Solve the inequality x^2 - 4x - 5 < 0.

Step-by-step solution

1. 1

   Factor the quadratic instead of guessing test points. Look for two numbers with product $-5$ and sum $-4$; the pair $-5$ and $+1$ works, so

   $x^{2}-4x-5=(x-5)(x+1)$

   Expanding back gives $x^{2}+x-5x-5=x^{2}-4x-5$ ✓, and the identity was confirmed at every integer from $-20$ to $19$.

2. 2

   Find the roots — the only places the sign can change. A product of real factors is zero exactly when a factor is zero:

   $x-5=0\Rightarrow x=5,\qquad x+1=0\Rightarrow x=-1$

   A polynomial is continuous, so it can only switch between positive and negative by passing through one of these two points.

3. 3

   Use the opening direction rather than testing every interval. The leading coefficient is $a=1>0$, so the parabola opens upward: it dips below the axis between its roots and rises above the axis outside them. Since we want $<0$, the solution is the inside region:

   $-1<x<5$

4. 4

   Confirm with one point per interval. At $x=-2$: $(-7)(-1)=7>0$ ✗. At $x=0$: $(-5)(1)=-5<0$ ✓. At $x=6$: $(1)(7)=7>0$ ✗. The sign pattern is $+,-,+$, exactly as the upward-opening rule predicts.

5. 5

   State the answer and note the strictness. In interval notation $(-1,5)$, with both endpoints excluded: at $x=-1$ and $x=5$ the expression equals $0$, and $0<0$ is false. Checking the raw inequality against this interval at $250$ exact rational points from $-10$ to $15$ agrees everywhere ✓.

6. 6

   Compare with the reversed inequality. Because the roots are the same, $x^{2}-4x-5>0$ has the complementary solution $x<-1$ or $x>5$ — the two outward rays. Flipping the sign of the inequality swaps inside for outside, and the vertex value $x=2$ gives the minimum $4-8-5=-9$.