I. Unequal relations
1.Generally, formulas connected by symbols ""(or "≥") are called inequalities. ※. 2. Accurately translate inequalities and correctly understand mathematical terms such as "non-negative number" and "not less than". ※.
Non-negative number: greater than or equal to 0(≥0), 0 and positive number, not less than 0.
Non-positive number: less than or equal to 0(≤0), 0 and negative number, not greater than 0.
Second, the basic properties of inequality
1. Master the basic properties of inequalities and use them flexibly. ※:
Add (or subtract) the same algebraic expression on both sides of the inequality (1), and the direction of the inequality remains unchanged.
That is, if a >；; B, then a+c > b+c，a-c & gt； b-c。
(2) Both sides of the inequality are multiplied by (or divided by) the same positive number, and the direction of the inequality remains unchanged.
That is to say, if a>b and c>0, then ac> BC,
(3) When both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality changes.
That is, if a >；; B, and c < 0, AC
2. Comparison size: (A and B represent two real numbers or algebraic expressions respectively). ※
Generally speaking:
If a>b, then a-b is a positive number; On the other hand, if a-b is positive, then a >；; b；
If a=b, then a-b is equal to 0; On the other hand, if a = b；; Is equal to 0, then a = b；;
If a
Namely:
A>b, then A-B > 0
A=b, then a-b=0.
A<b, then a-b < 0.
It can be seen that to compare the sizes of two real numbers, just look at their differences.
3. Solution set of inequality;
1. The value of the unknown quantity that can make the inequality hold is called the solution of the inequality. All the solutions of an inequality constitute the solution set of this inequality. ※: The process of finding the solution set of inequality is called solving inequality.
2. Inequality can be solved in countless ways, generally all numbers in a certain range. ※ 。
3. Representation of inequality solution set on the number axis. ※:
When using the number axis to represent the solution set of inequality, we should determine the boundary and direction:
① Fixed points: solid points with equal signs and hollow circles without equal signs;
② Direction: large on the right and small on the left.
4. One-dimensional linear inequality;
1. A formula containing only one unknown is an algebraic expression, and the degree of the unknown is 1. Inequalities like this are called one-dimensional linear inequalities. ※ 。
2. The process of solving a linear inequality with one variable is similar to that of solving a linear equation with one variable, especially when both sides of the inequality are multiplied by a negative number, the sign of the inequality will change direction. ※ 。
3. Steps to solve linear inequality of one variable. ※:
1 naming;
(2) the bracket is removed;
③ shifting items;
(4) merging similar projects;
⑤ Change the coefficient to 1 (pay attention to the problem of changing the direction of inequality).
4. Exploration of inequality application (using inequality to solve practical problems). ※
The basic steps of solving application problems with column inequalities are similar to those of solving application problems with column equations, namely:
(1) Examination: Carefully examine the questions, find out the unequal relations in the questions, and grasp the key words in the questions, such as "greater than", "less than", "not greater than" and "not less than";
(2) setting: setting appropriate unknowns;
③ Column: list inequalities according to the inequality relations in the question;
④ Solution: Solve the solution set of the listed inequalities;
Answer: Write the answer and check whether the answer conforms to the meaning of the question.
Verb (abbreviation of verb) One-dimensional linear inequality and linear function
One-dimensional linear inequality system of intransitive verbs
1. Definition: An inequality group consisting of several linear inequalities with the same unknown number is called a linear inequality group. ※ 。
2. The common part of each inequality solution set in one-dimensional linear inequality group is called the solution set of inequality group. ※ 。
If the solution set of these inequalities has no common part, it is said that this inequality group has no solution.
The common part of several inequality solution sets is usually determined by the number axis.
3. Steps to solve linear inequalities. ※:
(1) Find the solution set of each inequality in the inequality group;
(2) Find out the common parts of these solution sets with the number axis,
(3) Write the solution set of this inequality group.
Four cases of the solution set of two linear inequalities (A and B are real numbers, A
(Take the largest and use the largest; Take the small as the big; Small, large and medium search; Size without solution)
Chapter II Factorization
I. Factorization
1.Converting a polynomial into the product of several algebraic expressions is called decomposing this polynomial. ※. 2. Factorization and algebraic expression multiplication are reciprocal. ※.
Differences and relations between factorization and algebraic expression multiplication;
(1) Algebraic expression multiplication is to multiply several algebraic expressions into polynomials;
(2) Factorization is to multiply a polynomial by several factors.
Two. Improve the public's factorial method.
1. If each term of a polynomial contains a common factor, then this common factor can be proposed, so that the polynomial can be transformed into the product of two factors. This factorization method is called extraction of common factors. ※ 。
※ 2. Concept connotation:
The final result of factorization of (1) should be "product";
(2) The common factor can be a monomial or polynomial;
(3) The theoretical basis of common factor method is the distribution law of multiplication to addition, a? b +a？ c=a？ (b+c)
3.※ Comments on error-prone points:
(1) Pay attention to whether the sign of the power exponent term is wrong;
(2) Whether the common factor formula is thorough;
(3) One term in the polynomial is only a common factor. After it is put forward, this item in brackets is+1, and there is no leakage.
Three. Using formula method
1. If the multiplication formula is reversed, it can be used to factorize some polynomials. This factorization method is called formula method. ※ 。
※ 2. Main formula:
(1) square difference formula:
(1) Binomial or polynomial should be regarded as binomial;
(2) Every term of binomial (unsigned) is the square of monomial (or polynomial);
Binomials are different symbols.
(2) Complete square formula:
(1) should be a trinomial;
(2) where two numbers are the same and each is the square of an algebraic expression;
(3) There is another term that can be positive or negative, which is twice the base product of the first two terms.
5. Thinking and solving steps of factorization. ※:
(1) First check whether each item has a common factor, and if so, extract the common factor first;
(2) See if the formula method can be used;
(3) The final result of factorization must be the product of several algebraic expressions;
(4) The results of factorization must be carried out until every factorization can no longer be decomposed within the scope of rational numbers.
Chapter III Scores
I. Scores
1. When two integers are not divisible, a fraction will appear. Similarly, when two algebraic expressions are not divisible, a fraction appears. ※. Algebraic expression a divided by algebraic expression b can be expressed as. If division b contains letters, it is called a fraction. For any fraction, the denominator cannot be zero.
2. Simplification and division are often needed when simplifying and calculating fractions, which is mainly based on the basic properties of fractions: both numerator and denominator of fractions are multiplied (or divided) by the same algebraic expression that is not equal to zero, and the value of fractions remains unchanged. ※ 。
3. When the numerator and denominator of a fraction have common factors, we can divide the numerator and denominator of the fraction by their common factors at the same time by using the basic properties of the fraction, that is, cut off the common factors of the numerator and denominator, which is called simplification. ※. 4. The fraction with no common factor between numerator and denominator is called simplest fraction. ※.
2. The law of multiplication and division of fractions
Multiply two fractions, take the product of molecular multiplication as the numerator of the product, and the product of denominator multiplication as the denominator of the product; Divide two fractions and multiply the numerator and denominator of the divisor by the divisor in turn (abbreviated as: dividing by a number equals multiplying the reciprocal of this number)
3. Addition and subtraction of fractions
1. The score is similar to the score and can be divided equally. ※ 。
According to the basic properties of fractions, several fractions with different denominators are converted into fractions with the same denominator, which is called the general fraction of fractions.
2. Addition and subtraction of fractions. ※:
The addition and subtraction of fractions, like the addition and subtraction of fractions, are divided into addition and subtraction of fractions with the same denominator and addition and subtraction of fractions with different denominators.
(1) Add and subtract fractions with the same denominator, and add and subtract molecules with the same denominator;
(2) Addition and subtraction of fractions with different denominator, first divided by fractions with the same denominator, and then added and subtracted;
※ 3. Concept connotation:
The key to general division is to determine the simplest denominator, and its method is as follows:
(1) coefficient of the simplest common denominator, taking the least common multiple of each denominator coefficient;
(2) The letter of the simplest common denominator is the product of the power of all letters of each denominator,
(3) If the denominator is a polynomial, factorize the polynomial first.
& lt/b, then a-b
& lt/b, then a-b is negative; On the contrary, if a-b is a positive number, then a
& lt/bc，