If `|-3a+15|=6` and `|-2b+12|=4`, what is the greatest possible value of ab ?

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) from the given equations involving absolute values, and then determine the greatest possible value of the product \( ab \).

### Step 1: Solve the first equation \( |-3a + 15| = 6 \)

The absolute value equation can be split into two cases:

1. Case 1:
\[ -3a + 15 = 6 \]
Rearranging gives:
\[ -3a = 6 - 15 \implies -3a = -9 \implies a = \frac{-9}{-3} = 3 \]

2. Case 2:
\[ -3a + 15 = -6 \]
Rearranging gives:
\[ -3a = -6 - 15 \implies -3a = -21 \implies a = \frac{-21}{-3} = 7 \]

Thus, the possible values for \( a \) are \( 3 \) and \( 7 \).

### Step 2: Solve the second equation \( |-2b + 12| = 4 \)

Similarly, we split this absolute value equation into two cases:

1. Case 1:
\[ -2b + 12 = 4 \]
Rearranging gives:
\[ -2b = 4 - 12 \implies -2b = -8 \implies b = \frac{-8}{-2} = 4 \]

2. Case 2:
\[ -2b + 12 = -4 \]
Rearranging gives:
\[ -2b = -4 - 12 \implies -2b = -16 \implies b = \frac{-16}{-2} = 8 \]

Thus, the possible values for \( b \) are \( 4 \) and \( 8 \).

### Step 3: Calculate the products \( ab \)

Now we have the possible pairs of \( (a, b) \):
- \( (3, 4) \)
- \( (3, 8) \)
- \( (7, 4) \)
- \( (7, 8) \)

Calculating the products:
1. \( ab = 3 \times 4 = 12 \)
2. \( ab = 3 \times 8 = 24 \)
3. \( ab = 7 \times 4 = 28 \)
4. \( ab = 7 \times 8 = 56 \)

### Step 4: Determine the greatest possible value of \( ab \)

From the calculated products, the greatest possible value of \( ab \) is:
\[ \text{Greatest value of } ab = 56 \]

### Final Answer:
The greatest possible value of \( ab \) is \( \boxed{56} \).

---