For part(a) I have worked out that the answer is -3.4 by reading the value for x on the graph where y = -5. For part(b), what I did was to draw the line of y = -5x onto the graph and where this linear graph intersects with cubic graph (at x = -2.5) that is the solution.
I am struggling to know the intuition behind that. Why do I need to draw that linear graph and why does it give the solution to the cubic equation in part(b). Looking for intuition here.
Well, let $f(x)=x^3-0.2x^2-9x+7$ and let $g(x)=-5x$. Then, the first one has a graph as the one shown in your picture while the other one is a straight line passing through $O(0,0)$ and $A(1,-5)$. Note that, in order to find where these two curves intersect, you have to find any $x$ such that the points $F(x,f(x))$ and $G(x,g(x))$ coincide, that is, we have o find any $x$ such that $f(x)=g(x)$. Note that: $$f(x)=g(x)\Leftrightarrow x^3-0.2x^2-9x+7=-5x\Leftrightarrow x^3-0.2x^2-4x+7=0,$$ which is the equation you are requested to solve. So, the interpretation you request is that the solution is the intersection point of the two curves.