11. Solve the mathematical problem involving absolute value.
3 - I 2/3 x -6 I + 2
*
2 points
9
-3
1
-21
To solve this mathematical problem involving absolute value, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: When the expression inside the absolute value is positive
3 - 2/3x - 6 + 2 = 5 - 2/3x
Case 2: When the expression inside the absolute value is negative
-(3 - 2/3x - 6) + 2 = 1 + 2/3x
Now let's simplify both cases further:
Case 1: 5 - 2/3x
Case 2: 1 + 2/3x
Since absolute value is defined as the magnitude of a number without regard to its sign, we have two equations:
1. |3 - 2/3x - 6| + 2 = 5 - 2/3x
2. |3 - 2/3x - 6| + 2 = 1 + 2/3x
To solve equation 1, we can eliminate the absolute value by considering two cases within it:
Case 1a: 3 - 2/3x - 6 + 2 = 5 - 2/3x
Simplifying this equation gives us:
-1 = 0
Case 1b: -(3 - 2/3x - 6) + 2 = 5 - 2/3x
Simplifying this equation gives us:
-1 = 0
Since both cases result in an inconsistent equation, there is no solution for equation 1.
To solve equation 2, we can also eliminate the absolute value by considering two cases within it:
Case 2a: 3 - 2/3x - 6 + 2 = 1 + 2/3x
Simplifying this equation gives us:
-1 = 1/3x
Case 2b: -(3 - 2/3x - 6) + 2 = 1 + 2/3x
Simplifying this equation gives us:
-1 = 5/3x
To solve for x in case 2, we can cross multiply:
Case 2a: -1 * 3 = 1/3x
-3 = 1/3x
x = -9
Case 2b: -1 * 3 = 5/3x
-3 = 5/3x
x = -9/5
Therefore, the solutions for equation 2 are x = -9 and x = -9/5.
In summary, there is no solution for equation 1, and the solutions for equation 2 are x = -9 and x = -9/5.