# System of Inequalities

• Nov 20th 2010, 05:39 PM
Reiner
System of Inequalities
$t^2 - k^2 < 6$
$t + k > 4$
$t > k$

Both t and k are positive integers. What is the value of t? Multiple choice: 1, 2, 3, 4, or 5

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I'm confused by the way this is worded... It seems to say that there is only one value of t which satisfies this system. But if I graph this, I get an entire region, not just one value? In fact, it would seem if I graph this I get no overlap of shaded regions. I don't get it.

I guess one way is to just start trying numbers, but there must be a better way?
• Nov 20th 2010, 06:12 PM
dwsmith
t can't be 1 or 2 by the 2nd and 3rd inequality. Also, k > 4-t which tell us t can't be 4 so t is 1, 2, or 3 but we know t isn't 1 or 2.
• Nov 20th 2010, 06:33 PM
Wilmer
Quote:

Originally Posted by Reiner
$t^2 - k^2 < 6$ [1]
$t + k > 4$ [2]
$t > k$ [3]

[1] (t + k)(t - k) < 6 [1A]
[2] t + k = 5 minimum [2A]

Substitute [2A] in [1A]:
5(t - k) < 6
So t - k < 2

[3] t > k
So t = 3, k = 2