To determine when particles $P$ and $Q$ are moving towards each other, we need to compare the velocities of both particles.

1. Particle P's Velocity:

   • From the graph $x_P(t)$:
     • The slope from $t = 0$ to $t = 4$ is negative.
     • The slope from $t = 4$ to $t = 7$ is positive.
     • The slope from $t = 7$ to $t = 10$ is negative.

2. Particle Q's Velocity $v_Q(t)$:

   • At $t = 1$, $v_Q = 3$ (positive).
   • At $t = 3$, $v_Q = 0$.
   • At $t = 7$, $v_Q = -2$ (negative).
   • At $t = 9$, $v_Q = 3$ (positive).

Analysis:

• For $t = 1$:

  • $P$ has negative velocity (moving left).
  • $Q$ has positive velocity (moving right).
  • They are moving towards each other.

• For $t = 3$:

  • $P$ still has negative velocity.
  • $Q$ is stationary (velocity $v_Q = 0$).
  • They are not moving towards each other.

• For $t = 7$:

  • $P$ has positive velocity (moving right).
  • $Q$ has negative velocity (moving left).
  • They are moving towards each other.

• For $t = 9$:

  • $P$ has a negative velocity.
  • $Q$ has positive velocity.
  • They are moving towards each other.

Particles $P$ and $Q$ are moving towards each other at $t = 1$, $t = 7$, and $t = 9$.