What is the sum of all integers $n$ that satisfy $$-3n < 7n-20 < 3n~?$$

$($In other words, what is the sum of all integers $n$ that satisfy both of the inequalities
$$-3n<7n-20\text{~~and~~}7n-20&l t;3n~?)$$

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Looks like the inequality is

-3n < 7n - 20 < 3n

Add 3n to each side:

0 < 10n - 20 < 6n

Solve the right inequality.

10n - 20 < 6n

Add -10n to both sides:

-20 < -4n

Divides both sides by -4:

5 > n

Now solve the left inequality, and take the intersection of the two solution intervals.

0 < 10n - 20

Add 20 to both sides:

20 < 10n

Divide both sides by 10:

2 < n

So 2 < n < 5. There are only 2 integers in this range (3 and 4), whose sum is 7.

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