2 If x>y, y & gtz；; Then x & gtz；; (transitivity)

③ if x>y and z is any real number or algebraic expression, then x+z >; y+z； (addition principle, or additivity of inequality in the same direction)

4 If x>y, z>0, then xz & gtyz；; If x>y, z<0, and then xz.

⑤ If x>y, m>n, then X+M > y+n； (Sufficiently unnecessary conditions)

6. If x>y>0, m>n>0, then xm & gtyn；;

⑦ If x>y>0, then the n power of X >; The n power of y (n is a positive number), the n power of x < the n power of y (n is a negative number).

In other words, the basic nature of inequality is:

① symmetry;

② Transitivity;

③ monotonicity of addition, that is, additivity of inequality in the same direction;

④ Monotonicity of multiplication;

⑤ Multiplicity of positive inequality in the same direction;

⑥ Positive inequalities can be multiplied;

⑦ Positive inequalities can be squared;

⑧ Reciprocity rule.

In addition, inequality has three properties:

(1) Inequality 1: both sides of the inequality add (or subtract) the same number (or formula) at the same time, and the direction of the inequality remains unchanged;

2 Inequality 2: Both sides of the inequality are multiplied (or divided) by the same positive number at the same time, and the direction of the inequality remains unchanged;

③ Inequality 3: Both sides of inequality are multiplied (or divided) by the same negative number at the same time, and the direction of inequality changes.

Precautions editing

sign

Add and subtract the same number or formula on both sides of inequality, and the direction of inequality remains unchanged. (Move items to change symbols)

When both sides of inequality multiply or divide the same positive number, the direction of inequality remains unchanged. (The equivalent coefficient is 1, which is a positive number that can be used.)

When both sides of inequality are multiplied or divided by the same negative number, the direction of inequality changes. (Change sign when ÷ or × 1 negative)

relax

Determine the solution set:

(1) If it is greater than two values, it is greater than the larger one (whichever is greater);

(2) If it is less than two values, it is less than the smaller one (the same as the smaller one);

(3) It is bigger than the big one, smaller than the small one, and there is no solution (neither big nor small can be taken);

(4) It is bigger than the small one, smaller than the big one, and there is a solution in the middle (the small one takes the middle).

An inequality group consisting of three or more inequalities can be analogized.