2008 iTest Problems/Problem 23

Problem

Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities

$4n-18 < 2008$
$7n + 17 > 2008$.

Solution

Solve the first inequality to get \[4n-18 < 2008\] \[4n < 2026\] \[n < 506 \frac{1}{2}\] Solve the second inequality to get \[7n+17>2008\] \[7n>1991\] \[n > 284 \frac{3}{7}\] The integers in the range are $285,286,287 \cdots 505,506$, so there are $\boxed{222}$ integral solutions.


See Also

2008 iTest (Problems)
Preceded by:
Problem 22
Followed by:
Problem 24
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