High School Mathematics Absolute Geometry, Advanced Mathematics Absolute Value

It is recommended to use the most basic method (the most secure and error-free method for everyone to master):

Break down the absolute value and discuss it with positive and negative signs.

Other methods, such as geometry, are ingenious and can be mastered.

Absolute value Geometrical significance: On the number line, the distance between a number and the origin is called the absolute value of the number

Algebraic significance: The absolute value of a positive number is itself, the absolute value of a negative number is its opposite, and the absolute value of 0 is 0

The absolute values of two numbers that are opposite to each other are equal.

Absolute values are marked with "|".a |" means read as "the absolute value of a".

Such as: |－2|Reads as an absolute value of 2.

The absolute value of a positive number is itself, the absolute value of a negative number is its opposite, and the absolute value is a non-negative number of 0.

The absolute value of the particular zero is both his own and his opposite, written |0|=0

The two negative numbers are compared to the size, and the absolute value is larger than the smaller.

For example: if |2(x—1）—3|+(2y—4) =0, then x= y= is an absolute value).

Answer: 2(x-1)-3=0

x=5/22y-4=0

y=2 The absolute value of a pair of opposite numbers is equal:

Example: The absolute value of +2 is equal to the absolute value of -2 [because they are equal in length per unit from the origin on the number line].

Geometric and algebraic meanings of absolute values:

Geometric definition: The distance between the point representing the number a and the origin point on the number line is called the absolute value of the number a. (The distance between the point representing the number a and the origin on the number line must be non-negative).

Algebra definition: |a|={a>0 a=a

a<0 a=-a

a=o a=0

Question about absolute values: Known |x|=3，|y|=1 2, and |x-y|=y-x, find y-x

Solution: Because |x-y|>0 or =0, and |x-y|=y-x, so x<0, x can only be equal to or =1 2. Let y=1 2, then the original formula = 1 2-(-3) = 3 and 1 2.

Let y=-1 2, then the original formula = (-1 2)—(3)=2 and 1 2.

Answer: y-x is equal to 3 and 1 2 or 2 and 1 2.

x-1|+|x-2|+|x-3|..x-5|If you want to be the smallest, you need to take the middle one, which is x-3 = 0, that is, x = 3 original formula = 6, which is the minimum value.

x-1|+|x-2|+|x-3|+|x-4|Then take any point in the middle of 2 and 3 to get 4

Formula|m-n|-|n-m|=0

m n can be any number.

Vector operation is used to prove that although the writing process of the proof process is longer, it reduces the amount of thinking and the time of thinking because there is no need to add auxiliary lines

Formula: a + b a+b ; When a and B have the same number, take the equal sign, and take the unequal sign when the different number].

Thus from f(x)-a <1, f(x)-a + a < a +1; [Add a on both sides of the inequality];

and f(x)-a + a f(x)-a+a = f(x) [Think of f(x)-a as a in the formula; Think of a as b in the formula and write it backwards: f(x) f(x)-a + a ;

Absolute value f(x) = absolute value f(x)-a + a absolute value f(x)-a + absolute value a

Discussion of this type of question type:

3x-2|-|2x+4|

If x>0, then.

f(x)=3x-2-2x-4=x-6

In this case, there is no maximum or minimum value.

If x>0, then.

f(x)=2-3x+2x-4=x+2

In this case, there is no maximum or minimum value.

If x=0, then f(x

6, then the maximum and minimum values are both 6

You can draw images with the x-axis and y-axis to be more intuitive.

X-500 5 or X-500 co-cavity-5

So 495 x 505

2x>2 or 2x Ode to Chaos-2

So x> Kai Zheng 1 or x<-1

The first one can't read what you wrote. The second answer is that x is greater than 1 or less than 6

x-500|5 gets 495 x 505|2x+5|7 x 1 or x -6

If it is an equality group, it will intersect.

495 ≤x≤505

I want to remove the absolute value in conformity.

x-500|≤5

Equivalent to (x-500) 5

Again relatively. X-500 5 and -(X-500) 5

You can solve these 2 formulas.

x≤505 x≥495

Take the collection to find x495 x 505

2x+5|7 Same.

The geometric meaning of the absolute value: On the number line, the distance from a number to the origin is called the absolute value of the number. Represents the distance between a point representing a and a point representing b on the number line.

Applications: |5|Refers to the distance between 5 and the origin on the number line, this distance is 5, so the absolute value of 5 is 5. Similarly, it means that the distance between -5 and the origin is represented on the number line, and this distance is 5, so the absolute value of -5 is also 5.

The distance between -3 and -2 points on the exponential axis, the sub-value of this equation is 1. The same also indicates the distance between 3 and 2 points.

This kind of question. There are 3 ideas. Draw a wavy line (you can do it once), and return to the pure type such as |x+1|+|x-2|0, first, find the two roots that make a part of the two equations zero.

Find these two points on the number line, from top to bottom and right to left. The part on the number line is greater than zero, and the bottom is less than zero, and if there is a constant outside, the default point is 0 point. It seems that the wave thread can solve your above.

What is the meaning of a few omissions: Classification Discussion Method: It is also to find the two roots that make a part of two equations zero.

Classification: x -1, when -1 x 2, when 2 x. It is the trouser button that divides the number axis into three sections, and classifies them to discuss and remove absolute values.

And then solve inequalities or equations, which is the general law. :Function Method:

Let this equation be equal to y, and draw the image in the coordinate system, and the solution can be obtained directly.

If you want to practice this kind of question, you can buy a new method of Peiyou competition, Mr. Huang Dongpo's, seventh grade, which summarizes a large number of question types in detail. You can practice. You're a freshman in high school, but you didn't understand it in your first year of junior high school?

I don't learn this in high school...

On the number line, the distance from x to the origin is less than or equal to 6, and x is between -6 and 6.

In fact, the negative sign can be removed, that is, lxl is less than or equal to 6, and the result is -6 less than or equal to x less than or equal to 6