We are given the following conditions:

• $P$, $Q$, and $R$ are points on a line with $Q$ between $P$ and $R$.
• $PQ = 2w - 3$
• $QR = 4 + w$
• $PR = 34$

Since $Q$ is between $P$ and $R$, we know that:

$PR = PQ + QR$

Substitute the given expressions for $PQ$ and $QR$:

$34 = (2w - 3) + (4 + w)$

Now, solve for $w$:

$34 = 2w - 3 + 4 + w$ $34 = 3w + 1$ $34 - 1 = 3w$ $33 = 3w$ $w = 11$

Finding the lengths of $PQ$ and $QR$:

1. $PQ = 2w - 3 = 2(11) - 3 = 22 - 3 = 19$
2. $QR = 4 + w = 4 + 11 = 15$

Thus, the value of $w$ is $11$, and the lengths are:

• $PQ = 19$
• $QR = 15$

Would you like further details or clarification?

Here are 5 related questions:

1. How do you solve linear equations with one variable?
2. How does the concept of points on a line help in geometry?
3. What are some real-life applications of solving for unknowns in linear expressions?
4. How would this problem change if $QR$ was given as $4 - w$ instead of $4 + w$?
5. What happens if $Q$ is not between $P$ and $R$?

Tip: Always verify your solution by substituting the found values back into the original equation to check for consistency.