What is (-1)/2 <| x/((x^2-x-1))|<2 ?

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(-1)/2 <| x/((x^2-x-1))|<2

\frac{- 1}{2} < \frac{x}{( x ^{2} - x - 1 )} < 2

x < \frac{1 - 17}{4} or - \frac{1}{2} < x < \frac{1 + 17}{4} or x > 2

Interval Notation

(- \infty, \frac{1 - 17}{4}) \cup (- \frac{1}{2}, \frac{1 + 17}{4}) \cup (2, \infty)

Decimal Notation

x < - 0.78077 \dots or - 0.5 < x < 1.28077 \dots or x > 2

Solution steps

\frac{- 1}{2} < \frac{x}{( x ^{2} - x - 1 )} < 2

a < u < b

then

a < u and u < b

\frac{- 1}{2} < \frac{x}{( x ^{2} - x - 1 )} and \frac{x}{( x ^{2} - x - 1 )} < 2

\frac{- 1}{2} < \frac{x}{x ^{2} - x - 1} : x < \frac{1 - 5}{2} or \frac{1 - 5}{2} < x < \frac{1 + 5}{2} or x > \frac{1 + 5}{2}

\frac{x}{x ^{2} - x - 1} < 2 : x < \frac{1 - 17}{4} or - \frac{1}{2} < x < \frac{1 + 17}{4} or x > 2

Combine the intervals

(x < \frac{1 - 5}{2} or \frac{1 - 5}{2} < x < \frac{1 + 5}{2} or x > \frac{1 + 5}{2}) and (x < \frac{1 - 17}{4} or - \frac{1}{2} < x < \frac{1 + 17}{4} or x > 2)

Merge Overlapping Intervals

x < \frac{1 - 17}{4} or - \frac{1}{2} < x < \frac{1 + 17}{4} or x > 2

- 1 < \frac{( x + 2 )}{( x - 1 )} \leq 1

x + 1 < 3 x - 5 < 4 x - 7

- 12 < 4 - 3 \frac{x}{- 5} \leq 2

44 < x < 45

25 < x < 35

What is (-1)/2 <| x/((x^2-x-1))|<2 ?
The solution to (-1)/2 <| x/((x^2-x-1))|<2 is x<(1-sqrt(17))/4 \lor-1/2 <x<(1+sqrt(17))/4 \lor x>2