Let's analyze Avni's game where a player either gains or loses 4 points during each turn.
1. Possible Points After the First Turn: After completing the first turn, there are two possible outcomes:
- If the player wins the turn, they gain 4 points.
- If the player loses the turn, they lose 4 points.
2. Representing the Possible Outcomes:
- Win: The player would have [tex]\( p = 4 \)[/tex] points.
- Lose: The player would have [tex]\( p = -4 \)[/tex] points.
3. Absolute Value to Describe Both Outcomes: To cover both scenarios (whether winning or losing), we need an equation that reflects both [tex]\( +4 \)[/tex] and [tex]\( -4 \)[/tex]. The absolute value function [tex]\( |p| \)[/tex] serves this purpose. The expression [tex]\( |p| = x \)[/tex] means [tex]\( p \)[/tex] can be either [tex]\( x \)[/tex] or [tex]\( -x \)[/tex].
4. Matching Our Requirements:
- [tex]\( |p| = 4 \)[/tex] covers both possible states of [tex]\( p \)[/tex]:
- [tex]\( p = 4 \)[/tex]
- [tex]\( p = -4 \)[/tex]
Let’s evaluate the given options to find which matches the above conclusions:
- Option 1: [tex]\( p = 4 \)[/tex] (Only accounts for when the player wins)
- Option 2: [tex]\( p = -4 \)[/tex] (Only accounts for when the player loses)
- Option 3: [tex]\( |p| = 4 \)[/tex] (Accounts for both outcomes)
- Option 4: [tex]\( |p| = -4 \)[/tex] (Absolute value cannot be negative, so this is incorrect)
Therefore, the equation that represents all possible numbers of points [tex]\( p \)[/tex] a player may have after the first turn of the game is:
[tex]\[ |p| = 4 \][/tex]
