The basic properties of inequality, the basic properties of inequality

Basic properties of inequality: symmetry; transitivity; additive monotonicity, i.e., the additiveness of codirectional inequality; multiplicative monotonicity; multiplicability of positive inequalities in the same direction; positive inequalities are multipliable; Positive inequalities can be squared; The law of reciprocal counts.

An inequality is a mathematical formula that is connected by greater than, less than, greater than or equal to, and less than or equal to.

Unary Inequalities: Inequalitys that contain one unknown number and the number of unknowns is one order, such as 3-x>0.

Similarly, a binary inequality is an inequality that contains two unknowns and the number of unknowns is one.

Common theorems:

The inequality f(x)< g(x) is the same as the inequality g(x) > f(x).

inequality f(x) If the inequality f(x) definition domain is contained by the definition domain of the parsed formula h(x), and h(x) > 0.

The inequality f(x)g(x)>0 is the same solution as the inequality.

Property 1: The same number or formula is added (or subtracted) to both sides of the inequality, and the direction of the inequality sign does not change.

Property 2: Both sides of the inequality are multiplied (or divided) by the same positive number, and the direction of the inequality sign does not change.

Property 3: When both sides of the inequality are multiplied (or divided) by the same negative number, the direction of the inequality sign changes.

Concept of inequality: The formula that uses an inequality sign to express the relationship between size and size is called an inequality.

Property 1: If a b, b c, then a c (transitivity of the inequality).

Property 2: If a b, then a c b c (additive property of inequality).

Property 3: If a b, c 0, then ac bc; If a b, c 0, then ac bc (multiplicative property of inequality).

If x>y, then yy; (Symmetry).

If x>y, y>z; Then x>z; If x>y and z is an arbitrary real number or integer, then x+z>y+z, i.e., the same integer is added or subtracted from both sides of the inequality at the same time, and the direction of the inequality is unchanged;

If x>y,z>0, then xz>yz, i.e., both sides of the inequality are multiplied (or divided) by the same integer greater than 0 at the same time, and the direction of the inequality sign is unchanged;

If x>y,z<0, then xzIf x>y,m>n, then x+m>y+n;

If x>y>0, m>n>0, then xm>yn;

Inequality property 1:The same number (or equation) is added (or subtracted) on both sides of the inequality, and the direction of the inequality sign remains the same

Inequality property 1:

Inequality property 2: Multiply (or divide) the same positive number on both sides of the inequality, and the direction of the inequality sign does not change

Inequality property 2

Inequality property 3:When both sides of the inequality are multiplied (or divided) by the same negative number, the direction of the inequality sign changes

Inequality property 3:

Summary. Hello dear, the properties of inequality are: symmetry; transitivity; additive monotonicity, i.e., the additiveness of codirectional inequality; multiplicative monotonicity; multiplicability of positive inequalities in the same direction; positive inequalities are multipliable; Positive inequalities can be squared; The law of reciprocal counts.

Kiss, wild state Hello, the properties of inequality are: symmetry; Closure transitivity; additive monotonicity, i.e., the additiveness of codirectional inequality; multiplicative monotonicity; multiplicability of positive inequalities in the same direction; Songyan is squared by inequalities; Positive inequalities can be squared; The law of reciprocal counts.

If x>y, then yy, y>z; Then x>z; If x > y, and z is an arbitrary real number or integer, then x z>y z, i.e., both sides of the inequality are added or subtracted at the same time by a dusty pre-integer of the same distribution, and the direction of the inequality sign remains unchanged; If x>y,z>0, then x*( z>y*( z, i.e., both sides of the inequality are multiplied (or divided) by the same integer greater than 0 at the same time, and the direction of the inequality sign is unchanged; If x>y,z<0, then x*(z)z, i.e., both sides of the inequality are multiplied (or divided) by the same integer less than 0 at the same time, and the direction of the inequality sign changes; If x>y,m>n, then x+m>y+n; If x>y>0, m>n>0, then xm>yn; If x>y>0, then the nth power of ny of x (n is positive), and the nth power of x.

Kiss, the above are all inequalities, can you understand it? If you don't understand, you can ask me.

If x>y, then yy; (symmetry) if x>y, y>z; Then x>z; If x>y and z is an arbitrary real number or integer, then x+z>y+z, i.e., the same integer is added or subtracted from both sides of the inequality at the same time, and the direction of the inequality is unchanged; If x>y,z>0, then x*( z>y*( z, i.e., both sides of the inequality are multiplied (or divided) by the same integer greater than 0 at the same time, and the direction of the inequality sign is unchanged; If x>y,z<0, then x*( zy,m>n, then x+m>y+n; If x>y>0, m>n>0, then xm>yn; If x>y >0, then the nth power of x" the nth power of y (n is positive), and the nth power of x <>

In other words, another expression of the fundamental properties of inequality is: symmetry; transitivity; additive single-lead blind tonality, i.e., the additiveness of co-directional inequality; multiplicative monotonicity; multiplicability of positive inequalities in the same direction; positive inequalities are multipliable; Huai Xiaokong is a positive inequality can be squared; The law of reciprocal counts.

The basic properties of inequalities are as follows:

1.If x>y, then yy; (Symmetry).

2.If x>y, y>z; Then x>z; transitivity;

3.If x>y and z is an arbitrary real number or integer, then x+z>y+z, i.e., the same integer is added or subtracted from both sides of the inequality at the same time, and the direction of the inequality sign remains the same.

4.If x>y,z>0, then xz>yz, i.e., both sides of the inequality are multiplied (or divided) by the same integer greater than 0 at the same time, and the direction of the inequality sign is unchanged;

5.If x>y,z<0, then xz6If x>y,m>n, then x+m>y+n;

7.If x>y>0, m>n>0, then xm>yn;

8.If x>y>0, then the nth power of x is the nth power of y (n is positive), and the sail of x is simple to the nth power of y (n is negative).

1. Symmetry;

2. Transitivity;

3. Addition monotonicity, that is, the additiveness of co-directional inequality;

4. Multiplication monotonicity;

5. Multiplicability of positive inequalities in the same direction;

6. The positive inequality can be multiplied;

7. The square can be opened for the inequality of positive values;

8. The law of countdown auspiciousness.

Inequality 8 properties:

If x>y, then yy, y>z; Then x>z;

If x>y and z is an arbitrary real number or integer, then x+z>y+z, i.e., the same integer is added or subtracted from both sides of the inequality at the same time, and the direction of the inequality sign remains the same.

If x>y,z>0, then xz>yz, i.e., both sides of the inequality are multiplied (or divided) by the same integer greater than 0 at the same time, and the direction of the inequality sign is unchanged;

If x>y,z<0, then x*( zy,m>n, then x+m>y+n;

If x>y>0, m>n>0, then xm>yn;

If x>y>0, then the nth power of ny of x (n is positive), and the nth power of x.

Inequality is an important concept in mathematics, which is a mathematical statement that compares the relationship between the magnitude of two numbers. The basic properties of inequalities include the following:

Addition and subtraction: A number is added (or subtracted) on both sides of the inequality at the same time, and the relationship between the inequalities remains unchanged. For example, for inequality a

Positive and negative: both sides of the inequality are multiplied (or divided) by a positive number at the same time, and the relationship of the inequality remains unchanged; Multiply (or divide) a negative number on both sides at the same time, and the relationship of the inequality is reversed. For example, for inequality ABC.

Inversion: Both sides of the inequality are negated at the same time (i.e., multiplied by -1), and the relationship of the inequality is reversed. For example, for inequalities a-b.

Transmissibility: If a

Reflexivity: When any number is larger than oneself, its size relationship is equal, i.e., a=a.

These basic properties are the basis for the study of the rapid collapse and the application of inequalities, through which the operation and derivation of inequalities can be carried out, and various methods and techniques of inequalities can be further mastered and applied.

Hello, there are 3 basic properties of inequalities:

The two sides of the inequality are added (or subtracted) with the same phase as the same imaginary integer, and the sign direction of the inequality does not change.

For example, if x>y, add or subtract m from both sides, then x+m>y+m, x-m>y-m

The two sides of the inequality multiply (or divide) the same positive number, and the sign direction of the inequality does not change.

For example, if x>y, m>0, then x*( m>y*( m is multiplied (or divided) by the same negative number on both sides of the inequality, and the sign direction of the inequality changes.

For example: x>y, m<0, then x*( m

Added: The rest of the properties of inequality:

If x>y, then yy; (Symmetry).

If x>y, y>m; then x>m; (transitivity) if x>y, m>n, then x+m>y+n;

If x>y>0, m>n>0, then xm>yn;

If x>y >0, then the nth power of x" the nth power of y (n is positive), and the nth power of x <>

Basic Inequality Formula:

1）(a+b)/2≥√ab

2）a^2+b^2≥2ab

3）(a+b+c)/3≥(abc)^(1/3)

4）a^3+b^3+c^3≥3abc

5）(a1+a2+…+an)/n≥(a1a2…an)^(1/n)

6）2/(1/a+1/b)≤√ab≤(a+b)/2≤√[a^2+b^2)/2]

Basic properties of inequality:

If x>y, then yy; (Symmetry).

If x>y, y>z; Then x>z; transitivity;

If x >y and z is an arbitrary real number or integer, then x+z>y+z. (The additive principle, or additivity of co-directional inequalities).

If x>y,z>0, then xz>yz. If x>y,z<0, then xz

If x>y,m>n, then x+m>y+n; (sufficient and unnecessary).

Add or subtract the same number or formula on both sides of an inequality, and the direction of the inequality sign does not change. (Change the number of the move item).

Multiply or divide the two sides of an inequality by the same positive number, and the direction of the inequality sign does not change. (Equivalent to a coefficient of 1, which can only be used if it has to be positive).

Multiply or divide the two sides of the inequality by the same negative number and the direction of the unequal sign changes. (or 1 negative number).