"((x + 1)/(x − 1))² — If x Is Replaced by 1/x..." — GMAT® Worked Solution

From Episode 45 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind plugging in numbers, read: GMAT® Algebra: What to Do When You're Stuck.

The Problem

Source: Official Guide for GMAT® Review, 11th Edition

((x + 1) / (x − 1))²

If x ≠ 0 and x ≠ 1, and if x is replaced by 1/x everywhere in the expression above, then the resulting expression is equivalent to:

(A) ((x + 1) / (x − 1))²
(B) ((x − 1) / (x + 1))²
(C) (x² + 1) / (1 − x²)
(D) (x² − 1) / (x² + 1)
(E) −1 × ((x − 1) / (x + 1))²

Try it before reading on.

Why the Algebra Gets Messy Here

A lot of people start by substituting 1/x into the expression and then FOILing. That's valid math. But it leads somewhere frustrating.

If you FOIL after the substitution, you wind up with something like (1/x² + x + 1) in the numerator and (1/x² − x + 1) in the denominator. All correct. But it doesn't match any of the answer choices. You've spent a couple of minutes on accurate algebra and you're stuck.

About half of students who go the algebra route end up in that situation on this problem. They do correct work that doesn't connect to the answer.

The right algebraic approach is to find common denominators inside the fraction BEFORE simplifying — but that path requires you to resist your instinct to FOIL, and even then there are several places where a fraction error can creep in.

The Optimal Algebraic Approach (Quick Version)

Substitute 1/x for x. You get (1/x + 1) on top and (1/x − 1) on the bottom, all squared.

Find a common denominator on top: 1/x + x/x = (x + 1)/x.

Find a common denominator on the bottom: 1/x − x/x = (x − 1)/x. That simplifies to −(x − 1)/x, but let's keep going.

Wait — 1/x − 1 = 1/x − x/x = (1 − x)/x. So the bottom is (1 − x)/x.

Now you're dividing (x + 1)/x by (1 − x)/x. Dividing by a fraction means multiplying by the reciprocal. The x's cancel.

You get (x + 1) / (1 − x). Note that (1 − x) = −(x − 1). So this becomes −(x + 1)/(x − 1).

Square the whole thing: (−(x + 1)/(x − 1))² = ((x + 1)/(x − 1))².

The negative disappears when you square. You get exactly what you started with.

The answer is (A).

That's the clean version. In real time, under pressure, many of us don't see every one of those moves. So let's look at the backup approach.

Plugging In Numbers

Choose x = 2. It satisfies both constraints (x ≠ 0, x ≠ 1) and keeps the numbers simple.

Step 1: Plug 1/x into the original expression.

Remember — the problem says "x is replaced by 1/x." So you're actually plugging in 1/2.

Top: 1/2 + 1 = 1/2 + 2/2 = 3/2.

Bottom: 1/2 − 1 = 1/2 − 2/2 = −1/2.

The fraction: (3/2) ÷ (−1/2).

Dividing by a fraction means multiplying by the reciprocal: 3/2 × (−2/1) = −6/2 = −3.

Square it: (−3)² = 9.

That's your target number. Any correct answer choice should give you 9 when you plug in x = 2.

Step 2: Test the answer choices with x = 2.

(A) ((2 + 1)/(2 − 1))² = (3/1)² = 9. Match.

(B) ((2 − 1)/(2 + 1))² = (1/3)² = 1/9. Not 9. Eliminate.

(C) (4 + 1)/(1 − 4) = 5/(−3) = −5/3. Not 9. Eliminate.

(D) (4 − 1)/(4 + 1) = 3/5. Not 9. Eliminate.

(E) −1 × ((2 − 1)/(2 + 1))² = −1 × (1/3)² = −1/9. Negative, so it can never equal positive 9. Eliminate.

Only (A) matches.

The answer is (A).

What to Take Away

When the algebra stalls, plug in a number. You don't have to abandon algebra forever. But on this specific problem, plugging in x = 2 gets you to the answer in about 90 seconds with basic fraction arithmetic. The algebra requires several non-obvious moves and takes most people much longer.

Plug 1/x into the original expression, but plug x into the answer choices. This is the key detail that's easy to miss. The problem asks what happens when 1/x replaces x in the original — and the answer choices are written in terms of x. So you compute the original with 1/x, then check the answers with x.

Test at least one other answer choice as a safety check. If (A) matches and nothing else does, you can feel confident. If two answers had matched, you'd need a different value of x to break the tie. On this problem, only one matches — so you're done.

Ready for the toughest one from this episode? Roman numerals, square roots, and the "none of the above" trap: "If x and y Are Positive, Which Must Be Greater Than 1/√(x + y)?".

Want the full strategy? Read: GMAT® Algebra: What to Do When You're Stuck

From Episode 45 of Real GMAT® Problems (The GMAT® Strategy Podcast).