B)∈M, and for other elements (c, d) in m, there is always c≥a, then a = _ _ _.

Analysis: Understanding and revealing the mathematical essence of the problem will be the breakthrough to solve the problem. How to understand "for other elements (C, D) in M, there is always c≥a"? What are the characteristics of the elements in M?

Solution: According to the problem, this problem is equivalent to finding the function x=f(y)=(y+3)? |y- 1|+(y+3)

② When 1≤y≤3,

So when y= 1, xmin = 4.

Note: the form of set appears in conditions, so we should recognize the essential attributes of set elements, and then reveal their mathematical essence in combination with conditions, that is, find the satisfaction relationship of elements in set M.

Example 2. Solve inequalities about:

Analysis: This example mainly reviews the idea of solving inequality with absolute value and classified discussion. The key to this problem is not to discuss the parameters, but to discuss the unknowns when the absolute values are removed, and two groups of inequalities are obtained. Finally, the solution sets of the two groups of inequalities are combined to get the solution set of the original inequality.

Solution: When

Example 3. Three inequalities are known: ① ② ③.

(1) If the values of ① and ② satisfy ③ at the same time, find the value range of m;

(2) If the value of ③ satisfies at least one of ① and ②, find the value range of M. ..

Analysis: This example mainly reviews the algebraic expression, fractional inequality, the solution of inequality with absolute value, and the idea of combining numbers with shapes. The key to solve this problem is to find out the necessary and sufficient conditions for satisfying the values of ① and ② at the same time: ③ The two corresponding equations are in sum respectively. Inequality is closely related to its corresponding equation and function image. In the process of solving problems, we should contact their internal relations in time.

Solution: Remember that the solution set of ① is A, ② is B and ③ is C.

Solution ① obtains a = (- 1, 3); B= for solution ②

(1) Because the values of ① and ② satisfy ③ and AB C at the same time.

Assuming that the minor root is less than 0 and the major root is greater than or equal to 3, the equation can be satisfied.

(2) Because the value satisfying ③ satisfies at least one of ① and ②, because

This small root is greater than or equal to-1 and the big root is less than or equal to 4, so

Note: The necessary and sufficient conditions for the value of x to satisfy ① ② and ③ are as follows: ③ The corresponding two equations 2x +mx- 1=0 are respectively at (-∞, 0) and; (2)(-∞,- 1]∪[2,+∞); (3){2}; (4)[- 1,+∞).

5. Solve the inequality about.

6.(2002 Beijing dialect) series is determined by the following conditions:

(1) Proof: For,

(2) Proof: Yes.

7. Let p = (log2x)+(t-2) log2x-t+1. If t changes in the interval [-2,2], and P is always positive, try to find the range of X. 。

8. In the known series,

B 1= 1, and the point P(bn, bn+ 1) is on the straight line x-y+2=0.

I) find the sequence

Ⅱ) Let the sum of the first n terms be Bn, and try to compare them.

Ⅲ) Let Tn=

Verb (abbreviation of verb) refers to the answer

1. solution: draw an image, which can be obtained from the knowledge of linear programming, and choose D.

2. Solution: When the proposition P is true, that is, all real numbers greater than zero can be obtained from the real number part, then the discriminant of quadratic function, thus; When the proposition q is true,.

If p or q is true and p and q are false, only one of p and q is true and the other is false.

P is true, q is false, and there is no solution; If p is false and q is true, the result is 1

3. Analysis: This topic mainly reviews the solution of fractional inequality, the idea of classified discussion and the basic steps of solving inequality by using the ordered axis standard root method. The key to this problem is to decompose the denominator and turn the original inequality into

And compare the sum with the size of 3 to determine the classification method.

Solution: The original inequality is transformed into:

(1) When, from the diagram 1, we know that the solution set of inequality is

(2) When

(3) When

4. Analysis: The roots of equations, the properties of functions, and images are closely related to the solution of inequalities. We should be good at connecting them organically, transforming each other and communicating with each other.

Solution (1) According to the meaning of the question, A > 0 and-1, 2 is the root of the equation ax2+bx+a2- 1≤0, so

(3) From the meaning of the question, 2 is the root of the equation ax2+bx+a2- 1=0, so

4a+2b+a2- 1=0。 ①

And {2} is the solution set of inequality ax2+bx+a2- 1≤0, so

(4) according to the meaning of the question, A = 0. B < 0, and-1 is the root of the equation bx+a2- 1=0, that is, -b+a2- 1=0, so

a=0，b=- 1。

Explanation: Quadratic function, quadratic equation and quadratic inequality are closely related. When solving specific mathematical problems, we should pay attention to their mutual connection and infiltration, and they will be transformed under certain conditions.

5. Analysis: method of substitution and graphic method are commonly used skills in solving inequalities. By changing elements, more complex inequalities can be classified into simpler or basic inequalities. Combining the constructor with the number and shape, the solution of inequality can be classified as an intuitive and vivid image relationship. For inequalities with parameters, graphic method can also make the classification criteria clearer.

Solution: Suppose that the original inequality is transformed into a bifunctional image in the same coordinate system.

Therefore (1) when

(2)

(3) If the solution set of the original inequality is φ.

To sum up, when, the solution set is); When the solution set is

, the solution set is φ.

6. Proof: (1)

(2) When,

=

7. Analysis: Obviously, it is necessary to find out the inequality (group) containing X according to the conditions set in the question, so it is necessary to seriously think about the meaning of "when T changes in the interval [-2,2], P is always positive". How to understand it? If it is difficult to continue thinking, please think from another angle. In a given mathematical structure, the formula on the right contains two letters, X and T, and T changes in a given interval, but what you can think of is the range of X?

Solution: Let p = f (t) = (log2x-1) t+log22x-2log2x+1. Because p = f (t) is a straight line in the top rectangular coordinate system, it is a necessary and sufficient condition for p to be positive when t changes in the interval [-2,2].

Log2x > 3 or log2x

Explanation: change the perspective of the problem, construct a linear function about t, and use the idea of function flexibly to turn a difficult problem into a familiar one.

8. Analysis: This topic mainly reviews the knowledge about the general term, sum and inequality of sequence.

Brief description: I)

ⅱ)Bn = 1+3+5+…+(2n- 1)= N2

ⅲ)Tn =①

②

①-② Acquisition