Answer:
(a) m < n: sometimes
(b) m = n: sometimes
(c) m > n: sometimes
Explanation:
You want to know whether the equation mx +4 = 0.5nx +8 always, sometimes, or never has a solution for different relations between m and n.
Solution
The solution to the equation is ...
mx +4 = 0.5nx +8
x(m -0.5n) = 4
x = 4/(m -0.5n) = 8/(2m -n)
There is no solution to the equation for 2m -n = 0, or 2m = n.
(a) m < n
There is a solution in every case except when 2m = n. Such a case can exist in this domain when ...
m < 2m
m > 0 . . . . . . . subtract m from both sides
There will be solutions to the equation except when 2m = n > 0.
There are "sometimes" solutions to the equation in this domain.
(b) m = n
The case of 2m = n will exist in this domain when ...
m = 2m
0 = m . . . . . subtract m
There will be solutions to the equation except when m = n = 0.
There are "sometimes" solutions to the equation in this domain.
(c) m > n
The case of 2m = n will exist in this domain when ...
m > 2m
m < 0 . . . . . . . subtract m from both sides
There will be solutions to the equation except when 2m = n < 0.
There are "sometimes" solutions to the equation in this domain.
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Additional comment
Here are some specific instances of solutions and not.
(a) 0 = m < n = 2, solution x = 8/(2·0 -2 = -4. 1 = m < n = 2, no solution x = 8/(2·1-2) = 8/0 (undefined)
(b) 1 = m = n, solution x = 8/(2·1 -1) = 8. 0 = m = n, no solution x = 8/(2·0-0) = 8/0 (undefined)
(c) 0 = m > n = -2, solution x = 8/(2·0 -(-2)) = 4. -1 = m > n = -2, no solution x = 8/(2(-1)-(-2)) = 8/0 (undefined)
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