Consider the equation `p = 5-2q - 3`
Greatest sat of all possible values of `p` for `q in R` is

Options

A. `(-oo,5]`
B. `(-oo,5]`
C. `(-5,5)`
D. None of these

Answer

To solve the equation \( p = 5 - 2q - 3 \) and find the greatest set of all possible values of \( p \) for \( q \in \mathbb{R} \), we can follow these steps:

### Step 1: Simplify the equation
Start by simplifying the equation given:
\[ p = 5 - 2q - 3 \]
Combine like terms:
\[ p = 2 - 2q \]

### Step 2: Analyze the equation
The equation \( p = 2 - 2q \) is a linear equation in terms of \( q \). As \( q \) varies over all real numbers, we need to determine how \( p \) behaves.

### Step 3: Determine the range of \( p \)
Since \( p \) is expressed as \( p = 2 - 2q \), we can see that:
- When \( q \) increases, \( p \) decreases.
- When \( q \) decreases, \( p \) increases.

### Step 4: Find the limits of \( p \)
As \( q \) approaches \( \infty \) (positive infinity), \( p \) approaches \( -\infty \):
\[ \lim_{q \to \infty} p = -\infty \]
As \( q \) approaches \( -\infty \) (negative infinity), \( p \) approaches \( \infty \):
\[ \lim_{q \to -\infty} p = \infty \]

### Step 5: Conclusion
Thus, the greatest set of all possible values of \( p \) is:
\[ p \in (-\infty, \infty) \]
This means \( p \) can take any real number value.

### Final Answer
The greatest set of all possible values of \( p \) for \( q \in \mathbb{R} \) is \( \mathbb{R} \).

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