Solve `x^2-x-1 lt 0`

Explanation

Let us first factorize `x^(2)-x-1.`
For that let `x^(2)-x-1=0`
`implies x=(1+-sqrt(1+4))/(2)=(1+-sqrt(5))/(2)`
Now on the number line (x-axis), mark `x=(1+-sqrt(5))/(2)`

From the sign scheme of `x^(2)-x-1`, which is shown in the figure
we have
`x^(2)-x-1 lt 0`
`implies x in ((1-sqrt(5))/(2),(1+sqrt(5))/(2))`