(2) To solve the inequality group, generally, first find the solution set of each inequality in the inequality group, then find their common parts, and then find the solution set of the inequality group.
Enumerate one-dimensional linear inequalities (groups) to solve practical problems and master the steps to solve inequality application problems:
(1) Find out the inequality relation of practical problems, set unknown numbers and list inequalities (groups);
(2) Solving inequalities (groups);
(3) Find the answer that accords with the meaning of the question from the solution set of the inequality group.
, the solution of the linear equation of one variable and its three solutions: cough (1) The general steps to solve the linear equation of one variable are to remove the denominator, brackets, shift terms, merge similar terms, and change the unknown coefficient into1;
(2) The simplest linear equation ax=b can be solved in the following three situations:
① When a≠0, the equation has one and only one solution;
② When a=0 and b≠0, the equation has no solution;
③ When a=0 and b=0, the equation has infinite solutions.
Other mathematical problem-solving methods are developed with the in-depth study of mathematical objects. The sixth grade students are about to graduate from primary school, and the door of middle school has been opened to us. To learn mathematics well, we must master the characteristics of junior high school mathematics, especially the method of solving problems. The following are the most commonly used methods to solve problems in junior high school mathematics, and some methods are also required to be mastered in the middle school syllabus. These methods can also give you some help in your present study. Please save it as data. Of course, it is best to learn to understand it all in the future and keep it in your mind.
1, the so-called formula of matching method is to match some items of an analytical formula into the sum of positive integer powers of one or more polynomials by means of constant deformation. The method of solving mathematical problems with formulas is called matching method. Among them, the most common method is to make it completely flat. Matching method is an important method of constant deformation in mathematics. It is widely used in factorization, simplifying roots, solving equations, proving equality and inequality, finding extreme values of functions and analytical expressions.
2, factorization method
Factorization is to transform a polynomial into the product of several algebraic expressions. Factorization is the basis of identity deformation. As a powerful mathematical tool and method, it plays an important role in solving algebra, geometry and trigonometry problems. There are many methods of factorization, such as extracting common factors, formulas, grouping decomposition, cross multiplication and so on. Middle school textbooks also introduce the use of decomposition and addition, root decomposition, exchange elements, undetermined coefficients and so on.
3. method of substitution method of substitution is a very important and widely used problem-solving method in mathematics. We usually refer to unknowns or variables as elements. The so-called method of substitution is to replace a part of the original formula with new variables in a complicated mathematical formula, thus simplifying it and making the problem easy to solve.
4. Discriminant method and Vieta theorem.
The root discrimination of unary quadratic equation ax2+bx+c=0(a, B, C belongs to R, a≠0) and△ = B2-4ac is not only used to judge the properties of roots, but also widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations as a problem-solving method.
Vieta's theorem not only knows one root of a quadratic equation, but also finds another root. Knowing the sum and product of two numbers, we can find the symmetric function of the root, calculate the sign of the root of quadratic equation, solve the symmetric equation and solve some problems about quadratic curve. , has a very wide range of applications.
5, undetermined coefficient method
When solving mathematical problems, it is first judged that the obtained results have a certain form, which contains some undetermined coefficients, then the equations about undetermined coefficients are listed according to the problem setting conditions, and finally the values of these undetermined coefficients or some relationship between these undetermined coefficients are found. This method is called undetermined coefficient method to solve mathematical problems. It is one of the commonly used methods in middle school mathematics.
6. Construction method When solving problems, we often use this method to construct auxiliary elements through the analysis of conditions and conclusions. It can be graphs, equations (groups), equations, functions, equivalent propositions, etc. And establish a bridge connecting conditions and conclusions, so that the problem can be solved. This mathematical method of solving problems is called construction method. Using construction method to solve problems can make algebra, trigonometry, geometry and other mathematical knowledge permeate each other, which is beneficial to solving problems.
7. The reduction to absurdity is an indirect proof method. It is a way to put forward a hypothesis contrary to the conclusion of the proposition, and then proceed from this hypothesis and lead to contradictions through correct reasoning, thus denying the opposite hypothesis and affirming the correctness of the original proposition. The reduction to absurdity can be divided into reduction to absurdity (with only one opposite conclusion) and exhaustive reduction to absurdity (with more than one opposite conclusion). The steps to prove a proposition by reduction to absurdity can be roughly divided into: (1) reverse design; (2) return to absurdity; (3) conclusion.
Anti-design is the basis of reduction to absurdity. In order to make correct anti-design, we need to master some commonly used negative expressions, such as: yes/no; Existence/non-existence; Parallel/non-parallel; Vertical/not vertical; Equal to/unequal to; Large (small) inch/small (small) inch; Both/not all; At least one/none; At least n/ at most (n-1); At most one/at least two; Only/at least two.
Reduction to absurdity is the key to reduction to absurdity. There is no fixed model in the process of derivation of contradiction, but it must be based on reverse design, otherwise the derivation will become passive water and trees without roots. Reasoning must be rigorous. There are the following types of contradictions: contradictions with known conditions; Contradicting with known axioms, definitions, theorems and formulas; There are dual contradictions; Contradictions