Question: 1/10 and 1/2

Comment on 1/10 and 1/2

We can also solve for 1) as follow, instead of using all values and make calculation
we can rewrite expression in 1) as

10>4x+2>2, this means, 4x+2 can be 3,4,5,6,7,8 or 9
if we solve above, we get
divide by 4
that leave us option: x=1.

Can we do following?
1/x<1/y<1/z then x>y>z, i think so.
gmat-admin's picture


Your suggested rule (if 1/x <1/y < 1/z then x > y > z) is true AS LONG AS x, y, and z are all positive or all negative.

For example, 1/(-2) < 1/(-3) < 1/2, however we can't say that -2 > -3 > 2

Thank you so much.

I really liked your videos. The way that you solves and explains little facts are really amazing as these little facts can trick in simple questions.


Great video. Question: on test day, would we have to go through each of the integers from 3-9 to figure something like this out? Or is there a way we can identify a shortcut when we see something like this? If not, I guess just knowing this question type beforehand will save us time so we can get right to it.
gmat-admin's picture

Once we know that 4x+2 equals one of the integers from 3 to 9, we can automatically eliminate all of the ODD integers, since 4x will be EVEN and when we add 2, the result is EVEN.

This leaves us with 4, 6 or 8 top test.

Thank you!


Please help me, I solved the first stated as below.

1/10 < 1/4x+2 < 1/2

0.1 < 1/4x+2 < 0.5

Since there is no integer that lies between 0.1 & 0.5, I declared this statement to be insufficient.
gmat-admin's picture

Be careful. We are told that x is an integer. We are not told that 1/(4x+2) is an integer.

For example, ix x = 1, then 1/(4x+2) = 1/6 = 0.166...


Exercise 224 (Problem Solving - OG 2017)
Could you explain me in a different way from OG 2017 answer solution?

Thank you in advance.

gmat-admin's picture

At the risk of appearing to "pass the buck," I'd like to direct you to a very rich discussion of this question here:

In particular, Mitch (GMATGuruNY) provides a concise solution.
Also, Ceilidh (from Manhattan Prep) explains why this might be a great candidate for guessing and moving on.

If you have any questions about those various solutions, I'm happy to respond. I just don't think I'd do much better than Mitch's solution.

Hey Brent, on statement 2, when I got (x+6)(x-1), I concluded that x must be -6 or 1, and then made the mistake of checking for extraneous roots, which took extra time. How do you know when to check for extraneous roots?
gmat-admin's picture

You only need check for extraneous roots when dealing with equations with square roots and equations with absolute value.


To avoid testing all the values for statement 1, I just converted it into an inequality 10>4x+2>2 , then solve it to get 2>x>0, given x is integer, answer must be 1.
gmat-admin's picture

That works too. Nice job!

Hi Brent, for statement 1, would it be okay to split the inequality into two separate ones.

Such that 1/10 < 1/4x+2 < 1/2 becomes:

1) 1/10 < 1/4x+2 and 2) 1/4x+2 < 1/2

This gives us 1) x < 2 and 2) x > 0

If we combine 1) and 2) again to bring back us back to to the original equality, we get:

0<x<2. Since x is an integer, x must be 1.

Just want to know if this approach works for all inequality questions with 2 inequality signs.

gmat-admin's picture

That's a perfectly valid solution. Nice work.


In the video question above, why doesn't 0 count as an integer?
x could be 0 - couldn't it?
gmat-admin's picture

x is, indeed, an integer.
However, x = 0 is not a solution to either statement.

Let's see why:

Statement 1) 1/10 < 1/(4x + 2) < 1/2
Plug in x = 0 to get: 1/10 < 1/(4(0) + 2) < 1/2
Simplify to get: 1/10 < 1/2 < 1/2
Since it is NOT the case that 1/2 < 1/2, x = 0 is not a possible value of x.

Statement 2) x² + 5x - 6 = 0
Plug in x = 0 to get: 0² + 5(0) - 6 = 0
Simplify to get: 0 + 0 - 6 = 0 (doesn't work)
So, 0 is not a possible value of x.

Does that help?


Hi. I tried to actually solve for x. But I wanted to multiply all sides by the reciprocal in the middle. But when I take 1/(4x+2) *(4x +2)/1 then the whole thing turns into 1 and I no longer have an x at all. What is the problem with this method?
gmat-admin's picture

Let's see how that approach will work.

Given: 1/10 < 1/(4x+2) < 1/2
Multiply all sides by (4x+2) to get : (4x+2)/10 < 1 < (4x+2)/2
Multiply all sides by 10 to get : (4x+2) < 10 < 5(4x+2)
Expand to get : 4x + 2 < 10 < 20x + 10

We can treat this a two separate inequalities:
4x + 2 < 10 and 10 < 20x + 10

Let's solve each one.

Take: 4x + 2 < 10
Subtract 2 from both sides to get: 4x < 8
Divide both sides by 4 to get: x < 2

Take: 10 < 20x + 10
Subtract 10 from both sides to get: 0 < 20x
Divide both sides by 20 to get: 0 < x

If we combine x < 2 and 0 < x, we get: 0 < x < 2

Since x is an INTEGER, x must equal 1

Does that help?


Hi Brent! Could you tell me if the following approach is valid?

In respect of statement 1 can we assume 2 situations i.e. when x is positive and when x is negative.

When x is positive the equation becomes 10<4x+2<2 which is not possible and hence can be discarded.

When x is negative the equation becomes 10>4x+2>2 which simplified gives us 2>x>0 and the only integer value possible in this case is x=1.

Is this a valid approach?
gmat-admin's picture

That's a good idea, but that's not quite how the inequality works.

You write: "When x is positive the equation becomes 10 < 4x+2 < 2 which is not possible and hence can be discarded"

If x is positive, then it must be the case that 2 < 4x+2 < 10 (not 10 < 4x+2 < 2)
Since, x is an integer, we can be certain that x = 1.
So, we can't discard that scenario.
Conversely, the statement "When x is negative the equation becomes 10>4x+2>2 which simplified gives us 2>x>0 and the only integer value possible in this case is x=1" is contradictory.

If x is negative, then we can't then conclude that x = 1 (a positive number)

Does that help?


Hi Brent. Please help me to understand whether I did okay with statement 1:

1/10 < 1/(4x+2) < 1/2

I really struggled with x being in the denominator so I tried the following:

10 > 4x +2 > 2.

Then follows, 8 > 4x > 0 , and finally 2 > x > 0. As the only possible integer between these values is 1. x equals 1.

Is that correct or should I stop doing such things?
gmat-admin's picture

That approach is perfectly valid.
In fact, it's pretty much the same as my approach :-)
You concluded that 10 > 4x +2 > 2
I concluded that 4x + 2 = 3, 4, 5, 6, 7, 8, or 9 (since x must be an integer)


Hi Brent. Looking at statement 2, I get to the same two solutions as you. Taking negative 6 for x and putting it into the question stem, it becomes invalid as the answer is negative (so not between 1/10 and 1/2). I then concluded that x must equal 1 and so the statement was sufficient. Please can you explain why I'm not able to make that conclusion (that x = -6 is invalid). Thanks very much for your help!
gmat-admin's picture

Hi tan27,

You're making a very common mistake. When we analyze statement 2, we must be sure to IGNORE information from statement 1.

The following video addressed that mistake:


Hi there,

For working out part 1 I did:

1/10 < 1/4x+2 < 1/2
10 > 4x+2 > 2
8 > 4x > 0
2 > x > 0

Therefore, from reciprocal-ing everything and inverting the signs I was able to determine x = 1

Is that a correct approach?

gmat-admin's picture

That approach will work AS LONG AS the fractions are ALL positive or ALL negative.
For example, it is true that -1/2 < 1/4 < 1/3.
However, we can't then conclude that -2 > 4 > 3

In the above question, we can see that 1/10 and 1/2 are both positive. Since 1/4x+2 lies BETWEEN 1/10 and 1/2, we can be certain that 1/4x+2 is also positive. So, your strategy works perfectly!


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