Simplify [1 ÷ (x + 1)] - [2 ÷ (x - 1)] + [3 ÷ (x2 - 1)]

Step 1: Analyze the question We need to combine three fractions into one. The denominators are (x + 1), (x − 1) and x² − 1. Since x² − 1 = (x + 1)(x − 1), the common denominator is x² − 1.

Step 2: Concept Difference of two squares: x² − 1 = (x − 1)(x + 1). Rewrite each fraction over the LCM, then combine the numerators.

Step 3: Solution

1/(x + 1) − 2/(x − 1) + 3/(x² − 1)

= [1(x − 1) − 2(x + 1) + 3] / (x² − 1)

Expand the numerator:

• 1(x − 1) = x − 1
• −2(x + 1) = −2x − 2
• +3

Add: x − 1 − 2x − 2 + 3 = −x + 0 = −x

So the expression becomes:

−x / (x² − 1)

Step 4: Answer

Tip: In problems like this, factorise the denominators first to find the common denominator. Be careful with the negative sign when you multiply −2 across (x + 1). Forgetting to distribute it to both terms is a very common mistake.