Q: Points D, E, and F are collinear with DE < EF and DF < EF. If DE = 2x+6, DF = 3x-4, and EF = 22, what is DE?
First, recall that ‘DE’ refers to the length of the segment with endpoints D and E. It is the distance between D and E. Next, DON’T ASSUME that E is between the D and F! Since DE and DF are each less than EF, and all three points lie on the same line (collinear), D must be between E and F.
Draw the diagram:
E D F
Now label the distances as they are given in the problem:
E 2x+6 D 3x-4 F
Use the concept: Part + Part = Whole (also called the Segment Addition Postulate) to write the equation:
2x+6 + 3x-4 = 22
5x + 2 = 22
5x = 20
x = 4
Plug x=4 back in to the ‘DE’ part:
DE = 2x+6 = 2(4) + 6 = 14.
Tip: To be sure your answer is correct, plug x=4 into the ‘DF’ part, too:
DF = 3x-4 = 3(4)-4 = 8. Now, see that the sum of the two ‘parts’ does equal the ‘whole’:
14 + 8 = 22.