<p>I'm begging you. Please don't answer if you're not gonna explain step by step.</p>
<p>I'm lost in the world of inequalities. It is my ultimate weakness.</p>
<p>Please, assume I am a 7th grader and that I've recently been introduced to inequalities.. lol</p>
<p>t^2 - k^2 < 6
t + k > 4</p>
<p>If t and k are positive integers, and t > k , what is the value of t?</p>
<p>(a^2 - b^2) = (a + b)(a - b) - basic math theorem. If you don’t know it, learn it;
(t^2 - k^2) = (t + k)(t - k) - just an application of the theorem to our problem;
(t + k) must be greater than 4, so let’s just make it 5;
(5)(t - k) = (t^2 - k^2), which must be something less than 6, let’s call it 5 too.
(5)(t - k) = 5 - simplify those fives out.
(t - k) = 1 - you have an equation! t is one greater than k.</p>
<p>t = 1 + k;
t = 3; k = 2; I think that fulfills both conditions.</p>
<p>If you are a 7th grader, or just approaching the test like a smart 7th grader (which is often a good idea) then the best way to handle this problem is to play around and make up numbers that fit. You need positive integers. They have to add up to more than 4, one bigger than the other. Literally, the first pair I tried was 3 and 2. 3 + 2 is more than 4. And when I checked 3^3 - 2^2 it came out to 9 - 4 = 5 which is less than 6. So I was done. If my first numbers didn’t work, I’d try others. Once I found numbers that DID work, I would answer the question and move on.</p>
<p>But if you want to know WHY 3 and 2 are the only numbers that work…since we are limited to integers, look at the squares of consecutive integers. After 2^2 and 3^2, you will see that the squares of consecutive integers all differ by more than 6. So when they said that t + k >4, they guaranteed that 2 and 3 were the only pair that you could use.</p>
<p>But if that last paragraph didn’t make sense, it really doesn’t matter. I tell students all the time: play with numbers, find numbers that work and then go with them.</p>
<p>I concur with pckeller. Obviously he’s much more clever than me, but I agree that you have to get comfortable playing around with numbers to be successful on the SAT math. </p>
<p>Case in point, when I was solving that, I had no idea where I was actually doing, and ultimately came to the equation t = 1 + k. From there, I tried a few random values and came to the conclusion that both conditions are satisfied when t = 3. If both conditions are satisfied and there’s no glaring caveat, you have arrived at the correct answer!</p>
<p>^ you may be mistaking old age for cleverness :)</p>
<p>I was able to get this answer by playing around with the numbers. I just tried all answers until one worked… this took me around a minute. That’s why I wanted the mathematical explanation, since I really just don’t get inequalities in general.</p>
<p>I truly appreciate your answers, but I still don’t feel comfortable with this question. Is the only way to answer it really just putting in random numbers? I would have no idea where to start, and it would end up taking a minute just like this did…</p>
<p>Could someone please explain this by mostly using inequalities and substituting the least amount possible? </p>
<p>P.S. the answer is t = 3</p>
<p>Here’s a video that solves the problem systematically, without having to make guesses. I’m not saying that this is necessarily a quick way to solve the problem but I think it’s good to understand the math and then look for tricks to get the answer quicker.</p>
<p><a href=“http://wildaboutmath.com/2012/03/22/an-sat-math-problem-with-logic-algebra-and-inequalities/[/url]”>http://wildaboutmath.com/2012/03/22/an-sat-math-problem-with-logic-algebra-and-inequalities/</a></p>