Final answer:
To find the values of x for which 2x and 3x are between 1 and 2, we solve two inequalities and look at the overlapping interval. The overlapping interval where both conditions are met is (1/2, 2/3). Hence, x must be between 1/2 and 2/3 for both 2x and 3x to be in the interval (1,2).
Step-by-step explanation:
The question is asking to find all values of x for which both 2x and 3x fall within the interval (1,2). To find these values, we need to set up two separate inequalities because 2x and 3x need to be greater than 1 and also less than 2 simultaneously.
For 2x, we have:
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- 1 < 2x < 2
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- 0.5 < x < 1 (Dividing each part by 2)
For 3x, we have:
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- 1 < 3x < 2
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- 1/3 < x < 2/3 (Dividing each part by 3)
To satisfy both inequalities at the same time, x must be in the overlap of these two intervals. Therefore, upon comparing the two intervals (0.5 < x < 1) and (1/3 < x < 2/3), we can see that the overlap is (1/2 < x < 2/3) since this is the only section where both conditions are true simultaneously.
Therefore, the solution to the problem is that x must be in the interval (1/2, 2/3) for both 2x and 3x to be in the interval (1,2).