First solution:

Let x be the length of the shorter piece of rope in feet. Then (x + 18) is the length of the longer piece of rope in feet. The rope consists of two of these pieces so 40 = x + (x + 18).
Step 1: Open brackets: 40 = x + x + 18
Step 2: Subtract 18 from both parts of equation: 40 - 18 = x + x
Step 3: Simplify: 22 = 2x
Step 4: Divide sides sides of the equation by 2: 11 = x
Thus the length of the shorter piece is 11 feet.

Second solution:

If we cut 18 feet from the longer piece of the rope we receive 2 pieces of the same length. The total length of the pieces is
40 - 18 = 22 feet and the length of one piece is = 11 feet.

Third solution:

The shorter piece will be shorter than half of the rope = 20. Take answers that fit this constraint and check whether the total length of the rope is 40 feet. For instance, if the shorter piece of the rope is 9 feet long then the longer piece is 9 + 18 = 27 feet long and the total length of the rope is 9 + 27 = 36 40 feet. But if the shorter piece of the rope is 11 feet long then the longer piece is 11 + 18 = 29 feet long and the total length of the rope is 11 + 29 = 40 feet.

Rationale

Modeling the problem helps to understand it clearly.

Pitfalls:

Read the question carefully and do not miss the word "shorter" (answer E is wrong)

Useful formulas:

Linear equation