Solve the mathematical problem involving absolute value. 3 - I 2/3 x -6 I + 2 * 2 points 9 -3 1 -21
To solve the problem, we need to evaluate the absolute value expression and then simplify the entire equation.
First, let's evaluate the absolute value expression:
|2/3x - 6|
Now, let's solve for the two cases when the expression inside the absolute value is positive and negative:
Case 1: 2/3x - 6 is positive
2/3x - 6 ≥ 0
2/3x ≥ 6
x ≥ 6 * 3/2
x ≥ 9
Case 2: 2/3x - 6 is negative
-(2/3x - 6) ≥ 0
2/3x - 6 ≤ 0
2/3x ≤ 6
x ≤ 6 * 3/2
x ≤ 9
Now, let's solve the equation based on these cases:
Case 1: x ≥ 9
3 - |2/3x - 6| + 2
3 - (2/3x - 6) + 2
3 - 2/3x + 6 + 2
9 - 2/3x
Case 2: x ≤ 9
3 - |2/3x - 6| + 2
3 - (-(2/3x - 6)) + 2
3 + 2/3x - 6 + 2
-1 - 2/3x
Therefore, the solution to the mathematical problem involving absolute value is:
When x ≥ 9, the answer is 9 - 2/3x.
When x ≤ 9, the answer is -1 - 2/3x.
What’s the answer
Apologies for the confusion in my previous response. Let's recheck the calculations.
The given equation is: 3 - |2/3x - 6| + 2
To solve this, we need to consider two cases based on the sign of the expression inside the absolute value.
Case 1: 2/3x - 6 is positive or zero
In this case, the absolute value can be removed, and the equation becomes:
3 - (2/3x - 6) + 2
= 3 - 2/3x + 6 + 2
= 11 - 2/3x
Case 2: 2/3x - 6 is negative
In this case, we need to change the sign inside the absolute value and solve, so the equation becomes:
3 - (-(2/3x - 6)) + 2
= 3 + 2/3x - 6 + 2
= -1 + 2/3x
Therefore, the solution to the given equation involving absolute value is:
When 2/3x - 6 is positive or zero, the answer is 11 - 2/3x.
When 2/3x - 6 is negative, the answer is -1 + 2/3x.
These are the answers
9
-3
1
-21
Apologies for the incorrect response again. Let's go through the equation more carefully to provide the correct answers.
The given equation is: 3 - |2/3x - 6| + 2.
To solve this, let's first simplify the absolute value expression.
|2/3x - 6|
To simplify the expression inside the absolute value, we need to consider two cases based on its sign.
Case 1: 2/3x - 6 is positive or zero
In this case, we can remove the absolute value without changing the sign, so the expression becomes:
2/3x - 6.
Case 2: 2/3x - 6 is negative
In this case, we need to change the sign inside the absolute value, so the expression becomes:
-(2/3x - 6) = -2/3x + 6.
Now, let's substitute the simplified expressions back into the original equation and solve for x.
Case 1: 2/3x - 6 is positive or zero
3 - (2/3x - 6) + 2
= 3 - 2/3x + 6 + 2
= 11 - 2/3x = 0
To solve for x, let's isolate the variable:
-2/3x = -11
x = -11 / (-2/3)
x = -11 * (3/2)
x = 33/2
x = 16.5
Case 2: 2/3x - 6 is negative
3 - (-(2/3x - 6)) + 2
= 3 + 2/3x - 6 + 2
= -1 + 2/3x = 0
To solve for x, let's isolate the variable:
2/3x = 1
x = 1 / (2/3)
x = 1 * (3/2)
x = 3/2
x = 1.5
Therefore, the solutions to the given equation involving absolute value are x = 16.5 and x = 1.5.