Simplify $\frac{2}{2 + x} + \frac{2}{2 - x}$

Analyze the question: We need to add two algebraic fractions with different denominators, so we must find a common denominator first.

Concept: To add fractions, use the common denominator. Here the denominators are (2 + x) and (2 − x), and their product is a difference of two squares:

(2 + x)(2 − x) = 4 − x²

Solution:

Expand the numerator:

2(2 − x) = 4 − 2x 2(2 + x) = 4 + 2x

Add them: (4 − 2x) + (4 + 2x) = 8 (the x terms cancel)

So:

Check with a number: Let x = 1. The left side is 2/3 + 2/1 = 8/3. Option B gives 8/(4 − 1) = 8/3 ✓. (Option C would give 4/3, which is wrong.)

Tip: When the denominators are (a + b) and (a − b), the common denominator is a² − b². Also, watch for terms like −2x and +2x that cancel in the numerator. A quick substitution check, such as x = 1, can confirm your answer in seconds during the exam.