Question: Using the number line below, find the length of each segment. A B C D E -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 1. AB = ____ 2. AD = ____ 3. AE = ____ 4. CA = ____ 5. EC = ____ 6. ED = ____ Solve for the value of the variable and each segment length given. 7. In the figure below, what does the lines coming from the number 32 mean? x = ____ TV = ____ VW = ____ 32 T V W 3x x + 8

1. **State the problem:** Find the lengths of segments on a number line and solve for variable $x$ in a segment addition problem. 2. **Given points on the number line:** - $A = -9$ - $B = -6$ - $C = -4$ - $D = 0$ - $E = 2$ 3. **Formula for length of segment between two points $P$ and $Q$ on a number line:** $$\text{Length} = |Q - P|$$ 4. **Calculate each segment length:** 1. $AB = |-6 - (-9)| = |-6 + 9| = |3| = 3$ 2. $AD = |0 - (-9)| = |0 + 9| = 9$ 3. $AE = |2 - (-9)| = |2 + 9| = 11$ 4. $CA = |-4 - (-9)| = |-4 + 9| = 5$ 5. $EC = |2 - (-4)| = |2 + 4| = 6$ 6. $ED = |2 - 0| = 2$ --- 7. **Solve for $x$ in segment addition:** Given segment $TW = 32$, with $TV = 3x$ and $VW = x + 8$. By segment addition postulate: $$TW = TV + VW$$ $$32 = 3x + (x + 8)$$ Simplify: $$32 = 3x + x + 8$$ $$32 = 4x + 8$$ Subtract 8 from both sides: $$32 - 8 = 4x + \cancel{8} - \cancel{8}$$ $$24 = 4x$$ Divide both sides by 4: $$\frac{24}{\cancel{4}} = \frac{4x}{\cancel{4}}$$ $$6 = x$$ Calculate segment lengths: $$TV = 3x = 3 \times 6 = 18$$ $$VW = x + 8 = 6 + 8 = 14$$ **Final answers:** 1. $AB = 3$ 2. $AD = 9$ 3. $AE = 11$ 4. $CA = 5$ 5. $EC = 6$ 6. $ED = 2$ 7. $x = 6$, $TV = 18$, $VW = 14$