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

Step-by-step solution

1. 1

   Keep it factored. A product is positive when both factors share the same sign and negative when they differ. Expanding to $x^2-2x-3$ would throw away exactly the information that makes this easy.

2. 2

   Find the zeros of each factor.

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

   The product can only change sign at these two points, so they cut the line into $(-\infty,-1)$, $(-1,3)$ and $(3,\infty)$.

3. 3

   Test one value in each interval. At $x=-2$: $(-1)(-5)=5>0$ (both negative). At $x=0$: $(1)(-3)=-3<0$ (signs differ). At $x=4$: $(5)(1)=5>0$ (both positive). The pattern is positive, negative, positive.

4. 4

   Confirm the pattern from the shape of the graph. Expanded, the leading coefficient is $+1$, so the parabola opens upward: it dips below the axis between its roots and rises above outside them. That is exactly the sign pattern found by testing — a useful independent check.

5. 5

   Select the intervals and exclude the roots. The inequality is strict, so the points where the product is exactly $0$ do not count:

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

6. 6

   Verify by scanning. Evaluating the product at $2001$ points from $-10$ to $10$ and comparing with the claimed solution set gives agreement at every single point ✓.