characteristics. < P > 7. Inequality knowledge permeates all branches of middle school mathematics. It has a wide range of applications. Therefore, the application of inequality reflects a certain degree of comprehensiveness, flexibility and diversity, which has played a very good role in promoting the students to integrate all the knowledge they have learned in mathematics. When solving problems, they should choose appropriate solutions according to the structural characteristics and internal relations of the questions, and finally come down to the solution or proof of inequality. The application scope of inequality is very wide. It runs through the whole middle school mathematics, such as the set problem, the discussion of the solution of the equation (group), the study of the monotonicity of the function, the determination of the definition domain of the function, the maximum and minimum problems in triangle, sequence, complex number, solid geometry and analytic geometry, all of which are closely related to inequality, and many problems can finally be attributed to the solution or proof of inequality.
8. The problem of inequality application embodies a certain comprehensiveness. This kind of problem can be roughly divided into two categories: one is to establish inequality and solve inequality; The other is to establish a function formula to find the maximum or minimum value. When using the mean inequality to find the maximum value of a function, we should pay special attention to the three conditions of "positive number, fixed value and equality", and sometimes we need to put them together properly to make them meet these three conditions. The basic steps of solving application problems by using inequality are: 1 to examine the problem, 2 to establish an inequality model, 3 to solve mathematical problems and 4 to answer.
9. Notes:
(1) The basic idea of solving inequalities is transformation and reduction, which are generally transformed into the simplest unary linear inequality (group) or unary quadratic inequality (group) to solve.
(2) When solving inequalities with parameters, we should pay special attention to the idea of combining numbers with shapes, the idea of functions and equations, and the vivid application of classified discussion ideas.
(3) There are many ways to prove inequality. We should not only pay attention to the applicable scope of all kinds of proofs, but also pay attention to choosing some special skills on the basis of mastering the conventional proofs. For example, when using scaling method to prove inequality, we should pay attention to adjusting the scaling degree.
(4) according to the structural characteristics of the topic, it is often an effective way of thinking.
References: Types and methods of inequality problems.