If `(|a+3|)/(2)=1 and 2|b+1|=6`, then `|a+b|` could equal any of the following EXCEPT

Options

A. `1`
B. `3`
C. `5`
D. `7`

Answer

To solve the problem step by step, we need to break it down into manageable parts.

### Step 1: Solve for \( a \)
We start with the equation:
\[ \frac{|a + 3|}{2} = 1 \]
To eliminate the fraction, we multiply both sides by 2:
\[ |a + 3| = 2 \]
This absolute value equation means that:
\[ a + 3 = 2 \quad \text{or} \quad a + 3 = -2 \]
Now, we solve each case:
1. \( a + 3 = 2 \)
\[ a = 2 - 3 = -1 \]
2. \( a + 3 = -2 \)
\[ a = -2 - 3 = -5 \]

Thus, the possible values for \( a \) are:
\[ a = -1 \quad \text{or} \quad a = -5 \]

### Step 2: Solve for \( b \)
Next, we solve the equation:
\[ 2|b + 1| = 6 \]
Again, we eliminate the coefficient by dividing both sides by 2:
\[ |b + 1| = 3 \]
This gives us two cases:
\[ b + 1 = 3 \quad \text{or} \quad b + 1 = -3 \]
Now, we solve each case:
1. \( b + 1 = 3 \)
\[ b = 3 - 1 = 2 \]
2. \( b + 1 = -3 \)
\[ b = -3 - 1 = -4 \]

Thus, the possible values for \( b \) are:
\[ b = 2 \quad \text{or} \quad b = -4 \]

### Step 3: Calculate \( |a + b| \)
Now we find the possible values of \( |a + b| \) using the combinations of \( a \) and \( b \):
1. If \( a = -1 \) and \( b = 2 \):
\[ a + b = -1 + 2 = 1 \quad \Rightarrow \quad |a + b| = 1 \]
2. If \( a = -1 \) and \( b = -4 \):
\[ a + b = -1 - 4 = -5 \quad \Rightarrow \quad |a + b| = 5 \]
3. If \( a = -5 \) and \( b = 2 \):
\[ a + b = -5 + 2 = -3 \quad \Rightarrow \quad |a + b| = 3 \]
4. If \( a = -5 \) and \( b = -4 \):
\[ a + b = -5 - 4 = -9 \quad \Rightarrow \quad |a + b| = 9 \]

### Step 4: Possible values of \( |a + b| \)
The possible values of \( |a + b| \) are:
\[ 1, 3, 5, 9 \]

### Conclusion
The question asks for the values that \( |a + b| \) could equal except for one. The possible values are \( 1, 3, 5, 9 \), and the only value that cannot be achieved is \( 7 \).