The solutions of the equations are:
53) x = - 7 / b. If - 7 / b > 0, then b < 0: b ∈ (- ∞, 0).
54) x = 3 / 4 - a. If 3 / 4 - a > 0, then a < 3 / 4: a ∈ (- ∞, 3 / 4).
55) x = - 6.5 · c. If - 6.5 · c > 0, then c < 0: c ∈ (- ∞, 0).
56) x = - b · (a / c). If - b · (a / c) > 0 and b > 0, then a / c < 0: (a > 0 and c < 0) or (a < 0 and c > 0), but - b · (a / c) > 0 and b < 0, then a / c > 0: (a > 0 and c > 0) or (a < 0 and c < 0): 1) [b ∈ (0, + ∞) ∩ [[a ∈ (0, + ∞) ∩ b ∈ (- ∞, 0)] ∪ [a ∈ (- ∞, 0) ∩ b ∈ (0, + ∞)]]] ∪ [b ∈ (- ∞, 0) ∩ [[a ∈ (0, + ∞) ∩ b ∈ (0, + ∞)] ∪ [a ∈ (- ∞, 0) ∩ b ∈ (- ∞, 0)]]]
What is the solution of a function such that its domain is the set of all positive numbers?
In this problem we must clear the variable x within each expression and determine the possible values of constants a, b, c such that x is a positive number. Now we proceed to resolve on each equation:
53) b · x = - 7
x = - 7 / b
If - 7 / b > 0, then b < 0: b ∈ (- ∞, 0).
54) x + a = 3 / 4
x = 3 / 4 - a
If 3 / 4 - a > 0, then a < 3 / 4: a ∈ (- ∞, 3 / 4).
55) - x / c = 6.5
x = - 6.5 · c
If - 6.5 · c > 0, then c < 0: c ∈ (- ∞, 0).
56) (c / a) · x = - b
x = - b · (a / c)
If - b · (a / c) > 0 and b > 0, then a / c < 0: (a > 0 and c < 0) or (a < 0 and c > 0), but - b · (a / c) > 0 and b < 0, then a / c > 0: (a > 0 and c > 0) or (a < 0 and c < 0): 1) [b ∈ (0, + ∞) ∩ [[a ∈ (0, + ∞) ∩ b ∈ (- ∞, 0)] ∪ [a ∈ (- ∞, 0) ∩ b ∈ (0, + ∞)]]] ∪ [b ∈ (- ∞, 0) ∩ [[a ∈ (0, + ∞) ∩ b ∈ (0, + ∞)] ∪ [a ∈ (- ∞, 0) ∩ b ∈ (- ∞, 0)]]]
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