Solve the mathematical problem involving absolute value.
3 - I 2/3 x -6 I + 2
To solve the mathematical problem involving absolute value:
Step 1: Start by solving the expression inside the absolute value bars.
In this case, the expression is: 2/3x - 6
Step 2: Set up two separate equations, one with the expression inside the absolute value bars as positive, and one with it as negative.
Equation 1: 2/3x - 6 = 3
Equation 2: 2/3x - 6 = -3
Step 3: Solve both equations separately.
For Equation 1:
Add 6 to both sides:
2/3x = 9
Multiply both sides by 3/2 to isolate x:
x = 9 * 3/2
x = 27/2
For Equation 2:
Add 6 to both sides:
2/3x = 3
Multiply both sides by 3/2 to isolate x:
x = 3 * 3/2
x = 9/2
Step 4: Check the solutions in the original absolute value expression.
For x = 27/2:
3 - |2/3(27/2) - 6| + 2
Simplify the expression inside the absolute value bars:
3 - |9 - 6| + 2
Simplify the absolute value:
3 - |3| + 2
Evaluate the absolute value:
3 - 3 + 2 = 2
For x = 9/2:
3 - |2/3(9/2) - 6| + 2
Simplify the expression inside the absolute value bars:
3 - |3 - 6| + 2
Simplify the absolute value:
3 - |-3| + 2
Evaluate the absolute value:
3 - 3 + 2 = 2
Step 5: Both solutions x = 27/2 and x = 9/2 satisfy the original absolute value expression.
To solve the mathematical problem involving absolute value, we need to consider two cases:
Case 1: When the expression inside the absolute value is greater than or equal to zero.
In this case, the absolute value is unnecessary and we can simplify the expression as follows:
3 - (2/3)x - 6 + 2
Combining like terms, we get:
-1 - (2/3)x
Case 2: When the expression inside the absolute value is less than zero.
In this case, we have to negate the expression inside the absolute value and then simplify.
-(3 - (2/3)x - 6 + 2)
Again, combining like terms, we get:
-5 + (2/3)x
Therefore, the solution to the problem involving absolute value is:
-1 - (2/3)x and -5 + (2/3)x
To solve the mathematical problem involving absolute value, let's break it down into steps.
Step 1: Simplify the expression inside the absolute value.
Start by multiplying the fractions and simplifying it:
2/3 x -6 = -12/3 = -4
Step 2: Rewrite the absolute value expression as two separate cases.
The expression |x| can be written as x or -x, depending on the value of x. In this case, we have:
|2 - 4| + 2 (Using x = 2 - 4)
And
|2 + 4| + 2 (Using x = 2 + 4)
Step 3: Simplify the expressions inside the absolute value.
For the first case:
|2 - 4| + 2
|-2| + 2
2 + 2
4
For the second case:
|2 + 4| + 2
|6| + 2
6 + 2
8
Therefore, the solution to the given expression is 4 or 8, depending on the case.