For all nonzero values of a and b, the value of which of the following expressions is always negative?

Options

A. a-b
B. `-a-b`
C. `|a| + |b|`
D. `-|a| - |b|`

Answer

To determine which of the given expressions is always negative for all nonzero values of \( a \) and \( b \), we will analyze each option step by step.

### Step 1: Analyze the modulus function
The modulus function \( |x| \) is defined as:
- \( |x| = x \) if \( x \geq 0 \)
- \( |x| = -x \) if \( x < 0 \)

Since \( a \) and \( b \) are nonzero, we know:
- \( |a| > 0 \)
- \( |b| > 0 \)

### Step 2: Evaluate each option
Let's evaluate each option provided in the question.

#### Option A: \( a - b \)
- This expression can be positive, negative, or zero depending on the values of \( a \) and \( b \). For example, if \( a = 3 \) and \( b = 2 \), then \( a - b = 1 \) (positive). If \( a = 2 \) and \( b = 3 \), then \( a - b = -1 \) (negative). Thus, this option is not always negative.

**Hint:** Check if the expression can yield both positive and negative results.

#### Option B: \( -a - b \)
- This expression can also be positive or negative depending on the values of \( a \) and \( b \). For instance, if \( a = 1 \) and \( b = 2 \), then \( -a - b = -1 - 2 = -3 \) (negative). However, if \( a = -1 \) and \( b = -2 \), then \( -a - b = 1 + 2 = 3 \) (positive). Hence, this option is not always negative.

**Hint:** Consider specific values of \( a \) and \( b \) to see if the expression can be positive.

#### Option C: \( |a| + |b| \)
- Since both \( |a| \) and \( |b| \) are always positive for nonzero \( a \) and \( b \), their sum \( |a| + |b| \) is also always positive. Thus, this option cannot be negative.

**Hint:** Remember that the sum of two positive numbers is always positive.

#### Option D: \( -|a| - |b| \)
- Since both \( |a| \) and \( |b| \) are positive, \( -|a| \) and \( -|b| \) will both be negative. Therefore, their sum \( -|a| - |b| \) will also be negative. This expression is always negative for any nonzero values of \( a \) and \( b \).

**Hint:** Consider the effect of negating positive quantities.

### Conclusion
The expression that is always negative for all nonzero values of \( a \) and \( b \) is:

**Final Answer: Option D: \( -|a| - |b| \)**