The image presents a number line with three points: $A$, $B$, and $C$, located at coordinates $-3$, $1$, and $6$, respectively. There are four statements, and we need to identify which ones are true based on this number line.

Let’s evaluate each statement:

A. The length of $\overline{AB}$ is $-3$.

• The length of $\overline{AB}$ is calculated as the distance between $A$ and $B$. This is $|1 - (-3)| = |1 + 3| = 4$. Length cannot be negative, so this statement is false.

B. $d(B, C) = BC = |6 - 1|$

• The distance between $B$ and $C$ is $|6 - 1| = 5$. This statement is true.

C. $AB + AC = BC$

• We know that $AB = 4$ and $AC = |6 - (-3)| = |6 + 3| = 9$, but $BC = 5$.
• So, $AB + AC = 4 + 9 = 13$, which is not equal to $BC$. This statement is false.

D. $AB + BC = AC$

• Here, $AB = 4$, $BC = 5$, and $AC = 9$. Since $AB + BC = 4 + 5 = 9$, and $AC = 9$, this statement is true.

Conclusion:

• The correct answers are B and D.

Would you like further clarification or additional details?

Here are 5 related questions for further practice:

1. How do you calculate the distance between two points on a number line?
2. Why can’t the length of a line segment be negative?
3. What is the formula for the distance between two points in a plane?
4. How can we check if three points are collinear?
5. What is the significance of the absolute value in distance calculations?

Tip: Always use absolute value when calculating distances, as distance is always non-negative.