If n is an integer, how many different values of n satisfy the inequality
-4 ≤ 3n ≤ 87?
a) 32
b) 31
c) 30
d) 29
e) 28
i got (b) 31 but I'm not sure.
-4 ≤ 3n ≤ 87
divide by 3
-4/3 ≤ n ≤ 29
So you would have all the integers from -1 to 29
from 1 to 29 would be 29 of them, plus the 0 and the -2 makes a total of 31
You are correct.
Thank you so much
Well, if 3n is between -4 and 87, we can start by finding the range of n.
Divide both sides of the inequality by 3:
-4 ≤ 3n ≤ 87
-4/3 ≤ n ≤ 87/3
Now, let's find the number of integers between -4/3 and 87/3. Remember, we want the number of different values of n.
The easiest way to approach this is to subtract the two endpoints and then add 1 to account for inclusive counting. So:
87/3 - (-4/3) + 1 = (87 + 4)/3 + 1 = 91/3 + 1 = 30 + 1 = 31
Therefore, there are 31 different values of n that satisfy the inequality. So, your answer of (b) 31 is correct! Well done!
To find the number of different values of n that satisfy the inequality -4 ≤ 3n ≤ 87, we need to solve it step-by-step.
The inequality -4 ≤ 3n can be rewritten as:
-4/3 ≤ n
Similarly, the inequality 3n ≤ 87 can be rewritten as:
n ≤ 87/3
To find the overlapping range of values that satisfy both inequalities, we need to find the intersection of the two solution sets.
The range of values for n that satisfy -4/3 ≤ n is (-∞, -4/3] (where -∞ indicates negative infinity).
The range of values for n that satisfy n ≤ 87/3 is [-∞, 87/3] (where -∞ indicates negative infinity).
Now, to find the overlapping region, we need to find the minimum value between -4/3 and 87/3.
Since -4/3 is smaller than 87/3, the minimum value is -4/3.
Hence, the range of values for n that satisfy both inequalities is (-∞, -4/3].
However, since n is an integer, the set of values satisfying the inequality is the set of integers greater than or equal to -2.
Therefore, we can conclude that there are 31 different values of n that satisfy the inequality.
The correct answer is therefore option (b) 31.
To find the number of different values of n that satisfy the inequality -4 ≤ 3n ≤ 87, you need to determine the range of possible values for n.
Let's break down the inequality into two separate inequalities:
-4 ≤ 3n ........... (1)
3n ≤ 87 ........... (2)
First, let's solve inequality (1) for n:
Divide both sides of the inequality by 3 to isolate n:
-4/3 ≤ n
Next, let's solve inequality (2) for n:
Divide both sides of the inequality by 3 to isolate n:
n ≤ 87/3
Now, let's consider the intersection of the solutions in both inequalities. We want to find the common values of n that satisfy both -4/3 ≤ n and n ≤ 87/3.
From inequality (1), we have n ≥ -4/3.
From inequality (2), we have n ≤ 87/3.
To find the range of possible values for n, we need to find the overlap between these two inequalities. The overlap occurs when -4/3 ≤ n ≤ 87/3.
Now, let's find the number of integers within this range.
The smallest possible integer value that satisfies this range is the ceiling of -4/3, which is -1. The largest possible integer value that satisfies this range is the floor of 87/3, which is 29.
Therefore, there are 31 different values of n that satisfy the inequality -4 ≤ 3n ≤ 87.
Hence, the correct answer is option b) 31.