Solve the inequality x² − 4x − 5 > 0.

Step-by-step solution

1. 1

   Factor the quadratic — never divide an inequality by an expression in x. Look for two numbers multiplying to $-5$ and adding to $-4$: those are $-5$ and $+1$.

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

   so the inequality becomes $(x-5)(x+1) > 0$. Factoring is essential because the sign of a product is decided factor by factor.

2. 2

   Find the critical points. 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, so they cut the number line into three intervals:

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

3. 3

   Test one convenient point per interval. The sign is constant on each interval, so a single test value settles it:

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

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

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

4. 4

   Decide whether the endpoints belong. The inequality is strict ($> 0$, not $\ge 0$), and at $x = -1$ and $x = 5$ the expression is exactly $0$. So both roots are excluded and the intervals are open.

5. 5

   Write the solution set.

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

   This matches the shape of the graph: an upward parabola is above the x-axis outside its two roots and below between them. Spot check: at $x = -1.5$ the value is $3.25 > 0$ and at $x = 5.5$ it is $3.25 > 0$, while at $x = 0$ it is $-5$.