Simplify $\frac{1}{(x + 1)} + \frac{1}{(x − 1)}$

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 LCM of the denominators: a/b + c/d = (ad + bc)/(bd)

Solution:

1. The denominators are (x + 1) and (x − 1). The common denominator is (x + 1)(x − 1).

2. Rewrite each fraction: • 1/(x + 1) = (x − 1) / [(x + 1)(x − 1)] • 1/(x − 1) = (x + 1) / [(x + 1)(x − 1)]

3. Add the numerators: (x − 1) + (x + 1) = 2x

4. Result: 2x / [(x + 1)(x − 1)]

This can also be written as 2x/(x² − 1), since (x + 1)(x − 1) is a difference of two squares.

Checking the options:

• A has (x − 3), which is wrong, since our factor is (x − 1).
• B is 2x/(x + 1)(x − 1), which matches our result.
• C has (x + 1)², which would only come from adding 1/(x+1) to itself in some form, so it is wrong.
• D is not even a fraction, so it is wrong.

Tip: When adding fractions with different denominators that share no common factor, the common denominator is simply the product of the two. Also, quickly test by substituting a value: let x = 2. Then 1/3 + 1/1 = 4/3. Option B gives 4/(3×1) = 4/3 ✓. Substituting a number is a fast way to eliminate wrong options in JAMB.