Question: 1. Point Q is on line segment PR. Given
PQ = 3x - 9, PR = 2x, and QR = 5, determine
the numerical length of PQ.
1. **State the problem:** We need to find the length of segment $PQ$ given that point $Q$ lies on segment $PR$ with $PQ = 3x - 9$, $PR = 2x$, and $QR = 5$.
2. **Recall the segment addition postulate:** For points $P$, $Q$, and $R$ on a line, $PR = PQ + QR$.
3. **Set up the equation:**
$$2x = (3x - 9) + 5$$
4. **Simplify the right side:**
$$2x = 3x - 9 + 5$$
$$2x = 3x - 4$$
5. **Isolate $x$:**
$$2x - 3x = -4$$
$$\cancel{2x} - 3x = -4$$
$$-x = -4$$
6. **Solve for $x$:**
$$x = 4$$
7. **Find $PQ$ by substituting $x=4$ into $PQ = 3x - 9$:**
$$PQ = 3(4) - 9 = 12 - 9 = 3$$
**Final answer:** The length of $PQ$ is $3$.