Absolute value equations, inequalities

1. Outline
   1. The problem is not the absolute value of Chaogang in the comprehensive examinations, and complex problem-solving process, can be used as the subject of a later qualifying exam
   2. The core idea of ​​solving this problem is to go to the absolute value, the absolute value of methods
      1. Category talk
         1. Solve simple problems can be classified discussion, when faced with complex problems to be divided into many layers, the process may be very complicated (fear of death is a multi-classification); but sometimes in a multi-layer classification can be obtained special conditions, regardless of some Happening
         2. In solving the equation or inequality, this method is often simpler expression
      2. Function Thought
         1. Equation, the inequality can be converted to a function of the relationship, meaning the absolute value may be changed according to x = a x-axis or flanging
         2. This method is suitable and known formulas containing absolute value side, the other side having a simple structure but the problem parameters
      3. Geometric meaning
         1. When the absolute value of the principal component elements are the same: two-dimensional absolute value may be converted into a distance, the square containing the absolute value of the difference in the two-dimensional distance may be converted into 2
   3. Note: After discussing classification must write, "In summary," to write out the solution

2. Absolute value equations

   1. Linear equation

      1. A plurality of single absolute value ( segment zero method step)
         1. Looking for the absolute value of zero
            1. Write the respective absolute value of x is the algebraic value of 0
         2. Zero segment discussion
            1. The number of shaft segments, discussed
         3. Segment solving equations
            1. Solving equations in each classification discussion, and then were tested
         4. Case
            
            
      2. Single multilayer absolute value
         1. From the inside to the absolute value symbol
            1. Zero fractionation method according to the absolute value of a layer, then test
            2. Case
               
         2. From outside to inside to the absolute value symbol
            1. The absolute value of a single place on the left, will put the rest of the other side, the right part can take positive and negative
            2. Case
               
               
      3. Function Method
         1. For equations containing parameters, the discussion classification is difficult, it is known can be represented by a function part, to convert the problem to find the intersections
            
         2. For a plurality of absolute value, write the first piecewise function, then the function draw
            

             

   2. Quadratic equation

      1. Similarly with the linear equation were looking for zero, you can factorization
         

3. Absolute value inequality

   1. Basic properties

      1. $|a|\geq|b|\Leftrightarrow a\geq|b|$或$a\leq-|b|\Leftrightarrow-|a|\leq b \leq |a| $

      2.  

         $|a|-|b|\leq|a\pm b|\leq|a|+|b|$

   2. Direct squares method
      1. 绝对值的部分平方后可以忽略绝对值。例如
         
   3. 分式法
      1. 对于$|a_1x^2+b_1x+c|=|a_2x+b_2|$，只要使$|a_2x+b_2|$不为零，就可以转化为$\displaystyle \frac{|a_1x^2+b_1x+c|}{|a_2x+b_2|}$，因式分解后可以化简
   4. 零点分段法
      1. 分类讨论
         
         
         

         

   5. 含参不等式
      1. 求条件不等式范围：分段考虑
         
      2. 几何意义