Use a difference graph, two graphs or a factored sign line

Move the right side to the left: $(x^2-9)-(x-3)(5-x^2)>0$. The difference graph has roots −2,1,3. It is above the x-axis for −2<x<1 or x>3.

Sketch y=x²−9 and $y=(x-3)(5-x^2)$. They intersect at x=−2,1,3. Select the intervals where the first graph is above the second: −2<x<1 or x>3.

$(x-3)(x+3)-(x-3)(5-x^2)>0$ becomes $(x-3)[(x+3)-(5-x^2)]>0$, then $(x-3)(x^2+x-2)>0$, hence $(x-3)(x-1)(x+2)>0$. Signs between −2,1,3 are −,+,−,+. Thus the solution set is $\{x\in\mathbb R:-2<x<1\text{ or }x>3\}$.

The source’s Equivalence note connects algebraic and graphical representations: all three methods describe the same input set. Its speech bubble asks whether x−3 can simply be divided out. It cannot be cancelled without cases: x−3 has unknown sign, and dividing reverses the inequality for x<3. At x=3 the original strict inequality is false. The source asks for a set, so present the result in set notation; its remarks allow inequality, interval or set notation for a request phrased as a range.