The numbers 3, 6, 12, 14 have frequencies 𝑥+6x+6, 𝑥+2x+2, 𝑥−3x−3 and 𝑥x respectively. If their mean is 5.97, find the value of 𝑥x. (Enter the value as next highest integer)

The mean of a set of numbers is calculated by dividing the sum of the numbers by the total count of the numbers. In this case, the mean is given as 5.97 and we know that the frequencies of the numbers are given by expressions in terms of x.

We can set up the equation for the mean as follows:

(3*(x+6) + 6*(x+2) + 12*(x-3) + 14*x) / (4x + 5) = 5.97

Solving this equation will give us the value of x.

First, simplify the numerator:

3x + 18 + 6x + 12 + 12x - 36 + 14x = 35x - 6

Then, substitute this back into the equation:

(35x - 6) / (4x + 5) = 5.97

Cross-multiply to get rid of the fraction:

5.97 * (4x + 5) = 35x - 6

23.88x + 29.85 = 35x - 6

Rearrange the equation to isolate x:

23.88x - 35x = -6 - 29.85

-11.12x = -35.85

Finally, solve for x:

x = -35.85 / -11.12

x = 3.22

Since the question asks for the next highest integer, we round 3.22 up to 4. So, the value of x is 4.