Question

N is an integer such that 3n 4 ? 19 and 10n n� 16 > 1 find all the possible values of n. note: please make your final answer clear by typing n =

Answer 1

Solving the inequalities 3n + 4 ≤ 19 and 10n - n² + 16 > 1 for integer solutions, we find that n can only take the values 0, 1, 2, or 3.

Step-by-step explanation:

The question involves solving two inequalities to find the possible integer values of n. The first inequality is 3n + 4 ≤ 19, and the second inequality is 10n - n² + 16 > 1. We solve these step by step:

• For the first inequality, we subtract 4 from both sides to get 3n ≤ 15, and then divide by 3 to find n ≤ 5.
• For the second inequality, we simplify it to 10n - n² + 15 > 0. To find the intervals where this inequality holds, we need to factor if possible or find the roots by other methods.

To factor the quadratic expression, we can look for two numbers that multiply to +15 and add up to +10. These numbers are 5 and 3. Therefore, we can express the quadratic as (n - 5)(n - 3) > 0. This inequality holds when n is outside the interval [3, 5]. However, we must consider only integer values of n.

Now, combining both inequalities and considering only integer solutions, we get that n must be an integer less than or equal to 5 due to the first inequality, but not between 3 and 5 inclusive due to the second inequality. Hence, the possible integer values for n are 0, 1, 2, and 3. Summarizing, n = 0, 1, 2, or 3.

Answer 2

The possible values for n for the compound inequalities 3n + 4 ≤ 19 and
`(10n)/(n^2+16) > 1` can be expressed as 2 < n ≤ 5.

Solving compound inequalities.

The solution to the compound inequalities 3n + 4 ≤ 19 and
`(10n)/(n^2+16) > 1` can be determined by solving the inequalities separately and finding out the critical points if necessary.

From 3n + 4 ≤ 19

3n ≤ 19 - 4

3n ≤ 15

n ≤ 15/3

n ≤ 5

From
`(10n)/(n^2+16) > 1`, let multiply both sides with n² + 16, we get:

10 n > n² + 16

n² - 10n + 16 < 0

By factorization;

(n - 8) (n - 2) < 0

The roots of the equation is n = 2 and n = 8. Now, the solution to the inequality can be written as 2 < n < 8.

Therefore, the possible values for n for the compound inequalities 3n + 4 ≤ 19 and
`(10n)/(n^2+16) > 1` can be expressed as 2 < n ≤ 5.

The complete question

n is an integer such that 3n + 4 ≤ 19 and
`(10n)/(n^2+16) > 1`. Find all the possible values of n.