Solve the inequality x^2 - 2x - 3 < 0.

Step-by-step solution

1. 1

   Factor before doing anything else. Look for two numbers with product $-3$ and sum $-2$. Since the product is negative the numbers have opposite signs, and $-3$ and $+1$ work:

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

   so the inequality becomes $(x-3)(x+1)<0$.

2. 2

   Find the critical points.

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

   These split the line into $(-\infty,-1)$, $(-1,3)$ and $(3,\infty)$; the product cannot change sign inside any of them.

3. 3

   Test one value per interval. At $x=-2$: $(-5)(-1)=5>0$. At $x=0$: $(-3)(1)=-3<0$. At $x=4$: $(1)(5)=5>0$. So the pattern is positive, negative, positive — negative only in the middle.

4. 4

   Confirm from the graph's shape. The leading coefficient is $+1$, so the parabola opens upward: it dips below the axis between its roots and stays above outside them. That matches the sign test exactly, and is the faster way to see the answer once the roots are known.

5. 5

   Exclude the endpoints. The inequality is strict, so the roots themselves — where the expression is exactly $0$ — are not included:

   $-1<x<3,\qquad(-1,3)$

6. 6

   Verify by scanning. Evaluating $x^2-2x-3$ with exact fractions at $4001$ points on $[-10,10]$ and comparing the sign against $(-1,3)$ gives agreement at every point ✓. As a further check, the vertex is at $x=1$ (the midpoint of the roots) with value $1-2-3=-4$, the most negative the expression gets.