The positive value of x that satisfies the equation (1 + 2x)

i will go with A as per graph they would be meeting at x=0;y=1

hence 0 must be there .

@ kraizada84 : which figure is this ?

I did a mistake here 0 is not positive hence its wrong.

By the graph of x^4 and x^5 between 1 and 2 they meet again in the 1st quadrant. rest we need to verify the value of each function at 1 and 0.5

Bunuel gave a nice way. I tried to solve graphically.

The question is asking for the positive value of X ..
if i get basics correct ( which i wish to get an expert opinion ) "0" is even integer but
"0" cannot be positive or negative.
hence we cannot go for A.

Zero is an even number and it's neither positive nor negative - TRUE.

But that's not the reason to discard option A (notice that A covers some positive range the same way as other options).

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Hi Zuch,
how much time did it take for you to do this?
I guess on exam we might not get options as close as C and D.

Hi Cumulonimbus,

I understand your concern regarding time. Calculating values for 1.5 especially is tedious. I did not time my attempt but I guess it would take approx. 3 minutes.
But, the point here is for this question, I found this method to be more efficient i.e. time-wise. I think it would be one of the difficult ones on GMAT and hence some extra time dedicated to such problems can be justified. Anyways if such questions come, you have to attempt them unless you are short on time and want to guess and move on.

Its taking too long for me to solve it, any shorter methods please... ?

Excellent ad easy explanation...kudos to you Zuch!

Thank you.

Hi Zuch,

Can you help me understand the concept that why the answer will lie between the values in which the polarity of 'X' changes?

OFFICIAL SOLUTION

The two expressions in question, \((1 + 2x)^5\) and \((1 + 3x)^4\), 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 \(3^5 =?= 4^4\).

Compute the two sides: \(3^5 = 243\), while \(4^4 = 16^2 = 256\). So the right side is bigger.

Now, we need another benchmark. Try x = 2. The “equation” becomes \(5^5 =?= 7^4\).

Compute or estimate the two sides. \(5^4 = 25^2 = 625\), and multiplying in another 5 gives you something larger than 3,000 (3,125, to be precise). Meanwhile, \(7^4 = 49^2 < 50^2 = 2,500\), 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 \(\frac{3}{2}\). Plugging in, you get \(4^5 =?= (\frac{11}{2})^4\)

\(4^5 = 2^{10} = 1,024\) (it’s good to know your powers of 2 this high)

For the other side, first compute the denominator: 24 = 16. Now the numerator: \(11^4 = 121×11×11 = 1,331×11 = 14,641\) (also quick to do longhand)

So \((\frac{11}{2})^4 = \frac{14,641}{16} < 1,000\). The right side is bigger.

Let’s recap:

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

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

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

\((1 + 2x)^5 > (1 + 3x)^4\) 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,

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

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

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

\((1 + 2x)^5 = 2^5 = 32\)

\((1 + 3x)^4 = (\frac{5}{2})^4 = 2.5^4 = 6.25^2 > 36 (= 6^2) > 32\)

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.

Hi All,

While the question 'looks' a bit crazy, the wording and set-up for it implies that there's just 1 solution (and we won't have to actually calculate the exact value). Second, the answers are numbers, so we should TEST THE ANSWERS. Since the "math" in this question could get ugly, I'm going to stick to integers.

X = 1 is a great place to start. Plug that value into both calculations and you'll end up with...

3^5 and 4^4
243 and 256

So the numbers are real close; the answer has to be close to 1. The question is do we pick B or C? Notice that the SECOND value in the above calculation is BIGGER.

For the next test, you can decide between a few options (.5, 1.5 or 2 if you prefer using an integer).

X = 2 gives us...

5^5 and 7^4
3025 and (49)(49) = approx. (50)(50) = 2500

Notice now that the FIRST value is BIGGER.

This tells me that increasing the value of X makes the first part of the equation "grow" faster than the second part. By extension, at some point between X=1 and X=2, the values would have been equal and then a little past that point the first value became bigger than the second. Thus, I would have to choose an answer that was a little bigger than 1.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Just a thought ...Can we solve this problem using Binomial Expansion ?

like (a+b)^n = (a+nb) ( in binomial )

Using using function G(x)=(1+2x)^5-(1+3x)^4,
You can check the change of sign. It’ll happen b/w 1&1.5

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