Given that x^2-5x+1 lies between -5 and 15, find the range of values of x?

We can find the ranges of values for the two conditions separately, then combine results.
.. x^2 - 5x + 1 > -5
.. (x - 2.5)^2 - 5.25 > - 5 ... complete the square
.. (x - 2.5)^2 > 0.25 ... add 5.25
.. |x - 2.5| > 0.5 ... square root
.. -0.5 > x - 2.5 > 0.5 ... rewrite the absolute value inequality
.. 2 > x > 3 ... add 2.5. These are disjoint intervals.

Replacing the above constraint with the other one, we have
.. (x - 2.5)^2 - 5.25 < 15
.. (x - 2.5)^2 < 20.25 ... add 5.25
.. |x - 2.5| < 4.5 ... square root
.. -4.5 < x - 2.5 < 4.5 ... rewrite the absolute value equation
.. -2 < x < 7 ... add 2.5

The range of values of x is between -2 and 2 and again between 3 and 7.