To determine which equation represents all numbers of points [tex]\( p \)[/tex] a player may have after their first turn of the game, let's break down the options provided. In each turn, a player either wins 4 points or loses 4 points.
### Step-by-Step Analysis:
1. Possible Outcomes:
- If the player wins the turn, they gain 4 points. Thus, [tex]\( p = 4 \)[/tex].
- If the player loses the turn, they lose 4 points. Thus, [tex]\( p = -4 \)[/tex].
2. Absolute Value Analysis:
- Regardless of whether the player wins or loses, the absolute value of the points gained or lost is always 4. Therefore, [tex]\( |p| = 4 \)[/tex].
3. Checking Each Given Option:
- [tex]\( p = 4 \)[/tex]: This equation represents the scenario where the player wins the turn. However, it does not cover the scenario where the player loses the turn.
- [tex]\( p = -4 \)[/tex]: This equation represents the scenario where the player loses the turn. However, it does not cover the scenario where the player wins the turn.
- [tex]\( |p| = 4 \)[/tex]: This equation correctly represents both possible outcomes (winning 4 points or losing 4 points) because it considers the absolute value of [tex]\( p \)[/tex], which will always be 4 regardless of whether [tex]\( p \)[/tex] is 4 or -4.
- [tex]\( |p| = -4 \)[/tex]: This equation is invalid because an absolute value cannot be negative.
### Conclusion:
The correct equation that represents all numbers of points [tex]\( p \)[/tex] a player may have after their first turn of the game is [tex]\( |p| = 4 \)[/tex].
Thus, the proper representation is:
[tex]\[ |p| = 4 \][/tex]
