Given [tex]\( n \)[/tex] is an integer, write the values of [tex]\( n \)[/tex] such that [tex]\(-15 \ \textless \ 3n \leq 6\)[/tex].



Answer :

To determine the integer values of [tex]\( n \)[/tex] that satisfy the inequality [tex]\( -15 < 3n \leq 6 \)[/tex], we need to follow these steps:

1. Divide the entire inequality by 3 to isolate [tex]\( n \)[/tex]:

[tex]\[ -15 < 3n \leq 6 \][/tex]

Dividing everything by 3, we get:

[tex]\[ \frac{-15}{3} < \frac{3n}{3} \leq \frac{6}{3} \][/tex]

Simplifying this, we have:

[tex]\[ -5 < n \leq 2 \][/tex]

2. Find the integer values of [tex]\( n \)[/tex] within this range:

We need to find all integers [tex]\( n \)[/tex] that are greater than [tex]\(-5\)[/tex] and less than or equal to [tex]\( 2 \)[/tex]. These are the values of [tex]\( n \)[/tex] that satisfy the inequality:

[tex]\[ -5 < n \leq 2 \][/tex]

Listing the integers within this range, we start from the smallest integer greater than [tex]\(-5\)[/tex] and go up to [tex]\(2\)[/tex]:

[tex]\[ -4, -3, -2, -1, 0, 1, 2 \][/tex]

Therefore, the integer values of [tex]\( n \)[/tex] that satisfy the inequality [tex]\( -15 < 3n \leq 6 \)[/tex] are:
[tex]\[ n \in \{-4, -3, -2, -1, 0, 1, 2\} \][/tex]