Points P and Q are both in the line segment AB and on the same side of its midpoint. P divides AB in the ratio 2:3, and Q divides AB in the ratio 3:4. If PQ=2, then find the length of the line segment AB.

If the length of AB is x, then

AP = 2x/5
AQ = 3x/7

PQ = AQ-AP = 3x/7 - 2x/5 = 2
Now just solve for x

To find the length of line segment AB, we can create equations using the given information:

Let's assume that the length of line segment AP is 2x (2 times the ratio), and the length of line segment PB is 3x.
Similarly, assume that the length of line segment AQ is 3y, and the length of line segment QB is 4y.

Since the points P and Q are on the same line segment AB and on the same side of its midpoint, the sum of the lengths of AP and PB should be equal to the sum of the lengths of AQ and QB.

Therefore, we can write the equation:
AP + PB = AQ + QB

Substituting the lengths we assumed:
2x + 3x = 3y + 4y

Simplifying the equation:
5x = 7y

To find a relationship between x and y, we can use the length of PQ, which is given as 2.
We can write the equation:
PQ = AQ + QB
2 = 3y + 4y
2 = 7y

Solving for y:
y = 2/7

Substituting the value of y back into the equation:
5x = 7(2/7)
5x = 2
x = 2/5

Now, we can find the length of AB by adding the lengths of AP and PB:
AB = AP + PB
AB = 2x + 3x
AB = 5x
AB = 5 * (2/5)
AB = 2

Therefore, the length of line segment AB is 2 units.