SAT Math Problem

Test Preparation SAT Preparation
1

<p>I managed to solve this problem, but I do have one question: </p>

<p>Are there any other ways this problem can be attacked? </p>

<p>The Problem Statement:</p>

<p>Let ∇x = x + 1/x for all nonzero integers x. </p>

<p>If ∇x = t, where t is an integer, which of the following is a possible value for t? </p>

<p>A) 1
B) 0
C) -1
D) -2
E) -3</p>

<p>--</p>

<p>My Solution:</p>

<p>1) Combine x + 1/x. That yields (x^2 + 1) / x = t</p>

<p>2) Plug in for t. E.g. I plugged in t = 1:</p>

<p>x^2 + 1 = x (cross-multiplication)</p>

<p>x^2 -x + 1 = 0 (cannot be factored - move on to next answer choice)</p>

<p>t = 0 ... and so on.</p>

<p>--</p>

<p>Is there a better way to attack this problem?</p>

2

<p>EDIT: Disregard this post, I messed up somewhere. I’ll fix it later.</p>

3

<p>1 is not the answer. </p>

<p>t ≠ 1, and x ≠ 1.</p>

4

<p>This question is quite perplexing.
Let’s lay out the equation once more: x + 1/x = t. Given information includes x being a nonzero integer, and t being an integer, whether positive, negative, or zero. In this case, two answers would work (I think). 1 + 1/1 = 2, and -1 + 1/-1 = -2. Can someone please clarify? Didn’t really understand your solution either, IceQube, if you could further simply it please?</p>

5

<p>Sure. I looked at the equation as a quadratic.</p>

<p><a href=“http://min.us/mXgCICQQd[/url]”>http://min.us/mXgCICQQd&lt;/a&gt;&lt;/p&gt;

<p>1) I had to find a value for t that would make the equation factorable:</p>

<p>The equation is:</p>

<p>(x^2 - 1) / x = t </p>

<p>2) After I find a value for t, I cross multiply and get a certain x value. If I plug in 1 for t, I’ll cross multiply and get x on the right side of the equation:</p>

<p>x^2 - 1 = x </p>

<p>3) I attempt to solve the quadratic by setting the equation to 0:</p>

<p>x^2 - x - 1 = 0</p>

<p>Unfortunately, no integer value will satisfy the above equation, so I am forced to plug in the next answer choice, and so on. </p>

<p>–</p>

<p>My question is whether or not there is a method that doesn’t involve plugging in. I know xiggi recommends abstaining from plugging in #s.</p>

6

<p>I haven’t heard anyone here say never plug in…</p>

<p>This problem cries out for plugging in #s…so much so that I would guess that the vast majority of students who solve this probably try x=1, see if it works, then move on to try x = -1 (yielding t = -2 which is on the list), see that it DOES work, and move on to the next question.</p>

7

<p>Oh waiiiiiiiit…one possible value for (t)…not (x)…that’s where I went wrong.</p>

<p>So anyway, x + 1/x = t. x is a nonzero integer, t is an integer. The only way for x + 1/x to yield an integer value is for x to be equal to 1/x, since the sum of integer values results in an integer. Sound simple? Alright. What would make x = 1/x? Either 1, or -1. OK, 1+ 1/x = 2. On the answer list? No. -1 + 1/-1 = -2. On the answer list? Yes. There you have it. I’m sorry but I have no idea what combination or cross multiplication or quadratic solving you went through there, but this strategy took about 30 seconds to go through. As I said earlier, I was pretty tired and not concentrating, didn’t read the question correctly.</p>