Solution set of 1. 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.

Second, one-dimensional linear inequality:

1 contains only one unknown, the formula containing the unknown is an algebraic expression, and the number of the unknown is 1. Inequalities like this are called unary linear inequalities. ※.

2. The process of solving a linear inequality with one variable is similar to 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. ※.

Three 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;

⑤ Answer: Write the answer and check whether the answer conforms to the meaning of the question.

Thirdly, one-dimensional linear inequalities.

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 the solution set of each inequality in the linear inequality group is called the solution set of the 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.

2. Mid-term knowledge points in the second volume of eighth grade mathematics

※ 1. inequality relation1. Generally speaking, formulas connected by symbols (or) and (or) are called inequalities.

2. Accurately translate inequalities and correctly understand non-negative numbers, not less than and other mathematical terms. ※ 。

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 ab, then a+cb+c, A-CB-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, if ab and c0, then acbc,.

(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 ab and c0, then ac.

2. Comparison size: (A and B represent two real numbers or algebraic expressions respectively). ※

Generally speaking:

If ab, then a-b is a positive number; On the contrary, if a-b is a positive number, then a

3. Mid-term knowledge points in the second volume of eighth grade mathematics

Chapter 1 Fraction 1, the numerator and denominator of the fraction and its basic properties are multiplied (or divided) by an algebraic expression that is not equal to zero at the same time, and the fraction is only constant.

2. Fraction operation

(1) The law of multiplication, division and multiplication of fractions: Fractions are multiplied by fractions, the product of molecules is the numerator of the product, and the product of denominator is the denominator of the product. Law of division: a fraction is divided by a fraction, and the numerator and denominator of division are multiplied by the divisor in turn.

(2) Law of fractional addition and subtraction: fractional addition and subtraction with the same denominator, and numerator addition and subtraction with the same denominator; Fractions with different denominators are added and subtracted, first divided by fractions with the same denominator, and then added and subtracted.

Chapter II Inverse Proportional Function

Expressions, images and properties of inverse proportional function.

Image: hyperbola

Expression: y=k/x(k is not 0)

Nature: the increase and decrease of the two branches are the same;

Chapter III Pythagorean Theorem

1, Pythagorean theorem: the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.

2. Inverse theorem of Pythagorean theorem: If the sum of squares of two sides in a triangle is equal to the square of the third side, then this triangle is a right triangle.

The fourth chapter quadrilateral

1, parallelogram

Attribute: equilateral; Diagonally equal; Divide diagonally.

Judgment: two groups of quadrangles with equal opposite sides are parallelograms;

Two groups of quadrangles with equal diagonal are parallelograms;

Quadrilaterals whose diagonals bisect each other are parallelograms;

A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

Inference: The midline of a triangle is parallel to the third side and equal to half of the third side.

2. Special parallelogram: rectangle, diamond and square.

(1) rectangle

Properties: All four corners of a rectangle are right angles;

Diagonal lines of rectangles are equal;

A rectangle has all the characteristics of a parallelogram.

Judgment: a parallelogram with a right angle is a rectangle; Parallelograms with equal diagonals are rectangles;

Inference: The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.

(2) The nature of the diamond: all four sides of the diamond are equal; Diagonal lines of the rhombus are perpendicular to each other, and each diagonal line bisects a set of diagonal lines; A diamond has all the characteristics of a parallelogram.

Judgment: A set of parallelograms with equal adjacent sides is a diamond; Parallelograms with diagonal lines perpendicular to each other are diamonds; A quadrilateral with four equilateral sides is a diamond.

(3) Square: It is both a special rectangle and a special diamond, so it has all the properties of a rectangle and a diamond.

3. Trapezoids: right-angled trapezium and isosceles trapezium.

Isosceles trapezoid: two angles on the same bottom of isosceles trapezoid are equal; The two diagonals of isosceles trapezoid are equal; A trapezoid with two equal angles on the same base is an isosceles trapezoid.

Chapter V Data Analysis

Weighted average, median, mode, range, variance.

4. Mid-term knowledge points in the second volume of eighth grade mathematics

Five knowledge points: 1, definition of quadratic equation in one variable, general form of quadratic equation in one variable, concept and application of solution of quadratic equation in one variable.

2. Four solutions of quadratic equation in one variable (factorization method, Kaiping method and matching method, popularization and application of matching method, formula method).

3. Discrimination of roots

4. The application of one-dimensional quadratic equation (sales problem and growth rate problem, area problem and dynamic problem).

5. The relationship between the roots and coefficients of a quadratic equation (Vieta's theorem)

Textbook related knowledge points

1, unary quadratic equation: contains only unknowns, and the sum is 2. Such an integral equation is called an unary quadratic equation.

2. The unknown number that can make a quadratic equation is called the solution (or root) of a quadratic equation.

5. Mid-term knowledge points in the second volume of eighth grade mathematics

1, definition: two groups of parallelograms with parallel opposite sides are called parallelograms.

2. The properties of parallelogram

(1) The opposite sides of the parallelogram are parallel and equal;

(2) The adjacent angles of the parallelogram are complementary and the diagonal angles are equal;

(3) diagonal bisection of parallelogram;

3. Determination of parallelogram

Parallelogram is an important content in geometry, and how to judge whether a parallelogram is a parallelogram according to its properties is a key point. Here are five ways to judge a parallelogram:

The first category: related to the opposite sides of a quadrilateral.

(1) Two groups of parallelograms with parallel opposite sides are parallelograms;

(2) Two groups of quadrangles with equal opposite sides are parallelograms;

(3) A group of quadrilaterals with parallel and equal opposite sides are parallelograms;

The second category: related to the diagonal of quadrilateral.

Two groups of quadrangles with equal diagonal are parallelograms;

The third category: related to the diagonal of the quadrilateral.

Quadrilaterals whose diagonals bisect each other are parallelograms.