Final answer:
The equation |x+1/2|=3 2/3 has two solutions: x = 19/6, obtained by considering the positive case of the absolute value, and x = -25/6, obtained from the negative case. The two cases reflect the property of absolute values that they can be either positive or negative.
Step-by-step explanation:
To solve the equation |x+1/2|=3 2/3, we need to consider two cases because the absolute value of a number can be positive or negative. First case is when x+1/2 is positive, and the second case is when x+1/2 is negative.
For the first case:
- x + 1/2 = 3 2/3
- x + 1/2 = 11/3
- x = 11/3 - 1/2
- x = (22 - 3)/6
- x = 19/6
For the second case:
- x + 1/2 = -3 2/3
- x + 1/2 = -11/3
- x = -11/3 - 1/2
- x = (-22 - 3)/6
- x = -25/6
So, the solutions to the equation |x+1/2|=3 2/3 are x = 19/6 and x = -25/6.