Find all values of t such that [t] =2t+3. If you find more than one value, then list the values you find in increasing order, separated by commas. The brackets represent the floor sign function.
-7/2 and -3, in that respective order.
I can not.
n</= 2t+3 ???
is that not any n equal to or greater than -3
No it is equal
⌊t⌋ = 2t+3
Clearly, t must be a multiple of 1/2
If an odd multiple, then we have
t = (2k+1)/2
⌊t⌋ = k
2t = 2k+1
k = 2k+1+3
k = -4
t = -7/2
If t is an even multiple of 1/2, then
⌊t⌋ = 2k/2 = k
2t = 2k
k = 2k+3
k = -3
t = -3
See the graphs at
http://www.wolframalpha.com/input/?i=%E2%8C%8At%E2%8C%8B+%3D+2t%2B3
Oh, I see you want to find the values of t that satisfy the equation [t] = 2t + 3. Well, let's dive right into it!
Now, just to clarify, the floor sign function [t] gives you the greatest integer that is less than or equal to t. So, let's break it down.
Suppose t is a positive integer. In that case, [t] is simply t itself. So, the equation becomes t = 2t + 3. Solving for t, we get t = -3, which is not a positive integer. So, no luck here.
Next, let's consider t as a negative integer. In this case, [t] is equal to the next greatest integer, which will be t - 1. So, the equation becomes t - 1 = 2t + 3. If we solve this equation, we find t = -4.
Now, if we plug in t = -4 into the original equation [t] = 2t + 3, we get [-4] = 2(-4) + 3. Simplifying this, we find -4 = -5. But hey, that's not true!
So, unfortunately, there is no value of t that satisfies the equation [t] = 2t + 3. I apologize if you were expecting a more exciting answer, but sometimes math doesn't have a clown-worthy punchline.
To find all values of t that satisfy the equation [t] = 2t + 3, where [t] represents the floor sign function, we need to consider the different cases when t is an integer and when t is not an integer.
Case 1: t is an integer
When t is an integer, the floor sign function [t] is equal to t. Therefore, the equation becomes t = 2t + 3.
Simplifying the equation, we have:
t = 2t + 3
t - 2t = 3
-t = 3
t = -3
So, when t is an integer, the only value that satisfies the equation is t = -3.
Case 2: t is not an integer
When t is not an integer, the floor sign function [t] is equal to the greatest integer less than or equal to t. We can rewrite the equation as follows:
[t] = 2t + 3
Let n be the greatest integer less than or equal to t. We can express t as:
t = n + r
where r is the decimal part of t (0 <= r < 1).
According to the properties of the floor sign function, we have:
n <= t < n + 1.
Substituting t = n + r into the equation, we get:
[n] = 2(n + r) + 3.
Since n is an integer, [n] is equal to n. Simplifying the equation, we have:
n = 2(n + r) + 3
n = 2n + 2r + 3
-n = 2r + 3
2r = -n - 3
r = (-n - 3)/2.
Since r must be between 0 and 1, we have:
0 <= (-n - 3)/2 < 1.
Simplifying this inequality, we get:
0 <= -n/2 - 3/2 < 1
-3/2 < -n/2 < 1/2.
To satisfy the inequality, we need to consider two cases:
Case 2.1: -3/2 < -n/2
Multiply both sides of the inequality by 2 (note that since 2 is positive, the direction of the inequality remains the same):
-3 < -n.
To find the upper bound for n, we consider the inequality -n < 2 and multiply both sides by -1 (note that since -1 is negative, the direction of the inequality changes):
n > -2.
Combining the inequalities, we have:
-3 < -n < -2.
Since n is an integer, the possible values for n are -2 and -1.
Case 2.2: -n/2 < 1/2
Multiplying both sides of the inequality by 2 (note that since 2 is positive, the direction of the inequality remains the same), we get:
-n < 1.
To find the upper bound for n, we consider the inequality -n < 0 and multiply both sides by -1 (note that since -1 is negative, the direction of the inequality changes):
n > 0.
Combining the inequalities, we have:
0 < n < 1.
Since n is an integer, the possible value for n is 0.
Putting all the values of n together, we have -2, -1, and 0 as the possible values for n.
Finally, we substitute these values of n back into the equation t = n + r to find the corresponding values of t:
For n = -2, t = -2 + r.
For n = -1, t = -1 + r.
For n = 0, t = 0 + r.
Therefore, the values of t that satisfy the equation are:
t = -2 + r, t = -1 + r, and t = 0 + r for -2 <= n <= -1, n = 0, and 0 <= r < 1.
In conclusion, the values of t that satisfy the equation [t] = 2t + 3 are t = -3, t = -1 + r, and t = 0 + r, where r is a decimal between 0 and 1.