Which of the following describes the values of x for which `(1-5x)/(x^(2)+1)` is negative ?

Options

A. `x gt 0`
B. `x gt (1)/(5)`
C. `x lt (1)/(5)`
D. `0 lt x lt (1)/(5)`

Answer

B

Explanation

Since the square of any real number is nonnegative, `x^(2)` is nonnegative. Then `x^(2)+1` is the sum of the nonnegative quantity `x^(2)` and the positive quantity 1. So `x^(2)+1` is positive. The fraction `(1-5x)/(x^(2)+1)` will be negative when the numerator `1-5x` is negative. Solve the inequality `1-5x lt 0` :
`1-5x lt 0`
`1 lt 5x`
`(1)/(5) lt x`
Thus, the fraction `(1-5x)/(x^(2)+1)` will be negative when and only when `x ge (1)/(5)`.