1. **State the problem:** We have a segment $IJ$ with length 2, and it is dilated by a scale factor of $\frac{3}{4}$ to form segment $I'J'$. We need to find the length of $I'J'$.\n\n2. **Formula used:** The length of the dilated segment is given by multiplying the original length by the scale factor: $$\text{Length of } I'J' = \text{Scale factor} \times \text{Length of } IJ$$\n\n3. **Apply the formula:** Substitute the given values: $$\text{Length of } I'J' = \frac{3}{4} \times 2$$\n\n4. **Calculate:** Multiply the numbers: $$\frac{3}{4} \times 2 = \frac{3 \times 2}{4} = \frac{6}{4} = 1.5$$\n\n5. **Conclusion:** The length of segment $I'J'$ after dilation is $1.5$.