Is there any good way to simplify the absolute value of mathematics?

First of all, you should understand what is the geometric meaning of absolute value.

For example, the geometric meaning of | X| is the distance from X to the origin, while the geometric meaning of |x- 1| is the distance from X to 1.

|3-4| is the distance from 3 to 4, and | x+1| =| x-(-1) | = the distance from x to-1

After knowing the geometric meaning of absolute value, you'd better draw a number axis and mark some known points;

For example, | x-1 |+x+2 | = | x-1|+| x-(-2) | can be marked as1and-2.

If it is | x-1 |+| x+2 |+| x |, you can mark1,-2,4,-3,0 (| x | can be regarded as |x-0|).

Analogy in turn

After marking the points, the number axis is divided into segments by the points you marked, and then you do it one by one.

For example |x- 1|+|x+2|,

When x is less than or equal to 1, 1 is less than or equal to 2, and x is greater than 2, discuss and simplify the original formulas one by one, and remove the absolute value.

Add and subtract two rational numbers:

After simplifying the symbol, add the same symbol, take the same symbol, and add the absolute value;

Subtract different symbols, take the symbol with larger absolute value, and subtract the symbol with smaller absolute value with the larger absolute value.

The sum of two opposites is zero.

Adding a number to zero still gets this number.

Note that no matter how to add or subtract, the simplified symbol is regarded as a formula that omits the plus sign and leaves only the symbol and absolute value.

If -3+(+2) is reduced to -3+2, it is regarded as the sum of -3+2, and the plus sign is omitted and read as the sum of -3+2 or -3+2.

For another example, -3-(+2) is reduced to -3-2, which is regarded as the sum of -3 and -2, and the plus sign is omitted and read as "minus 3 plus minus 2" or "minus 3 plus minus 2".

In this way, the calculation of -3-2 is to add the negative sign, take the same symbol "-",and add the absolute value (the absolute value here is directly consistent with the number learned in primary school, that is, the symbol is always positive), which is 3+2=5, and the result is -5.

The calculation of -3+2 is symbolic subtraction. Take the symbol "-"with large absolute value and subtract the one with small absolute value, that is, 3-2= 1, so the result is-1.

In the operation, zero can be omitted directly, such as: 0-3=-3, 0+3=3, 3+0=3, 3-0=3.

In the calculation process, only the nature symbols are considered, and the operation symbols are not considered, which reduces the errors caused by the confusion of the two symbols, and the absolute value is directly consistent with the numbers learned in primary school. Therefore, the key to adding and subtracting rational numbers is to identify symbols, and it is not difficult to do more problems carefully.

On the problem of removing the sign of absolute value. .

Symbolize the absolute value, turn the problem into a problem without absolute value sign, and determine the positive and negative parts of the absolute value sign (that is, the absolute value of non-negative numbers is equal to itself; The absolute value of a non-positive number is equal to its inverse number), so there are roughly three ways to remove the absolute value sign.

First, set the conditions according to the topic (the symbol of the absolute value formula can be directly determined by knowing the value range of letters)

Example 1: Let x <-1 and simplify the result of 2-| 2-| x-2 ||| |.

Determined by X.

As long as we know whether the algebraic expression whose absolute values will be combined is negative or zero, we can remove the absolute value symbol smoothly according to the absolute value meaning, which is a conventional idea to solve this kind of problem.

Second, with the help of teaching axis

This type of question is to mark the known conditions on the number axis, so that people can observe them with the information provided by the number axis, thus determining:

1. The left side of the origin is negative and the right side is positive.

2. The number represented by the right point is always greater than that represented by the left point.

3. The absolute value of the points far from the origin is large, so you can easily solve the problem by remembering these points.

Thirdly, the zero-point subsection discussion method is adopted.

"Zero method":

(1) Make each absolute value in the formula zero, and save the value of letters to get "zero point";

(2) representing each "zero point" on the number axis, and dividing the number axis into several parts to represent the range of each part;

(3) Simplify the absolute value according to each part.

The general steps of adopting this method are:

1. Find zero: make the algebraic sign in each absolute value zero, and find zero (not necessarily two).

2. Segmentation: According to the zero obtained in the first step, the points on the number axis are divided into several segments, so that the positive and negative parts in each absolute value symbol in each segment can be determined.

3. Investigate the problems in each part.

4. Synthesize the situation of each section and get the answer to the question.

Another understanding of absolute value (the distance between two points on the number axis)

For example, | -8 | means the distance from -8 to the origin 0, that is, the distance from -8 to 0 on the number axis, which can be expressed as |-8-0 | = 8.

For example, the distance between -4 and 2 on the number axis is 6, which means |-4-2 | = 6.

In the process of learning absolute values, we use the number axis, which embodies the idea of combining numbers with shapes.

We divide rational numbers into positive numbers, negative numbers or zero to study and discuss, and apply the thinking method of classified discussion.