Question

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
$$-3n<7n-20\text{~~and~~}7n-20<3n~?)$$

If you can't figure out the latex here is a screenshot ( in the comments)

Answer 1

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.

Answer 2

7

Explanation:

We solve each of the two inequalities separately. To solve the first inequality, -3n<7n-20, we add 3n to both sides:

0<10n-20, then we add 20 to both sides to get

20<10n. Finally, dividing both sides by 10 gives 2<n, so n must be greater than 2.

For the second inequality, 7n-20<3n, we subtract 3n from both sides:

4n-20<0, then add 20 to both sides to get

4n<20. Finally, dividing both sides by 4 gives n<5, so n must be less than 5.

The only integers that are greater than 2 and less than 5 are 3 and 4. So, the sum of all possible integer values of n is 3+4=7.