The task is to determine the signs of the expressions $b - a$, $a - c$, and $b - c$ by comparing them to zero, based on the positions of points $a$, $b$, and $c$ on the number line.

Step-by-Step Analysis

1. Identify Order on the Number Line:

   • From the image, the positions on the number line are:
     • $c$ is the smallest,
     • $b$ is in the middle,
     • $a$ is the largest.

   This means: $c < b < a$

2. Evaluate Each Expression:

   • $b - a$: Since $b$ is less than $a$, $b - a$ will be negative. $b - a < 0$

   • $a - c$: Since $a$ is greater than $c$, $a - c$ will be positive. $a - c > 0$

   • $b - c$: Since $b$ is greater than $c$, $b - c$ will also be positive. $b - c > 0$

Answer Summary

So the correct signs for the expressions are:

• $b - a < 0$
• $a - c > 0$
• $b - c > 0$

Would you like more detailed explanations on any part of this solution, or have any other questions?

Related Questions:

1. How can we determine the sign of an expression based on the number line?
2. What does it mean for an expression to be positive or negative in terms of ordering?
3. How can we generalize the process of comparing points on a number line?
4. Why does $b - a$ result in a negative number here?
5. What is the relationship between subtraction and ordering on a number line?

Tip: Remember that on a number line, moving to the right means increasing values, and to the left means decreasing values.