Solve the inequality x^2 - 2x > 3.

Step-by-step solution

1. 1

   Move everything to one side. A sign analysis only works once the inequality compares an expression with zero:

   $x^2-2x-3>0$

2. 2

   Factor the quadratic. Two numbers with product $-3$ and sum $-2$ are $-3$ and $1$:

   $x^2-2x-3=(x-3)(x+1)>0$

3. 3

   Locate the critical points. The factors vanish at $x=3$ and $x=-1$, splitting the line into three intervals: $(-\infty,-1)$, $(-1,3)$ and $(3,\infty)$.

4. 4

   Determine the sign in each interval. For $x<-1$ both factors are negative, so the product is positive; between the roots the factors have opposite signs, so the product is negative; for $x>3$ both are positive. The same conclusion follows from the shape: an upward parabola is above the axis outside its roots.

5. 5

   Write the solution and check. We want the product positive, so

   $x<-1\quad\text{or}\quad x>3,\qquad\text{i.e. }(-\infty,-1)\cup(3,\infty)$

   Test $x=-2$: $4+4=8>3$ $\checkmark$. Test $x=0$: $0>3$ false $\checkmark$. Test $x=4$: $16-8=8>3$ $\checkmark$. The endpoints are excluded because there $x^2-2x=3$ exactly.