$$\frac{2x - 5}6 < \frac{x^2 - 1}4 - \frac{x(x+1)}3$$
Should I solve it by turning it firstly into an equation of the form $ax^2+bx+c>0$ and then draw the graph or draw the graph of $\displaystyle y = \frac{2x - 5}6$ and in the same figure the one of $\displaystyle y=\frac{x^2 - 1}4 - \frac{x(x+1)}3$ and then watch where the equation is right?
You can do either way. I think the first one is simpler.
Rewriting we have $$4x - 10 < 3(x^2 - 1) - 4x(x + 1) \implies x^2 + 8x - 7 < 0 \implies (x + 4)^2 - 23 < 0$$
It's very easy to draw that parabola: you first draw $x^2$, then apply the two translations $x \mapsto x + 4$ and $y \mapsto y + 23$ and you are done.