Solve x^2-x-1<0

Let us first factorize $x^2-x-1$

For that let $x^2-x-1=0$

$⇒x=\frac{1±\sqrt{1+4}}{2}=\frac{1±\sqrt{5}}{2}$

Now on the number line (x-axis), mark $x=\frac{1±\sqrt{5}}{2}$

From the sign scheme of $x^2-x-1$, which is shown in the figure, we have

$x^2-x-1<0$

$⇒x∈\left(\frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2}\right)$