求解绝对值方程

绝对值的概念属于父主题 - 代数。它是数学的广泛领域之一。它涉及对数学符号和操作符号规则的研究。一般来说，绝对值的概念是用在距离上来表示任意两点之间的距离。

示例： |-7| = 7, |7| = 7, |0| = 0

绝对值的结果总是正的。不会是负面的。

求解绝对值方程

绝对值方程是变量在绝对值运算符之间的方程。示例： |x-2| = 0。解绝对值方程可以通过举个例子更好地解释，并解决这些问题以获得更好的理解。

问题 1：求解方程 |x + 4| = 1 这样 x 的可能值是多少。

解决方案：

Step: 1 First we need to write two equations from the given equation such a way that

x + 4 = 1            

x + 4 = -1

Step: 2 Solve the above formed equations to get possible values of x

x + 4 = 1 

x = 1 – 4  

x = -3           

x + 4 = -1

x = -1 – 4

x = -5

Step: 3 Verify the above results by substituting the x values in the given equation

|x + 4| = 1 

Substitute x = -3 in the above equation

|-3 + 4| = 1 

|4 – 3| = 1  

1 = 1

Hence x = -3 satisfies the given equation.

|x + 4| = 1

Substitute x = -5 in the above equation

|-5 + 4| = 1  

|4 – 5| = 1 

|-1| = 1 

1 = 1

Hence x = -5 satisfies the given equation.

So, the possible values of x are -3, -5.

求解绝对值方程的步骤

从上面的例子中，我们可以推导出求解绝对值方程的步骤，

1. 根据上面的例子，从给定的方程写出两个方程。
2. 求解步骤 1 形成的方程以获得变量的可能值。
3. 验证我们在给定方程中从第 2 步得到的结果。

让我们看一下要解决的不同类型的场景/方程式/问题。

示例问题

问题 1：求解 |2x – 5| = 9

解决方案：

Step-1 Make 2 equations from given equation

2x – 5 = 9            

2x – 5 = -9

Step-2 Solve the above equations

2x = 9 + 5          

2x = 14           

x = 14/2         

x = 7             

2x = -9 + 5

2x = -4

x = -2

Step-3 Verify the above results by substituting in the given equation

Given equation |2x – 5| = 9

Substitute x = 7 in given equation    

|2(7) – 5| = 9                                       

|14 – 5| = 9                                         

9 = 9                                               

Substitute x = -2 in given equation

|2(-2) – 5| = 9

|-4 – 5| = 9 

9 = 9

Hence x = 7, x = -2 are the possible values.

问题 2：求解表达式 2 |x + 1| – 2 = 4

解决方案：

In order to proceed to step-1 we can simplify the equation such that left side only has absolute value function and right side of equal to is a value.

2 |x + 1| -2 = 4 

2 |x + 1| = 4 + 2

2 |x + 1| = 6

|x + 1| = 6/2

|x + 1| = 3

This equation is used to write two equations,

Step-1:

x + 1 = 3      

x + 1 = -3

Step-2: Solve the above equations

x = 3 – 1

x = 2       

x = -3 – 1

x = -4

Step-3: Verification

2 |x + 1| – 2 = 4

Substitute x = 2 in the given equation      

2 |2 + 1| – 2 = 4                                         

2 |3| – 2 = 4                                             

6 – 2 = 4                                                    

4 = 4                                                      

Substitute x = -4 in the given equation

2 |-4 + 1| – 2 = 4

2 |-3| – 2 = 4

2 (3) -2 = 4

6 – 2 = 4  => 4 = 4

So the possible values of x after solving the equation 2 |x + 1| – 2 = 4 are 2, -4.

问题 3：求解 |2x – 5| = -1。

解决方案：

Note: The absolute value function can never be equal to negative value.

For the the given equation where the result is negative which indicates that there is no solution for this equation.

问题 4：求解 |2x – 5| = x + 4

解决方案：

Step-1 Write 2 equations from the above equation

2x – 5 = x + 4              

2x – 5 = -(x + 4)

Step-2 Solve these equations

2x – x = 5 + 4              

x = 9                     

2x – 5 = -x – 4

2x + x = 5 – 4

3x = 1

x = 1/3

Step-3 Verification

Given equation, |2x – 5| = x + 4

Substitute x = 9 in the given equation      

|2(9) – 5| = 9 + 4                                        

|18 – 5| = 13                                             

|13| =13                                                  

13 = 13                                                    

Substitute x=(1/3) in the given equation

 |2(1/3) – 5| = (1/3) + 4

 |(2/3) – 5| = (1/3) + 4

 |(2 – 15)/3| = (1 + 12)/3

 |-13/3| = 13/3 =>13/3 =13/3

So the possible values of x after solving the equation |2x – 5| = x + 4 are 9, 1/3.

问题 5：求解 |7 + 7x| = -8

解决方案：

There will be no way to solve the above equation because the result of absolute value equation will never be negative. In this equation the resultant value of absolute value equation is negative. So no solution for this equation.

问题 6：求解 |4t – 2| = |8t + 18|

解决方案：

Step-1 Write 2 equations from the above equation

4t – 2 = -(8t + 18)              

4t – 2 = 8t + 18

Step-2 Solve these equations

4t – 2 = -8t – 18                  

4t + 8t = -18 + 2                

12t = -16                          

t = -16/12                        

t = -4/3

4t – 8t = 18 + 2

-4t = 20

t = -20/4

t = -5 

Step-3 Verification

Substitute t = -4/3 in the given equation   

|4(-4/3) – 2| = |8(-4/3) + 18|                         

|(-16/3) – 2| = |(-32/3) +18|                         

|(-16 – 6)/3| = |(-32 + 54)/3|                         

|-22/3| = |22/3|                                          

22/3 = 22/3

Substitute t = -5 in the given equation

|4(-5) – 2| = |8(-5) + 18|

|-20 – 2| = |-40 + 18|

|-22| = |-22|

22=22

So the possible values of t for the given absolute value equation are -4/3 & -5.