Solve the inequality x^2 - 4x - 5 > 0 and write the solution in interval notation.

Step-by-step solution

1. 1

   Get zero on one side, then factor. The inequality already has $0$ on the right, so factor the quadratic. You need two numbers multiplying to $-5$ and adding to $-4$: those are $-5$ and $+1$, so

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

   and the inequality becomes $(x-5)(x+1)>0$.

2. 2

   Find the critical points — the zeros of the factors. Setting each factor to zero:

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

   These are the only places the expression can change sign, because a product changes sign exactly where one of its factors does.

3. 3

   Split the number line into the three resulting intervals.

   $(-\infty,-1),\qquad (-1,5),\qquad (5,\infty)$

   Inside each interval the sign of $(x-5)(x+1)$ is constant, so one test value per interval settles it.

4. 4

   Test one point in each interval.

   $x=-2:\ (-7)(-1)=7>0\quad\checkmark$

   $x=0:\ (-5)(1)=-5<0\quad\times$

   $x=6:\ (1)(7)=7>0\quad\checkmark$

   The pattern positive–negative–positive is what you expect from an upward-opening parabola with two distinct roots.

5. 5

   Write the solution set, excluding the roots. The inequality is strict ($>0$, not $\ge 0$), so the points $x=-1$ and $x=5$ — where the expression equals zero — are not included:

   $x<-1\ \text{ or }\ x>5,\qquad (-\infty,-1)\cup(5,\infty)$

   Note the union: the solution is two separate rays, not a single interval, so it cannot be written as a chained inequality like $-1>x>5$.