<p>If x > 2, what could be the value of x + 1 / x?</p>
<p>Correct answer: 11 / 8
My (wrong) answer: 1</p>
<p>X's should be cancelled, shouldn't they? </p>
<p>I think you’re supposed to do long division on these questions… not sure</p>
<p>no the x’s do not cancel because x and 1 are combined by an addition sign. This question was probably a grid in I assume? 11/8 does not make sense as an answer. The correct answer should be greater than 1.5. Is there any additional information they give in the question?</p>
<p>@realcoolman I assume you mean greater than 2.5.</p>
<p>@Actualizer The x’s don’t cancel. Try putting in a value to see why: does 3+1/3=1? If this were multiplication instead of addition, then you would be right.</p>
<p>Let’s think about why an answer has to be larger than 2.5 (this isn’t necessary for the SAT. Just plug in a value larger than 2 and be done with it) . Basically we want to show that 2<a then="" a+1="" a="">2.5 Well if a>2, then a-2>0. But we also know that 2a-1 is positive because if a>2, then a is also greater than 1/2 and 2a-1>0 is solved by any a>1/2. If we multiply these two expressions together we get (since the product of two positives is positive):</a></p><a then="" a+1="" a="">
<p>(a-2)(2a-1)>0
2a^2-5a+2>0</p>
<p>Dividing by 2a gives (this is allowed since 2a is positive)</p>
<p>a+1/a>5/2=2.5</p>
<p>Edit: Please use appropriate parentheses next time!! I interpreted this as x+(1/x) as per PEMDAS instead of (x+1)/x which seems to be in line with the sample answer. This is probably why @realcoolman gave the figure of 1.5, though then a possible answer would be LESS, not greater than.</p>
</a>
<p>@RandomHSer</p>
<p>yes 1.5 is the question is (x+1)/x or 2.5 if the question is x + (1/x). So we agree that the answer cannot be 11/8 ?</p>
<p>@Actualizer </p>
<p>@realcoolman @RandomHSer </p>
<p>Sorry, I didn’t provide all the information since I thought that was enough. The actual question provides 5 choices:</p>
<p>If x > 2, which of the following could be the value of (x +1) / x?</p>
<p>a: 3/5
b: 7/9
c: 1
d: 11/8
e: 12/5</p>
<p>We’re not solving for x but the value of that fraction, right? X can’t be 1.5 since the question says x is greater than 2. But how did you get that number, 1.5?</p>
<p>Btw, I discovered that everything else except for D which is 11/8 is less than or equal to 1 or greater than 2… . hehehee…can this be a clue…?</p>
<p>And I see many questions that I can’t get an accurate answer like this one. Is this what SAT math (not subject test) looks like?</p>
<p>Thanks in advance</p>
<p>uh the answer has to be greater than 1.5 and all the answers are less than 1.5. Are you sure you put the answer choices in right? I took the SAT Math 2 Subject Test if that is what you’re wondering about. I got a 800. It’s not that hard if you know your math up to Pre-Calc well. </p>
<p>You are all misinterpreting the question. X has to be greater than 2, but the limit as x goes to infinity of (x+1)/x is actually 1, since the 1 constant becomes less and less significant in the fraction. We also know that x>2, so the (x+1)/x has to be less than 3/2, not greater. This is because as we increase the value of x starting at 2, the value of (x+1)/x decreases. Choice D. 11/8 is the only answer that follows these restrictions.</p>
<p>Note that limits and calculus aren’t needed to solve this problem. Setting (x+1)/x=11/8 gives us that x = 8/3, clearly greater than 2.</p>
<p>oops yeah my bad @pizzabagel is right the answer has to fall between 1 and 1.5.</p>
<p>@Actualizer the 1.5 comes from the following:
x>2
x+1>3 (adding one to both sides)
At this point it would be naïve to say (x+1)/x > 3/2 because there are no rules of inequalities that allow us to do this.</p>
<p>In fact, the rule is that when we take the reciprocal of two values, their inequality will switch. In the case that x>2, 1/x<1/2.</p>
<p>There is, however, a rule that if a<b, and c<d, and all values are positive, then ac<bd. Letting a = 1/x, b= 1/2, c= x+1, and d= 3, we can determine that (x+1)/x<3/2.</p>
<p>@Actualizer looking into this question, I wouldn’t suspect this would show up on the SAT since it doesn’t test problems that can’t be solved algebraically or ones that require calculus.</p>
<p>This question can be solved using the common SAT math strategy of plugging in answers to see which ones work, but it requires calculus to be solved in the general in the sense that limits are the best way to explain why no matter how big x gets, (x+1)/x will never get less than 1 (assuming x>2).</p>
<p>@pizzabagel How are limits the best way to explain that? How about just saying that since x+1>x, (x+1)/x>1? We also know that 2x>x+1 (since this is solved by any x>1), which implies (x+1)/x<2. Together, this gives 1<(x+1)/x<2, which leaves D as the only answer.</p>
<p>Probably an even better way to explain it would be that (x+1)/x=1+1/x. We know that if x>2, then 0<1/x<1/2, so 1<(x+1)/x<3/2=1.5.</p>
<p>@RandomHSer well that might make it more obvious that the range of the numbers are 1<x<3 2,="" but="" the="" only="" way="" to="" show="" that="" 1="" x="" will="" never="" be="" less="" than="" zero="" for="" all="" real="" values="" of="">2 is to take the limit as x goes to infinity, yielding 1, and to show that the values of x in this range are monotonically decreasing (to prove this, taking the derivative of 1/x will yield -1/(x^2), which means that x will have a negative slope as long as x is nonzero). All of this requires calculus and whether or not one splits up the fraction still makes this question SAT-ineligible.</x<3></p>
<p>Huh? Of course 1/x is always positive for x>2. 1/x is the number that when multiplied by x gives 1 and since the product of two positives is positive whereas the product of a positive and negative is negative, 1/x must be positive. Please don’t tell me you need calculus to say that the product of two positive numbers is positive…</p>
<p>@RandomHSer Yeah that makes sense. You don’t really need calculus to say that 1/x will be positive, but knowing for 100% that 0 is the lower bound to the possible values of 1/x is what calculus is needed for (why isn’t it 1/4 or 1/8? Because x has to go to infinity for 1/x to approach 0) . Like I said before, this problem can be solved simply with intuition, but questions on the SAT need to have an algebraic solution, and this one requires too high of a level of math to make the cut.</p>
<p>Nope, you don’t need calculus for that either. If I want to make 1/x smaller than some positive number a, then I just pick x to be larger than 1/a. This will result in a smaller number because then a-1/x=(ax-1)/x. ax-1 is positive because x>1/a and we are working only with x positive. So this fraction is positive so a-1/x>0 or 1/x<a. This shows that I can make 1/x smaller than any positive number by making x large.</p>
<p>Well, I don’t want to create an international incident, but this isn’t about picking some arbitrary number a. It’s about finding the exact endpoint so that it’s obvious without using the multiple choice answers what the possible values of (x+1)/x can be. And by making x “large,” you are basically re-representing the concept of approaching infinity in different terminology. Even if calculus isn’t used, the College Board doesn’t expect students to arrive at a conclusion this obscure.</p>
<p>It is about picking some arbitrary number a. I just showed that I can make 1/x smaller than any given positive number by picking x right. This is why it doesn’t have a lower bound “1/4 or 1/8”. I don’t have to make reference to x being large, that was just to give some intuition.</p>