① The formula using symbol > connection = is called inequality.

② Add or subtract the same algebraic expression on both sides of the inequality, and the direction of the inequality remains unchanged.

③ Both sides of inequality are multiplied or divided by a positive number, and the direction of inequality remains unchanged.

④ Both sides of inequality are multiplied or divided by the same negative number, and the unequal numbers are in opposite directions.

Solution set of inequality;

(1) can make the value of an unknown inequality known as the solution of inequality.

(2) All solutions of an inequality with unknowns constitute the solution set of this inequality.

③ The process of finding the solution set of inequality is called solving inequality.

Unary linear inequality: An inequality with algebraic expressions on both sides and only one unknown with the highest degree of 1 is called unary linear inequality.

One-dimensional linear inequality system;

(1) Several linear inequalities about the same unknown quantity are combined into a linear inequality group.

② The common part of the solution set of each inequality in a linear inequality group is called the solution set of this linear inequality group.

③ The process of finding the solution set of inequality group is called solving inequality group.

2 1, square sum square root, an important knowledge point in senior one mathematics.

2. Area and square

(1) The sum of squares of any two positive numbers is equal to the sum of squares of these two numbers.

(2) The square of the difference between any two positive numbers is equal to the sum of the squares of these two numbers, and then twice the product of these two numbers is subtracted.

The square of the sum (or difference) of any two rational numbers is equal to the sum of the squares of these two numbers, plus (or minus) twice the product of these two numbers.

3. Square root

A positive number (1) has two square roots with opposite directions;

(2) Zero has only one square root, and the square root is zero itself;

(3) Negative numbers have no square root.

4. Real numbers

The decimal of infinite cycle is called irrational number.

Rational numbers and irrational numbers are collectively called real numbers.

5, the operation of the square root

6, the nature of the arithmetic square root

The square of the arithmetic square root of the attribute 1 non-negative number is equal to the number itself.

Property 2 The arithmetic square root of the square of a number is equal to the absolute value of this number.

7. Multiplication of arithmetic square root and Divison.

1) arithmetic square root multiplication

sqrt(a)sqrt(b)= sqrt(ab)(a & gt； =0，b & gt=0)

Square root division

sqrt(a)/sqrt(b)= sqrt(a/b)(a & gt； =0，b & gt0)

The way to rationalize the denominator is to multiply the numerator and denominator by a formula, remove the root sign in the denominator and burn the denominator in the root sign.

3) The exponent of each factor of the root sign is less than 2; (2) The root number does not contain letters. We call the square root satisfying these two conditions the simplest square root.

8' arithmetic square root addition and subtraction operations

If several square roots are converted into the simplest square roots and the number of roots is the same, then these square roots are called similar square roots.

9. One-variable quadratic equation and its solution

1) One-variable quadratic equation

An equation with only one unknown number and the highest degree of the unknown number is 2 is called a quadratic equation with one variable.

2) Solving the special quadratic equation with one variable.

3) Solving the general quadratic equation with the unary collocation method.

The general steps of solving a quadratic equation with one variable by collocation method are as follows:

1, the quadratic coefficient is 1. Divide the two sides of the equation by quadratic coefficient and change the equation into the form of x 2+px+q = 0.

2. Move the constant term to the right of the equation and convert the equation into the form of x 2+px =-q.

3. Add "the square of half of the first coefficient" to both sides of the formula equation to form a complete square, with the unknown on the left and the constant on the right.

According to the definition of square root, we can see that

(1) when p 2/4-q >; 0, the original equation has two real roots;

(2) When p 2/4-q = 0, the original equation has two equal real roots (multiple roots).

3 1, monomial: In an algebraic expression, if only multiplication (including power) operations are involved. Or algebraic expressions that contain division but do not contain letters in division are called monomials.

2. The coefficient and times of single item: the non-zero numerical factor in single item is called the numerical coefficient of single item, which is simply referred to as the coefficient of single item; When the coefficient is not zero, the sum of all the letter indexes in the single item is called the number of times of the single item.

3. Polynomial: The sum of several monomials is called polynomial.

4. Number and degree of polynomials: the number of monomials contained in a polynomial is the number of polynomial terms, and each monomial is called a polynomial term; In polynomial, the degree of the term with the highest degree is called the degree of polynomial.

Through the study of this chapter, students should achieve the following learning objectives:

1. Understand and master the concepts of monomial, polynomial and algebraic expressions, and find out the differences and connections between them.

2. Understand the concept of similar items, master the method of merging similar items, master the changing law of symbols when removing brackets, and be able to merge and remove brackets correctly. On the basis of accurate judgment and correct combination of similar items, add and subtract algebraic expressions.

3. Understand that the letters in the algebraic expression represent numbers, and the addition and subtraction operations of the algebraic expression are based on numbers; Understanding the basis of merging similar items and removing brackets is the distribution law; Understanding the operation rules and properties of numbers is still effective in the addition and subtraction of algebraic expressions.

4. Be able to analyze the quantitative relationship in practical problems and express it with a formula with letters.

Important knowledge points of junior one mathematics 4 Summary of important knowledge points of junior one mathematics

1. bracket removal: Generally speaking, several algebraic expressions are added and subtracted. If there are brackets, remove them first, and then merge similar items. If the factor outside the brackets is positive, the symbols of the items in the original brackets are the same as the original symbols after the brackets are removed. If the factor outside the brackets is negative, the symbols of the items in the original brackets are opposite to those after the brackets are removed.

2. Merging similar items: Merging similar items in polynomials into one item is called merging similar items. After merging similar items, the coefficient of the obtained item is the sum of the coefficients of similar items before merging, and the letter part remains unchanged.

Induction of Important Knowledge Points in Junior One Mathematics

Addition and subtraction of algebraic expressions

1. monomial: in algebraic expressions, if only multiplication (including power) operations are involved. Or algebraic expressions that contain division but do not contain letters in division are called monomials.

2. The coefficient and times of single item: the non-zero numerical factor in single item is called the numerical coefficient of single item, which is simply referred to as the coefficient of single item; When the coefficient is not zero, the sum of all the letter indexes in a single item is called the number of times of the single item.

3. Polynomial: The sum of several monomials is called polynomial.

4. Number and degree of polynomials: the number of monomials contained in a polynomial is the number of polynomial terms, and each monomial is called a polynomial term; In polynomial, the degree of the degree term is called the degree of polynomial; Note: (If A, B, C, P and Q are constants) ax2+bx+c and x2+px+q are two common quadratic trinomials.

5. Algebraic expression: Any algebraic expression that does not contain division operation or contains division operation but does not contain letters in the division formula is called algebraic expression.

6. Similar items: monomials with the same letters and the same index are similar items.

7. Rules for merging similar items: When the coefficients are added, the letter index remains unchanged.

8. Rules for deleting (adding) brackets: When deleting (adding) brackets, if there is a "+"sign before the brackets, all items in the brackets remain unchanged; If there is a "-"before the brackets, all items in the brackets should be changed.

9. Algebraic addition and subtraction: Algebraic addition and subtraction is actually to combine similar terms of polynomials on the basis of removing brackets.

10. Ascending power and descending power arrangement of polynomials: arrange the items of a polynomial from small to large (or from large to small) according to the exponent of a letter, which is called ascending power arrangement (or descending power arrangement). Note: The final result of polynomial calculation should generally be ascending power (or descending power arrangement).

Sorting out the important knowledge points of junior one mathematics

Geometric definition of absolute value

Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of A, which is denoted as |a|.

2. Algebraic definition of absolute value

(1) The absolute value of a positive number is itself; (2) The absolute value of a negative number is its inverse; The absolute value of 0 is 0.

Can be expressed in letters as follows:

① If a>0, then | a | = a② If A

It can be summarized as ①: a ≥ 0, |a|=a (the absolute value of non-negative number equals itself; A number whose absolute value is equal to itself is nonnegative. ②a≤0, |a|=-a (the absolute value of a non-positive number is equal to its inverse number; A number whose absolute value is equal to its opposite number is not a positive number. ) classic exam questions

As shown on the axis, simplify the following figures.

|a|，|b|，|c|，|a-b|，|a-c|，|b+c|

Solution: We know from the problem that because a>0, b<0, c0, A-C > 0，b+ c & lt； 0,

So | a | = a, | b | =-b, | c | =-c, | a-b | = a-b, | a-c | = a-c, | b+c | =-(b+c) =-b-c.

3. The essence of absolute value

The absolute value of any rational number is non-negative, that is, the absolute value is non-negative. Therefore, if a takes any rational number, there is |a|≥0. That is, the absolute value of (1)0 is 0; A number with an absolute value of 0 is 0. That is, a = 0 | a | = 0.

The absolute value of a number is non-negative, and the number with the smallest absolute value is 0. That is | a | ≥ 0;

(3) The absolute value of any number is not less than the original number. Namely: | a | ≥ a;

(4) There are two numbers whose absolute values are the same positive number and the directions are opposite. That is, if | x | = a(a >；; 0), then x = a；;

5] The absolute values of two opposite numbers are equal. That is: |-a|=|a| or |a|=|b| If a+b = 0;

[6] Two numbers with equal absolute values are equal or opposite. That is: |a|=|b|, then a=b or a =-b;

Once, if the sum of the absolute values of several numbers is equal to 0, then these numbers are simultaneously 0. That is |a|+|b|=0, then a=0 and b=0.

(Common properties of non-negative numbers: if the sum of several non-negative numbers is 0, then only these non-negative numbers are 0 at the same time)

Classic examination questions

Given | a+3 | | 2b-2 | | c- 1 | = 0, find the value of a+b+c.

Solution: Because |a+3|≥0, |2b-2|≥0, |c- 1|≥0, and | A+3 |+2B-2 | | C- 1 | = 0.

So |a+3|=0, |2b-2|=0, |c- 1|=0.

That is, a =-3, b = 1 and c = 1.

So a+b+c =-3+1+1=-1.

4. Comparison of rational numbers

⑴ Compare the size of two numbers by using the number axis: when two numbers on the number axis are compared, the left one is always smaller than the right one;

⑵ Compare the size of two negative numbers with absolute values: two negative numbers compare the size, and the absolute value is larger than the small one; The comparison size of two positive numbers with different signs.