*Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:*

The two expressions in question, (1 + 2*x*)5 and (1 + 3*x*)4, could in theory be expanded, but you’ll wind up doing a lot of algebra, only to find an equation involving *x*5, *x*4, *x*3, *x*2, and *x*. Solving for the positive value of *x* that makes the equation true is nearly impossible.

So how on Earth can you answer the question? Notice that the question does not require you to find a precise value of *x*; you just need a range. So plug in good benchmarks and track the value of each side of the equation.

Start with *x* = 1. The “equation” becomes 35 =?= 44.

Compute the two sides: 35 = 243, while 44 = 162 = 256. So the right side is bigger.

Now, we need another benchmark. Try *x* = 2. The “equation” becomes 55 =?= 74.

Compute or estimate the two sides. 54 = 252 = 625, and multiplying in another 5 gives you something larger than 3,000 (3,125, to be precise). Meanwhile, 74 = 492 < 502 = 2,500, so the left side is now bigger. Somewhere between *x*=1 and *x*=2, then, the equation must be true.

The only benchmark left to try is 1.5, or 3/2. Plugging in, you get 45 =?= (11/2)4

45 = 210 = 1,024 (it’s good to know your powers of 2 this high)

For the other side, first compute the denominator: 24 = 16. Now the numerator: 114 = 121×11×11. 121×11 = 1,331 (this is quick to do longhand)

1,331×11 = 14,641 (also quick to do longhand)

So (11/2)4 = 14,641/16 < 1,000. The right side is bigger.

Let’s recap:

(1 + 2*x*)5 = (1 + 3*x*)4 at what value of *x*?

(1 + 2*x*)5 < (1 + 3*x*)4 when *x* = 1

(1 + 2*x*)5 > (1 + 3*x*)4 when *x* = 1.5

(1 + 2*x*)5 > (1 + 3*x*)4 when *x* = 2

So the value at which the two sides are equal must be between 1 and 1.5.

**The correct answer is C.**

Extra points:

As *x* grows past 2, the larger power (5) on the left takes over, so you can see that the left side will always be bigger.

What about when *x* is between 0 and 1? Well, first notice that at *x*=0, the two sides are again equal. If *x* is a tiny positive number (say, 0.001 or something), then you can ignore higher powers of *x* (*x*2, *x*3, etc.), and this simplifies the algebraic expansion:

For tiny positive *x*,

(1 + 2*x*)5 = 1 + 5(2*x*) + higher powers of *x* ? 1 + 10*x*

(1 + 3*x*)4 = 1 + 4(3*x*) + higher powers of *x* ? 1 + 12*x*

So right away, the right side is bigger than the left side. You can also check *x* = ½:

(1 + 2*x*)5 = 25 = 32

(1 + 3*x*)4 = (5/2)4 = 2.54 = 6.252 > 36 (= 62) > 32

Again, the right side is bigger than the left side. For every *x* between 0 and 1, in fact, the right side is larger than the left side. This fact isn’t particularly easy to prove, but you don’t need to do so.