If `|a-b| gt ||a|-|b||` then

Options

A. `a.b gt 0`
B. `a.b ge 0`
C. `a.blt 0`
D. `a.b le 0`

Answer

To solve the inequality \( |a - b| > ||a| - |b|| \), we will follow these steps:

### Step 1: Square both sides
We start by squaring both sides of the inequality to eliminate the absolute values. This gives us:
\[ |a - b|^2 > (||a| - |b||)^2 \]

### Step 2: Expand both sides
Using the formula for squaring a difference, we can expand both sides:
\[ (a - b)^2 > (|a| - |b|)^2 \]
This expands to:
\[ a^2 - 2ab + b^2 > |a|^2 - 2|a||b| + |b|^2 \]

### Step 3: Substitute \( |a|^2 \) and \( |b|^2 \)
Since \( |a|^2 = a^2 \) and \( |b|^2 = b^2 \), we can rewrite the inequality as:
\[ a^2 - 2ab + b^2 > a^2 - 2|a||b| + b^2 \]

### Step 4: Simplify the inequality
Now, we can simplify the inequality by canceling \( a^2 \) and \( b^2 \) from both sides:
\[ -2ab > -2|a||b| \]

### Step 5: Divide by -2
Dividing both sides by -2 (remember that this reverses the inequality):
\[ ab < |a||b| \]

### Step 6: Analyze the inequality
The inequality \( ab < |a||b| \) implies that the product \( ab \) is less than the product of their magnitudes. This occurs when \( a \) and \( b \) have opposite signs. Therefore, we conclude that:
\[ ab < 0 \]

### Final Conclusion
Thus, the condition \( |a - b| > ||a| - |b|| \) holds true when \( a \) and \( b \) have opposite signs.

---