I would really appreciate it, if someone helped me how to do this.
│x-4│ - │7-x│ = 1
| x - 4 |-| 7 - x | = 1
| x - 4 | = 1 + | 7 - x |
Eliminate the absolute values:
First equation :
( - ( x - 4 ) ) = ( 7 - x ) + 1
Second equation :
( x - 4 ) = ( 7 - x ) + 1
Third equation :
( x - 4 ) = ( - ( 7 - x ) ) + 1
Fourth equation :
( x - 4 ) = ( 7 - x ) + 1
You have equation ( x - 4 ) = ( 7 - x ) + 1 twice.
So you must solve 3 equations :
( - ( x - 4 ) ) = ( 7 - x ) + 1
( x - 4 ) = ( 7 - x ) + 1
( x - 4 ) = ( - ( 7 - x ) ) + 1
( - ( x - 4 ) ) = ( 7 - x ) + 1
- x + 4 = 7 - x + 1
- x + 4 = 8 - x
- x + x = 8 - 4
0 = 4
NO SOLUTION
( x - 4 ) = ( 7 - x ) + 1
x - 4 = 7 - x + 1
x - 4 = 8 - x
x + x = 8 + 4
2 x = 12 Divide both sides by 2
x = 12 / 2 = 6
( x - 4 ) = ( - ( 7 - x ) ) + 1
x - 4 = - 7 + x + 1
x - 4 = - 6 + x
x - x = - 6 + 4
0 = - 2
NO SOLUTION
Final solution :
x = 6
Thank you A LOT!
To solve the equation │x-4│ - │7-x│ = 1, we need to consider different cases based on the values of x.
To start, let's remove the absolute value signs by considering the positive and negative expressions inside them separately:
Case 1: (x - 4) is positive and (7 - x) is positive:
In this case, │x-4│ is equal to (x - 4), and │7-x│ is equal to (7 - x). Thus, the equation becomes:
(x - 4) - (7 - x) = 1
Now, simplify and solve for x:
x - 4 - 7 + x = 1
2x - 11 = 1
2x = 12
x = 6
So, in this case, x = 6 is a solution.
Case 2: (x - 4) is negative and (7 - x) is negative:
In this case, │x-4│ is equal to -(x - 4), and │7-x│ is equal to -(7 - x). Thus, the equation becomes:
-(x - 4) - -(7 - x) = 1
Simplify and solve for x:
-(x - 4) + (7 - x) = 1
- x + 4 + 7 - x = 1
-2x + 11 = 1
-2x = -10
x = 5
So, in this case, x = 5 is a solution.
Case 3: (x - 4) is positive and (7 - x) is negative:
In this case, │x-4│ is equal to (x - 4), and │7-x│ is equal to -(7 - x). Thus, the equation becomes:
(x - 4) - -(7 - x) = 1
Simplify and solve for x:
x - 4 + 7 - x = 1
3 = 1
This equation is not true, so there are no solutions in this case.
Case 4: (x - 4) is negative and (7 - x) is positive:
In this case, │x-4│ is equal to -(x - 4), and │7-x│ is equal to (7 - x). Thus, the equation becomes:
-(x - 4) - (7 - x) = 1
Simplify and solve for x:
-(x - 4) - 7 + x = 1
-2x - 11 = 1
-2x = 12
x = -6
So, in this case, x = -6 is a solution.
Overall, the solutions to the equation │x-4│ - │7-x│ = 1 are x = 6 and x = -6.