Story problem in algebra

Question

When a fraction has a numerator that is one less than twice an unknown number and a denominator that is that unknown number, the fraction will be greater than the unknown number increased by four. Find the solutions that satisfy this inequality.

Answer 1

2x - 1
-------- > x + 4
x


2x - 1
------ - x - 4 > 0
x

let f(x) = (2x-1)/x - x- 4

We are interested in values of x
where f(x)>0

The zeros of this function are

(2x-1)/x - x - 4 = 0

(2x-1)/x = x+4

2x - 1 = x(x+4)

2x - 1 = x^2 + 4x

0 = x^2 + 4x - 2x + 1

0 = x^2 + 2x + 1

0 = ( x + 1 )( x + 1 )

x = -1

There is a vertical asymptope at x=-1

x< -1

-1 < x < 0

x>0

Examining the graph of f(x) or using sample test values,
f(x) > 0 when x<-1 or -1<x<0