If `x + 3` is positive, then which one of the following must be positive?

Options

A. `x - 3`
B. `(x - 3)(x - 4)`
C. `(x - 3)(x+3)`
D. `(x + 3)(x + 6)`

Answer

To solve the problem, we need to determine which of the given options must be positive if \( x + 3 \) is positive.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We start with the condition given in the problem:
\[ x + 3 > 0 \]
This implies:
\[ x > -3 \]

**Hint**: Remember that when you isolate \( x \), you can subtract or add the same number to both sides of the inequality.

2. **Evaluating the Options**:
We have four options to evaluate. We will check each one to see if it must be positive given that \( x > -3 \).

**Option A: \( x - 3 \)**
We check if \( x - 3 > 0 \):
\[ x - 3 > 0 \implies x > 3 \]
Since \( x \) can be any value greater than \(-3\) (e.g., \( x = 0 \)), \( x - 3 \) can be negative. Thus, **Option A is not necessarily positive**.

**Hint**: Test values of \( x \) that satisfy \( x > -3 \) to see if the expression is positive.

**Option B: \( (x - 3)(x - 4) \)**
We check if \( (x - 3)(x - 4) > 0 \):
- If \( x = 0 \):
\[ (0 - 3)(0 - 4) = (-3)(-4) = 12 \quad (\text{positive}) \]
- If \( x = 5 \):
\[ (5 - 3)(5 - 4) = (2)(1) = 2 \quad (\text{positive}) \]
- If \( x = 3.5 \):
\[ (3.5 - 3)(3.5 - 4) = (0.5)(-0.5) = -0.25 \quad (\text{negative}) \]
Since we found a case where it is negative, **Option B is not necessarily positive**.

**Hint**: Consider values of \( x \) around the roots of the expression to determine the sign.

**Option C: \( (x - 3)(x + 3) \)**
We check if \( (x - 3)(x + 3) > 0 \):
- If \( x = 0 \):
\[ (0 - 3)(0 + 3) = (-3)(3) = -9 \quad (\text{negative}) \]
Since we found a case where it is negative, **Option C is not necessarily positive**.

**Hint**: Test extreme values of \( x \) to see if the product can yield a negative result.

**Option D: \( (x + 3)(x + 6) \)**
We check if \( (x + 3)(x + 6) > 0 \):
- Since \( x + 3 > 0 \) (given), we know \( x + 6 \) will also be positive for any \( x > -3 \) because:
\[ x + 6 > -3 + 6 = 3 > 0 \]
Therefore, both factors are positive, and their product is positive:
\[ (x + 3)(x + 6) > 0 \]
Thus, **Option D must be positive**.

3. **Conclusion**:
The correct answer is:
\[ \text{Option D: } (x + 3)(x + 6) \]