Answer:
To find the range of values of x for which 1-x < (x-1)(5-x) < 3, we can solve the two inequalities separately.
First, let’s solve 1-x < (x-1)(5-x). Expanding the right side gives 1-x < 5x - x^2 - 5 + x. Simplifying and rearranging gives x^2 - 7x + 6 > 0. Factoring gives (x-1)(x-6) > 0. This inequality is satisfied when x < 1 or x > 6.
Next, let’s solve (x-1)(5-x) < 3. Expanding the left side gives 5x - x^2 - 5 - x < 3. Simplifying and rearranging gives x^2 - 6x + 8 < 0. Factoring gives (x-2)(x-4) < 0. This inequality is satisfied when 2 < x < 4.
Combining the two solutions, we find that the range of values of x for which 1-x < (x-1)(5-x) < 3 is satisfied is when 2 < x < 4.
Explanation: