How to solve the absolute value of the inequality, how to solve the absolute value of the inequality

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The solution of two absolute inequalities.

1. Absolute value definition method.

For some simple absolute values with inequalities that are constant on one side.

It can be defined directly by absolute value, 1x|on the representation. The number line allows you to represent the solution set as a< x < a

2、|x|A can also be represented on the number line, so the solution set can be obtained as x a or x a

3、|ax +b|Type C, which is characterized as an inequality group c ax + b c by using the absolute value, and then solves the inequality group.

Second, the flat method.

When both sides of an inequality are absolute, you can square both sides of the inequality at the same time.

Solve inequality |x+ 3| >x− 1|After square both sides of the equation at the same time as (x + 3)2 > x 1)2 to get x2 + 6x + 9 > x2 2x + 1, the inequality can be solved, and x > 1 can be solved

3. Zero-point segmentation method.

For inequalities, there are two or more absolute values and constant terms.

, the zero-point segmentation method is generally used. Example of inequality |x + 1| +x − 3| >5

It can be seen on the number line that the number line can be divided into three intervals: x < 1, 1 x < 3, and x 3.

When x < 1, because x + 1 < 0, x 3 < 0, the inequality is solved as x 1 x + 3 > 5 to get x < 322 When 1 x < 3, the inequality is unsolved as x + 1 x + > 0 and x 3 >< 0.

When x 3 is solved as x + 1 > 0 , x 3 > 0 so the inequality is solved as x + 1 + x 3 > 5 to get x >72 In summary, the solution of the inequality is x < 32 or x >72.

To solve the absolute value inequality, we must try to remove the absolute value sign in the formula, and the solutions to the absolute value inequality include the geometric meaning method, the discussion method, the square sensitivity method, and the function image method.

(1) Geometric meaning

For example: find the inequality |x|1.

Inequalities|x|The solution set of 1 represents the set of points whose distance to the origin is less than 1, so the inequality |x|The solution set of 1 is.

(2) Discussion method

For example: find the inequality |x|1.

When x 0, the original inequality can be reduced to x 1, 0 x 1.

When x 0, the original inequality can be reduced to -x 1, -1 x 0.

To sum up, inequality |x|The solution set of 1 is.

(3) Flat method

For example: find the inequality |x|1.

Sqing the two sides of the original inequality yields: x2

1, i.e., x21 0, i.e., (x+1)(x-1) 0

i.e. -1 x less than 1, unequal group early |x|The solution set of 1 is.

(4) Functional image method

For example: find the inequality |x|1.

From a functional point of view, inequality |x|The solution set of 1 represents the function y=|x|The value range of x for the part below the image with y=1 corresponds. So inequality |x|The solution set of 1 is.

a|The distance between the point a on the number line and the origin point is called the absolute value of the number aa|-|b|≤|a±b|≤|a|+|b|。

Two important properties:

1、|ab|=|a||b|

a/b|=|a|/|b|（b≠0）

2、|a|<|b|Reversible rollout|b|>|a|

a|-|b| |a+b|≤|a|+|b|, if and only if ab 0 the left equal sign is true, and the right equal sign is true when ab 0 is true.

In addition, there are: |a-b|≤|a|+|b|=|a|+|1|*|b|=|a|+|b|

a|-|b| |a±b|≤|a|+|b|

Absolute inequalities are a class of shapes like |x|

an inequality where a is a real number and x is an unknown number. The key to solving the absolute value inequality is to determine the range of absolute values, and then classify and discuss them according to the definition of absolute values. The following are two common solutions to absolute inequalities.

1.etc. or spring effect deformation method.

For shapes such as |x|For example, we ask for a solution |2x + 1|<5. According to the equivalent deformation method, we deform it to -5 < 2x + 1 < 5. Then, reduce it to -3 < x

2.Therefore, the solution set is (-3, 2).

2.Taxonomy Discussion Method.

For shapes such as |x|> the inequality of the absolute value of a, we can classify and discuss the anecdote. When x > 0, |x| =x；When x > 0, |x| =x。Therefore, we can divide the original inequality into two inequalities, which are x > a and x < a.

Then, solve the solution set of these two inequalities and combine them to form the solution set.

For example, we ask for a solution |x - 2|>3. According to the taxonomic discussion, we divide it into two inequalities:

x - 2 > 3, i.e. x > 5;

x - 2 < 3, i.e. x < 1.

Therefore, the solution set is (-1) 5,

It is not difficult to solve the absolute value inequality, but it is necessary to pay attention to judging the range of absolute values, choosing the appropriate solution, and using mathematical skills such as equation deformation and classification discussion reasonably. Mastering these techniques will make it easier to solve a variety of absolute inequality problems.

1) Inequality|ax+b|The solution of C(C>0) in the Divine School: first turn it into an inequality group -c ax+b c, and then use the properties of inequality to find the solution set of the original inequality.

2) Unequal training delay|ax+b|Solving of c(c>0): first transform into the inequality groups ax+b -c and ax+b c, and then use the properties of the inequality to find the solution set of the original inequality.

The core task of solving inequalities with absolute values is to remove absolute values, deform the inequality identities into conventional inequalities without absolute values, and then use the mastered solution methods to solve them. Note that you can't blindly square the absolute value symbol.

4.|x-a|+|x-b|c and |x-a|+|x-b|Solution of type C inequality.

Solution 1: You can use the geometric meaning of the absolute value of Yuga. (referred to as geometric method).

Solution 2: Using the idea of classification discussion, take the "zero point" of the absolute value as the demarcation point, divide the number line into several intervals, and then determine the sign of the polynomial in each absolute value, and then remove the sign of the absolute value. (referred to as the Segmented Discussion Method).

Solution 3: You can use the constructor to use the function image to obtain the solution set of the inequality. (referred to as the image method).

From the above, it can be seen that the key to solving inequalities with absolute values is to use the meaning of absolute values to try to remove the sign of absolute values and convert them into one or several ordinary inequalities or groups of inequalities (i.e., inequalities without absolute symbols).

Special reminder for absolute inequalities|x-a|-|x-b|c and |x-a|-|x-b|c, the above three methods can also be used to solve.

Inequalities of absolute values are a common mathematical problem that can often be solved using the image method or the algebraic method. These two solutions are described below.

1.Image method

The image method is an intuitive solution to the problem of absolute values by drawing an image of the function. For example, for inequalities|2x-3|<5, we can translate it into two unequal bush acacia formulas: 2x-3<5 and 2x-3>-5, i.e.:

2x-3<5 =>2x<8 =>x<4 2x-3>-5 =>2x>-2 =>x>-1

We can then plot the interval between x<4 and x>-1 on the number line and find their intersection, which is -1<>

2.Algebraic method

Algebraic is a solution based on algebraic operations that can solve inequalities of absolute values by deforming them. For example, for inequalities|2x-3|<5, we can convert it into two inequalities: 2x-3<5 and 2x-3>-5, namely:

2x-3<5 =>2x<8 =>x<4 2x-3>-5 =>2x>-2 =>x>-1

We can then merge these two inequalities to get the <> of -1