Solve the inequality (x - 1)(x - 3) > 0.

Step-by-step solution

1. 1

   Use the fact that the expression is already factored. A product of real numbers is positive exactly when both factors share a sign, so instead of expanding you only need to know where each factor changes sign.

2. 2

   Find the critical points. Setting each factor to zero:

   $x-1=0\Rightarrow x=1,\qquad x-3=0\Rightarrow x=3$

   These two values cut the number line into three test intervals: $(-\infty,1)$, $(1,3)$ and $(3,\infty)$.

3. 3

   Build the sign chart. Pick one test value in each interval and record the sign of each factor:

   interval	$x-1$	$x-3$	product
   $(-\infty,1)$, e.g. $x=0$	$-$	$-$	$+$
   $(1,3)$, e.g. $x=2$	$+$	$-$	$-$
   $(3,\infty)$, e.g. $x=4$	$+$	$+$	$+$

   The test values give $(0-1)(0-3)=3$, $(2-1)(2-3)=-1$ and $(4-1)(4-3)=3$, confirming the last column.

4. 4

   Select the intervals matching the inequality. You need the product greater than zero, so take the two $+$ rows:

   $x<1\quad\text{or}\quad x>3$

5. 5

   Decide about the endpoints. The inequality is strict ($>0$, not $\ge 0$), and at $x=1$ and $x=3$ the product equals $0$. Both roots are therefore excluded and the intervals are open:

   $(-\infty,1)\cup(3,\infty)$

6. 6

   Cross-check with the parabola. Expanding gives $y=x^{2}-4x+3$, an upward parabola with vertex at $x=2$, $y=-1$. An upward parabola lies above the axis outside its roots — exactly the answer obtained.