E = {n ∈ N | (3 - 4)² + 3n ≤ 291}
Answer (1)
To solve the given problem, we start by understanding what the expression E = { n ∈ N ∣ ( 3 − 4 ) 2 + 3 n ≤ 291 } means. It describes a set E containing all natural numbers n that satisfy the inequality.
First, let's simplify the expression ( 3 − 4 ) 2 .
Calculate ( 3 − 4 ) 2 : ( 3 − 4 ) = − 1 and therefore tex ^2 = 1].
Substitute this back into the inequality: 1 + 3 n ≤ 291 .
We need to find (n[/tex] such that the inequality holds, so subtract 1 from both sides of the inequality: 3 n ≤ 290 .
Divide both sides by 3 to isolate n : n ≤ 3 290 .
Calculate the division: 3 290 ≈ 96.67 . Since n is a natural number, n must be less than or equal to 96.
Therefore, the solution set E is {1, 2, 3, \ldots, 96}.
So, all natural numbers from 1 to 96 inclusive are part of the set E . This is because these are the values of n that satisfy the condition ( 3 − 4 ) 2 + 3 n ≤ 291 .
