1. Generally speaking, formulas connected by the symbol ""(or "≥") are called inequalities.

The value of the unknown quantity that can make the inequality hold is called the solution of the inequality. The solution of inequality is not unique. All that satisfy the inequality are liberated together to form the solution set of the inequality. The process of finding the solution set of inequality is called solving inequality.

An inequality group consisting of several linear inequality groups is called a linear inequality group.

Solution set of inequality group: the common part of each inequality solution set in linear inequality group.

The basic property of equation 1: Add (or subtract) the same number or algebraic expression on both sides of the equation, and the result is still an equation. Basic property 2: the result of multiplying or dividing the same number on both sides of an equation (the divisor is not 0) is still an equation.

Second, the basic properties of inequality 1: the same algebraic expression is added (or subtracted) on both sides of the inequality, and the direction of the inequality sign remains unchanged. (Note: The shift term should be changed, but the equal sign remains unchanged. ) property 2: both sides of the inequality are multiplied (or divided) by the same positive number, and the direction of the inequality remains unchanged. Property 3: When both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality changes. Basic properties of inequality

Other properties of inequality: reflectivity: if a >；; B, then b < a；; Transitivity: If a>b and b>c, then a>c

Third, the steps of solving inequality: 1, denominator; 2. Remove the brackets; 3. Transfer projects and merge similar projects; 4. The coefficient is 1. Fourth, the steps to solve the inequality group: 1, the solution set of inequality 2, indicating the solution set of inequality on the same axis. 5. Enumerate the general steps of solving practical problems with linear inequality of one variable: (1) examining questions; (2) Set an unknown number and find an (unequal) relationship; (3) setting independent variables, setting inequalities (groups) (according to inequalities) (4) solving inequality groups; Test and answer.

6. Frequently asked questions: 1, find the nonnegative solution of 4x-6 7x- 12. 2. It is known that the solution of 3(x-a)=x-a+ 1r is suitable for 2(x-5) 8a, and the range of a is found.

The solution of 3.3x+m-2(m+2)=3m+x is between -5 and 5.

Chapter II Factorization

1. formula: 1, ma+mb+mc=m(a+b+c)2, A2-B2 = (a+b) (a-b) 3, A2+2ab+B2 = (a+b) 2. Convert a polynomial into the product of several algebraic expressions. 1. Turning the product of several algebraic expressions into polynomials is a multiplication operation. 2. Turning a polynomial into the product of several algebraic expressions is factorization. 3.ma+mb+mc m(a+b+c)4。 Factorization and algebraic expression multiplication are opposite deformations.

3. Let all terms of a polynomial contain the same factor, which is called the common factor of each term of this polynomial. To decompose a factor by the common factor method is to convert a polynomial into a monomial and then multiply it with this polynomial. The general steps to find the common factor are: (1) If each coefficient is an integer coefficient, take the greatest common factor of the coefficient; (2) Taking the same letter, the index of the letter is lower; (3) Take the same polynomial with lower exponent. (4) The product of all these factors is the common factor.

4. The general steps of factorization are as follows: (1) If there is a "-",first extract the "-",if the polynomial has a common factor, then extract the common factor. (2) If the polynomial has no common factor, choose the square difference formula or the complete square formula according to the characteristics of the polynomial. (3) Every polynomial must be decomposed until it can no longer be decomposed.

5. A formula in the form of A2+2ab+b2 or A2-2AB+B2 is called a completely flat mode. Method of factorization: 1, method of extracting common factor. 2. Use the formula method.

Chapter III Scores

Note: 1 For any fraction, the denominator cannot be zero.

The difference between fractions and algebraic expressions is that the denominator of fractions contains letters, while the denominator of algebraic expressions does not contain letters.

3 the value of the score is zero, which has two meanings: the denominator is not equal to zero; Molecule equals zero. (When B≠0, the score is meaningful; In the score, when B=0, the score is meaningless; When A=0 and B≠0, the value of the score is zero. )

Common knowledge points: 1, meaning of score, simplification of score. 2. Addition, subtraction, multiplication and division of scores. 3. The solution of fractional equation and its application.

Chapter IV Similar Figures

First of all, the formula that two ratios are equal is called proportion. If the ratio of A to B and the ratio of C to D are equal, then a: B = C: D, then the four numbers A, B, C and D that make up the proportion are called proportional terms, the two terms at both ends are called external terms, and the two terms in the middle are called internal terms. That is, A and D are external terms, and C and B are internal terms. If you choose the same one, then say the ratio of these two line segments AB: CD = m: n, or write =, where line segments AB and CD are called the first and last terms of the ratio of these two line segments respectively. If expressed as the ratio k, then =k or AB=k? CD. Among the four line segments A, B, C and D, if the ratio of A to B is equal to that of C and D, that is, these four line segments A, B, C and D are called proportional line segments for short. Definition of golden section: On line segment AB, point C divides line segment AB into AC and BC. If yes, then the line segment AB is called the golden section by point C, and the point C is called the golden section of the line segment AB, and the ratio of AC to AB is called the golden section ratio, where ≈0.6 18. Lemma: The three sides of a triangle cut by a line parallel to one side of the triangle and intersecting with the other two sides are proportional to the three sides of the original triangle. Similar polygons: Two polygons with equal corresponding angles and proportional corresponding sides are called similar polygons. Similar polygons: the angles are equal and the sides are proportional. Similarity ratio: the ratio of the corresponding edges of similar polygons is called similarity ratio.

Second, the basic nature of the ratio: 1. If AD = BC (A, B, C and D are not equal to 0), then. If (B and d are not equal to 0), then ad=bc.2. Combination properties: if, then. 3. Proportional nature: If =…= (b+d+…+n≠0), then. 4, beyond nature: if it is. 5. Inverse ratio property: If it is,

Third, the problems that should be paid attention to when calculating the ratio of two line segments: (1) The lengths of two line segments must be expressed in the same length unit. If the unit length is different, it should be converted into the same unit first, and then the ratio of them should be calculated; (2) The ratio of two line segments has no length unit, which has nothing to do with the adopted length unit; (3) The lengths of the two line segments are both positive numbers, so the ratio of the two line segments is always positive.

4. The nature of similar triangles (polygon): the corresponding angles of similar triangles are equal, and the corresponding sides are proportional; The ratio of similar triangles to height, the ratio of bisector to angle and the ratio of centerline are all equal to similarity ratio. The perimeter ratio of similar polygons is equal to the similarity ratio, and the area ratio is equal to the square of the similarity ratio.

Five, congruent triangles's judgment methods are: ASA, AAS, SAS, SSS, right triangle plus HL.

VI. similar triangles's judgment method: 1. Similarity of three sides corresponding to two proportional triangles; 2. Two triangles with equal corresponding angles are similar; 3. The two sides are proportional and the included angle is equal; 4. Definition method: Two triangles with equal corresponding angles and proportional corresponding sides are similar. 5. Theorem: A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle. In a special triangle, some are similar and some are not. 1, two congruent triangles must be similar. 2. Two isosceles right triangles must be similar. 3. Two equilateral triangles must be similar. 4. Two right triangles and two isosceles triangles are not necessarily similar.

7. The ratio of the distance between any pair of corresponding points on the similarity graph and the similarity center is equal to the similarity ratio. If two graphs are not only similar graphs, but also the straight lines of each group of corresponding points pass through the same point, then such two graphs are called potential graphs, and this point is called potential center. The similarity ratio at this time is also called the potential ratio.

Eight, common knowledge points: 1, the basic nature of proportion, the golden ratio, and the nature of graphics. 2. The nature and judgment of similar triangles. Properties of similar polygons.

Chapter V Data Collection and Processing

(1) Definition of census: This comprehensive survey of the respondents for a certain purpose is called census. (2) Population: The whole of the investigated object is called population. (3) Individuals: Each survey object constituting the population is called an individual. (4) Sampling survey: sampling some individuals from the population for investigation, which is called sampling survey. (5) Sample: Some individuals selected from the population are called samples of the population. (6) When there are a large number of individuals in the group, in order to save time, manpower and material resources, sampling survey can be adopted. In order to obtain more accurate survey results, we should pay attention to the representativeness and extensiveness of the samples and the size of the samples. (7) We weigh the frequency of each object. And the ratio of the number of times each object appears to the total number of times is the frequency.

Statistics of data fluctuation: extreme range: refers to the difference between the largest data and the smallest data in a group of data. Variance: it is the average of the square of the difference between each data and the average. Standard deviation: the arithmetic square root of variance. Remember its formula. The smaller the range, variance or standard deviation of a set of data, the more stable it is. We should also know the definitions of average, mode and median.

Use: average, mode and median to describe the average level. Range, variance and standard deviation are used to describe the degree of dispersion.

Frequently tested knowledge points: 1. Make frequency distribution table and frequency distribution histogram. 2. Compare the stability of data with variance. 3. Average, median, mode, range, variance and standard deviation. 3. Definition of frequency and sample

Chapter VI Evidence

1. A sentence that judges a thing is called a proposition. That is, a proposition is a sentence that judges a thing. Generally speaking, interrogative sentences are not propositions. Graphic method is not a proposition. Each proposition consists of two parts: conditions and conclusions. Conditions are known things, and conclusions are inferred from known things. Generally speaking, a proposition can be written in the form of "If …, then …". The part that begins with "if" is a condition. The "then" part is the conclusion. To show that a proposition is a false proposition, you can usually give an example to make it have the conditions of the proposition, not the conclusion of the proposition. This example is called counterexample.

Second, the theorem of the sum of internal angles of a triangle: the sum of the three internal angles of a triangle is equal to 180 degrees. 1. The idea of proving the theorem of the sum of internal angles of a triangle is to "gather" the three angles in the original triangle into a right angle. It is generally necessary to make auxiliary lines. It can be a parallel line or an angle equal to an angle in a triangle. 2. The outer angle of a triangle and its adjacent inner angle are complementary angles.

3. The relationship between the external angle of a triangle and its non-adjacent internal angles is: (1) One external angle of a triangle is equal to the sum of its two non-adjacent internal angles. (2) An outer angle of a triangle is greater than any non-adjacent inner angle.

4. The basic steps to prove a proposition is: (1) Draw a graph according to the meaning of the topic; (2) Write the known and verified according to the conditions and conclusions combined with the graphics; (3) Through analysis, find out the verified method derived from the known, and write the proof process. In the proof, it should be noted that under normal circumstances: (1). 30。 A right-angled side is half of a hypotenuse. The height on the hypotenuse is half the height of the hypotenuse.

Frequently tested knowledge points: 1, triangle interior angle sum theorem, triangle exterior angle theorem. 2 the nature and judgment of two parallel lines. Proposition and its conditions and conclusions, the definition of true and false propositions.

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