Query from Equations, Inequations and VIC strategy guide

[sarmabrk]

Chapter 6:
Page: 101
Question 1:
If 4x-12 >=x+9, which of the following must be true?
a) X > 6 b) X < 7 c) X > 7 d) X > 8 e) X >8

when we solve the in equation:
4x-12 >=x+9
we get X >=7.

So according to me the answer should be C (X > 7) and D (X > 8).

But the answer given in the book is A (X > 6)
My understanding is since, it has not be given X is an integer, X can be something like 6.5 which doesn't satisfy X >=7.

So A is not the ans. But if X>=7, it should X >7 and 8.

Please clarify.

[tim]

Okay so if i tell you i have at least 7 pogs, which statement are you sure is true?

A) i have more than 6 pogs
C) i have more than 7 pogs
D) i have more than 8 pogs

If i tell you i have at least 7, there is no way you can conclude that i have more than 8. You do know i have more than 6 though. Does that help?

[sarmabrk]

Yes it does. Thx..

[tim]

glad to hear it..

[amit.dhameja]

I am not so sure. When you say pogs... Those can be counted. But the variable 'x', unless specified as an integer, can be 6.5, thereby making the answer choices kinda vague.

[jnelson0612]

I am not so sure. When you say pogs... Those can be counted. But the variable 'x', unless specified as an integer, can be 6.5, thereby making the answer choices kinda vague.

amit, if you solve the equation you get that x is greater than or equal to 7. That means that x is any number starting at 7 on the number line and going on to infinity. Thus, of those answer choices, it MUST be true that x is also greater than 6, because every number 7 and above is greater than 6.

Make sense?

[abhishek.shanker]

When we take X > 6, then i can consider X = 6.25. This doesnt agree with X>=7

If i take X > 7, then i can consider all possible values for X and it will agree with X >=7 ALWAYS.

Your explanation is still unclear to me. can you please help explaining this again on why is A correct as per you and C is incorrect?

[messi10]

Hi Abhishek,

I think you are thinking about the question in the opposite way. The question states: "Given the equation, which of the following must be true"

You have correctly solved the equation to get x is greater than or equal to 7. So as Jamie's post above states that x is a number which is at least 7 and can go up to infinity. For this given range, if x = 7 or 34.4 or 56.3939 or 200 or 90540935 etc, we know for sure that x will never be less than 7, i.e. x will never be 6 or 6.25 or 6.30942984.

So according to the question, X > 6 MUST be true.

The problem with C is that it does not include 7 as a possible value. If x = 7 (which it could be according to our equation), then C is no longer true

Hope this helps

Regards

Sunil

[gmittal]

I see tim has considered integers while explaining the solution which seems logical whereas others are suggesting that x>6 is the right answer even when fractions are taken into account. Can someone correct me here if the question is missing additional information about x being an integer or fraction.

[JohnHarris]

I see tim has considered integers while explaining the solution which seems logical whereas others are suggesting that x>6 is the right answer even when fractions are taken into account. Can someone correct me here if the question is missing additional information about x being an integer or fraction.

It makes no difference so the question is not missing information. As varun_783 pointed out, consider the number 7 which is an integer and allowable as a solution to x>7. However 7 is not greater than 7, so (c) can not be the answer. Along the same lines 7.1 is greater than or equal to 7 but not greater than 8 so (d) can not be the answer. The point is that if you choose any number to test the answers, it first must be greater than or equal to 7 to satisfy the given circumstances [4x-12 >=x+9].

Also, paraphrasing what members of the staff have said here on the forum, if it is not stated that the solution is restricted to an integer, then all numbers (integers, rationals and irrationals including positive and negative) must be considered.

[jnelson0612]

Thanks Sunil and John! Great points!

[aset_kaz]

Hi there!

I am also stuck on this one. I understand tim's "pog logic", but could you please clarify the following.

ok, let's assume that (a) x>6 must be true. however, if we plug in say 6.1 instead of x, then our equation would result in 12.4>15.1, which is not true.

and, if say x>1 was among the answer choices, would x>1 would be the correct one?

[StaceyKoprince]

That's because this is really a logic question, not a "mathematical solution" question.

Take a look at Tim's "pog" thing again. He said he had at least 7 pogs, right? And then we concluded that it must be true that he has at least 6. You said that made sense to you.

Does he have exactly 6.1 pogs? No. I know he has at least 7. He could have 7, 7.1, 8, whatever - but he has more than 6.1.

He does have AT LEAST 6.1 pogs, though. He also has at least 2 pogs. He has AT LEAST any number smaller than 7.

In the problem, we don't know what x is, but we do know for sure that x is larger than 6. It's also larger than 5, or 4. That doesn't mean is IS 6 or even 6.1. It's just LARGER than that.

The answer is not telling us actual possible values for x - that is, it is NOT saying that x can be anything greater than 6. Rather, it's telling us a characteristic that must be true of x, whatever that x is. It is true that x must be bigger than 6, because x must be 7 or greater, and all of those numbers are greater than 6.