Answer:
x within the domain (-∞,0) ∪ (0,1/12) ∪ (1/6,∞)
Step-by-step explanation:
Let's find the solution by modifying the conditions like this:
First condition:
2/3x > 8
Notice that x needs to be a positive number because if x is negative number the expression 2/3x is always negative, and the given condition will be false. Also, x can't be 0 because dividing by 0 is not allowed.
Multiplying the first condition in both sides by (3x/2) we obtain:
(2/3x)*(3x/2) > 8*(3x/2) equal to:
1 > 12x, which dividing by 12 in both sides is:
1/12 > x, which means a domain: (0,1/12)
Second condition:
2/3x < 4
Notice that if x is negative, the expression (2/3x) is always negative, so the second condition will be always true, so negative numbers are not a problem for this conditions.
Let's find the highest positive number which allows the second condition to be a true statement.
Multiplying the second condition in both sides by (3x/2) we obtain:
(2/3x)*(3x/2) < 4*(3x/2) equal to:
1 < 6x, which dividing by 6 in both sides is:
1/6 < x which means a domain: (-∞,0) ∪ (1/6,∞), notice that 0 is not in the domain because dividing by 0 is not allowed, but also as mentioned before negative numbers makes the statement true.
Notice that the problem uses the word OR (condition one or condition two), which means that we need to create a domain for x which allows that always at least one condition is true.
Adding the x domains for the two conditions we obtain:
(-∞,0) ∪ (0,1/12) ∪ (1/6,∞); notice that all the boundary numbers are not included.
In conclusion, the statement 2/3x>8 or 2/3x<4 is true for x within the domain (-∞,0) ∪ (0,1/12) ∪ (1/6,∞).
