Chapter 1 Primary Inequalities and Groups of Primary Inequalities I. In general, equations connected by the symbols "＜" (or "≤"), "＞" (or "≥") is called an inequality. The value of the unknown that makes the inequality hold is called the solution of the inequality. The solution of an inequality is not unique, and the set of all solutions that satisfy the inequality constitutes the solution set of the inequality. The process of finding the solution set of an inequality is called solving an inequality. The set of inequalities formed by several sets of one-variable inequalities is called the set of one-variable inequalities The solution set of the set of inequalities :The common *** part of the solution set of each of the inequalities of the set of one-variable inequalities. Basic Property of Equality 1: By adding (or subtracting) the same number or integer on both sides of an equation, the result is still an equation. Basic property 2: on both sides of the equation are multiplied or divided by the same number (divisor is not 0), the result is still an equation. Second, the basic nature of inequality 1: inequality on both sides are added (or subtracted) the same integer, the direction of the inequality sign remains unchanged. (Note: shift the term to change the number, but the inequality sign does not change.) Property 2: When both sides of an inequality are multiplied (or divided) by the same positive number, the direction of the inequality sign remains the same. Property 3: When both sides of an inequality are multiplied (or divided) by the same negative number, the direction of the inequality sign changes. Basic Properties of Inequalities <1>, if a>b, then a+c>b+c; <2>, if a>b, c>0 then ac>bc if c<0, then ac<bc Other Properties of Inequalities: Reflective: if a>b, then b<a; Transitive: if a>b, and b>c, then a>c Third, the steps to solve the inequality: 1, remove the denominator; 2, remove brackets; 3, shift the terms to merge like terms; 4, coefficients to 1. Fourth, the steps to solve the set of inequalities: 1, solve the set of solutions to the inequality 2, in the same axis to express the set of solutions to the inequality. V. The general steps of solving practical problems by a set of one-variable inequalities: (1) examine the problem; (2) set the unknowns, find (unequal quantities); (3) set the dollar, (according to the unequal quantities) the relationship between the set of inequalities (set) (4) solve the set of inequalities; test and answer. Sixth, the common questions: 1, for 4x-6 7x-12 non-negative solutions. 2, it is known that the solution of 3 (x - a) = x - a + 1r is suitable for 2 (x - 5) 8a, find the range of a. 3, when the value of m, 3x + m - 2 (m + 2) = 3m + x solution between -5 and 5. Chapter 2 Decomposing Factors I. Formulas: 1. ma + mb + mc = m (a + b + c) 2. a2 - b2 = (a + b) (a - b) 3. a2 ± 2ab + b2 = (a ± b)2 2. 2. A polynomial is reduced to the form of the product of several integers, and this deformation is called factoring this polynomial. The transformation of a product of several integers into the form of a polynomial is a multiplication operation.2. The transformation of a polynomial into the form of a product of several integers is a factorization.3. ma + mb + mc m (a + b + c) 4. Factorization and multiplication of integers are transformations in opposite directions. Third, the same factorization that each term of a polynomial contains is called the common factor of each term of the polynomial. The common factorization method of factoring is to reduce a polynomial to the form of multiplying a monomial and a polynomial. The general steps to find the common factor are: (1) if the factors are integers, take the greatest common divisor of the factors; (2) take the same letter, and the exponent of the letter is the lower one; (3) take the same polynomial, and the exponent of the polynomial is the lower one. (4) The product of all these factors is the common factor. Fourth, the general steps of factoring: (1) if there is "-" first extract "-", if the polynomials have a common factor, then extract the common factor. (2) If the polynomials do not have a common factor, then according to the characteristics of the polynomials, choose the formula of the square difference or the formula of perfect squares. (3) Each polynomial should be factored until it can no longer be factored. V. The equation shaped like a2 + 2ab + b2 or a2 - 2ab + b2 is called a perfect square. Methods of factoring: 1. mention the common factor method. 2. use the formula method. Chapter 3 Fractions Note: 1° For any fraction, the denominator cannot be zero. 2° Fractions differ from whole formulas in that fractions have letters in the denominator and whole formulas have no letters in the denominator. 3° The value of a fraction is zero in two ways: the denominator is not equal to zero; the numerator is equal to zero. (In the B ≠ 0, the fractional formula has meaning; in the fractional formula, when B = 0 fractional meaningless; when A = 0 and B ≠ 0, the value of the fractional formula is zero.) Frequently tested knowledge points: 1, the meaning of the fractional formula, the simplification of the fractional formula. 2, the addition, subtraction, multiplication and division of the fractional formula. 3, the solution of the fractional equation and its use of fractional equations to solve the application problems. Chapter 4 Similar Figures I. Definitions An equation that expresses the equality of two ratios is called a proportion. If the ratio of a to b and the ratio of c to d are equal, then or a∶b=c∶d, then the four numbers a,b,c,d are called the terms of the proportion, the two ends of the two terms are called the outer terms, and the middle two terms are called the inner terms. That is, a, d is the outer term, c, b is the inner term. If you use the same length unit to measure the length of the two segments AB, CD are m, n, then the ratio of the two segments (ratio) AB: CD = m: n, or written as = , where the segments AB, CD are called the two segments than the former and latter terms. If expressed as a ratio k, then =k or AB=k?CD. Four line segments a,b,c,d, if the ratio of a and b is equal to the ratio of c and d, that is, ,then these four line segments a,b,c,d called proportional line segment, referred to as proportional line segment. Definition of golden section: on the line segment AB, the point C divides the line segment AB into two line segments AC and BC, if ,then the line segment AB is golden sectioned by the point C. The point C is called the golden section of the line segment AB, and the ratio of AC to AB is called the golden ratio. The ratio of AC to AB is called the golden ratio. where ≈ 0.618. Lemma: A line parallel to one side of a triangle and intersecting the other two sides intercepts a triangle whose sides are proportional to the sides of the original triangle. Similar polygons: Two polygons whose corresponding angles are equal and whose corresponding sides are proportional are called similar polygons. Similar Polygons: Two polygons whose angles are equal and whose sides are proportional are called similar polygons. Similarity ratio: the ratio of the corresponding sides of similar polygons is called the similarity ratio. Second, the basic properties of proportions: 1, if ad = bc (a,b,c,d are not equal to 0), then . If (b,d are not 0), then ad = bc. 2, combined than the nature of: if ,then . 3, is proportional to the nature of: if = ... = (b + d + ... + n ≠ 0), then . 4, more proportional to the nature of: if then . 5, inversely proportional to the nature of: if then three, to find the ratio of the two segments of the problem to pay attention to: (1) the two line segments of the length must be expressed in the same unit of length, if the unit length is different, should be reduced to the same unit, and then their ratio; (2) the ratio of the two line segments, there is no unit of length, it has nothing to do with the unit of length used; (3) the length of the two segments are positive, so the ratio of the two segments is always positive. Properties of similar triangles (polygons): similar triangles corresponding angles are equal, corresponding sides are proportional, similar triangles corresponding to the ratio of the height, corresponding to the ratio of the angle bisector and corresponding to the ratio of the median are equal to the similarity ratio. Similar polygon perimeter ratio is equal to the similarity ratio, area ratio is equal to the square of the similarity ratio. Five, the determination of congruent triangles: ASA, AAS, SAS, SSS, right triangles in addition to HL six, similar triangles, judgment methods: 1. three sides corresponding to the proportionality of the two triangles are similar; 2. two triangles corresponding to the equality of the two triangles are similar; 3. the two corresponding to the proportionality of the two sides and the angle of the same; 4. Definition of the method: corresponding to the angle of equality, corresponding to the proportionality of the two triangles. 5, the theorem of the two triangles are similar to the corresponding angle of the proportionality of the corresponding side. Theorem: If a line parallel to one side of a triangle intersects the other two sides (or the extension lines of the two sides), the resulting triangle is similar to the original triangle. In special triangles, some are similar and some are not similar.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. VII, similar figures on any pair of corresponding points to the similar center of the distance ratio is equal to the similarity ratio. If the two figures are not only similar figures, and each group of corresponding points in the line through the same point, then such two figures are called similar figures, the point is called similar to the center, the similarity ratio is also known as similarity ratio. VIII. Frequently tested knowledge points: 1, the basic nature of the ratio, the golden ratio, the nature of similar figures. 2, the nature of similar triangles and determination. Properties of similar polygons. CHAPTER V COLLECTION AND PROCESSING OF DATA (1) Definition of census: such a comprehensive survey of the subjects examined for a certain purpose is called a census. (2) Aggregate: The whole of the objects to be examined in it is called aggregate. (3) Individuals: the composition of the total of each object of study is called individual (4) sampling survey: (sampling investigation): from the overall part of the individual survey, this survey is called sampling survey. (5) sample (sample): which is taken from the overall part of the individual is called a sample of the overall. (6) When the number of individuals in the whole is large, in order to save time, manpower and material resources, sampling can be used. In order to obtain more accurate results, sampling should pay attention to the representativeness and breadth of the sample. Attention should also be paid to the size of the sample. (7) We call the number of times each object occurs a frequency. And the ratio of the number of occurrences of each object to the total number of occurrences is the frequency. Statistics of data fluctuation: Extreme variance: is the difference between the largest and smallest data in a set of data. Variance: is the average of the squares of the differences between the individual data and the mean. Standard deviation: the arithmetic square root of the variance. Recognize its formula. The smaller the extreme deviation, variance or standard deviation of a set of data, the more stable the set is. Also know the definitions of mean, multitude, and median. To characterize the average, use: mean, multitude, and median. Characterize the degree of dispersion with: extreme deviation, variance, standard deviation. Frequently tested knowledge points: 1, make frequency distribution table, make frequency distribution histogram. 2, the use of variance to compare the stability of the data. 3, the mean, median, plurality, extreme variance, variance, standard deviation of the method of finding. 3, frequency, the definition of the sample Chapter 6 Proofs a. Sentences that make judgments about things are called propositions. That is: a proposition is a sentence that judges a thing. In general: a question is not a proposition . A graphical work is not a proposition . Every proposition has a condition and a conclusion. The condition is what is known and the conclusion is what is inferred from what is known. In general, propositions can be written in the form "if ......, then ......". The "if" part is the condition and the "then" part is the conclusion. To show that a proposition is a false proposition, it is often possible to give an example that has the conditions of the proposition, but not the conclusion of the proposition. Such examples are called counterexamples. Second, the triangle interior angle sum theorem: the sum of the three interior angles of a triangle is equal to 180 degrees. 1, the idea of proving the triangle interior angle sum theorem is to "put" the three angles of the original triangle together to form a flat angle. Generally need to make auxiliary lines. Either as parallel lines, or as an angle equal to an angle in the triangle.2, the exterior angles of the triangle and its neighboring interior angles are complementary. Third, the relationship between the exterior angles of a triangle and its non-adjacent interior angles is: (1) An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it. (2) An exterior angle of a triangle is greater than any interior angle that is not adjacent to it. IV.The basic steps in proving that a proposition is true are (1) Draw the figure according to the question. (2) According to the conditions, the conclusion, combined with the graph, write the known, the proof. (3) After analysis, to find out the known from the way out of the proof, write the proof process. In the proof need to pay attention to: (1) in general, the process of analysis is not required to write out. (2) Each step in the proof of the reasoning must be based on. If two lines are parallel to a third line, then the two lines are also parallel to each other.30. The right-angled side opposite the hypotenuse is half of the hypotenuse. The height on the hypotenuse is half of the hypotenuse. Frequently Examined Points: 1. The interior angle sum theorem for triangles, and the exterior angle theorem for triangles.2 Properties and determinations of parallelism of two straight lines. Propositions and their conditions and conclusions, definitions of true and false propositions. (From the Internet after repeated comparisons to find you, adopt Oh!) %D%A