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The problem requires us to calculate the distances between points \( A, B, C, \) and \( D \) on a number line. The points have the following coordinates:

- \( A = -14 \)
- \( B = -\frac{1}{2} \)
- \( C = 7 \)
- \( D = \frac{1}{3} \)

We need to find the distances between:

1. \( AB \)
2. \( BD \)
3. \( BC \)
4. \( CD \)

### Solution:

The distance between two points on a number line is given by the absolute value of the difference between their coordinates:

1. **Distance \( AB \):**
   \[
   AB = |B - A| = \left| -\frac{1}{2} - (-14) \right| = \left| -\frac{1}{2} + 14 \right| = \left| \frac{27}{2} \right| = \frac{27}{2}
   \]

2. **Distance \( BD \):**
   \[
   BD = |D - B| = \left| \frac{1}{3} - \left(-\frac{1}{2}\right) \right| = \left| \frac{1}{3} + \frac{1}{2} \right| = \left| \frac{5}{6} \right| = \frac{5}{6}
   \]

3. **Distance \( BC \):**
   \[
   BC = |C - B| = \left| 7 - \left(-\frac{1}{2}\right) \right| = \left| 7 + \frac{1}{2} \right| = \left| \frac{15}{2} \right| = \frac{15}{2}
   \]

4. **Distance \( CD \):**
   \[
   CD = |D - C| = \left| \frac{1}{3} - 7 \right| = \left| \frac{1}{3} - \frac{21}{3} \right| = \left| -\frac{20}{3} \right| = \frac{20}{3}
   \]

### Summary of Distances:

- \( AB = \frac{27}{2} \)
- \( BD = \frac{5}{6} \)
- \( BC = \frac{15}{2} \)
- \( CD = \frac{20}{3} \)

Would you like further explanations or have any questions?

### Relative Questions:

1. How would you calculate the distance between points on a coordinate plane?
2. What is the importance of the absolute value in calculating distances?
3. How do different number representations (fractions, decimals) affect calculations?
4. Can the distance formula be extended to three-dimensional space?
5. What are some real-life applications of calculating distances between points?

**Tip:** Always convert to a common denominator when working with fractions to simplify arithmetic operations.

Sur une droite graduée, A, B, C, D sont les points d'abscisses respectives -14, -1/2, 7 et 1/3. Calculer les distances AB, BD, BC, CD.

Learn how to calculate distances between points A, B, C, and D on a number line. Points are given as -14, -1/2, 7, and 1/3. The problem covers key concepts such as absolute value, distance formula, and handling fractions. Suitable for students in Grades 8-10.

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