2008 iTest Problems/Problem 23
Problem
Find the number of positive integers 
that are solutions to the simultaneous system of inequalities
.Solution
Solve the first inequality to get
![\[4n-18 < 2008\]](http://latex.artofproblemsolving.com/3/1/3/3136b76e570da21c3f298dbef92e8de7b3e1b6af.png)
![\[4n < 2026\]](http://latex.artofproblemsolving.com/0/1/7/0171d501178e73416c6b626d7b4d343798254712.png)
![\[n < 506 \frac{1}{2}\]](http://latex.artofproblemsolving.com/a/4/9/a496a009813ca69829dac9350cf5166fdbea67c8.png)
Solve the second inequality to get
![\[7n+17>2008\]](http://latex.artofproblemsolving.com/2/3/b/23bc1b94c79c8848c765850363a2cb16cccc6d4e.png)
![\[7n>1991\]](http://latex.artofproblemsolving.com/7/2/f/72f7c777afd504b1b6f7fdacf710aadcb09be630.png)
![\[n > 284 \frac{3}{7}\]](http://latex.artofproblemsolving.com/3/1/6/316e9653edb4d4d01ef3920bcc6fdb82d142b8e7.png)
The integers in the range are 
, so there are 
integral solutions.
See Also
| 2008 iTest (Problems) | ||
| Preceded by: Problem 22 |
Followed by: Problem 24 | |
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