Solve the inequality x2 - 7x - 6 > 0.

Step-by-step solution

1. 1

   Try factoring, then give up quickly. We would need two integers with product $-6$ and sum $-7$: the candidates $(1,-6)$ sum to $-5$ and $(-1,6)$ sum to $5$, $(2,-3)$ to $-1$, $(-2,3)$ to $1$. None works, so the roots are not rational and the quadratic formula is required.

2. 2

   Compute the discriminant.

   $\Delta=(-7)^2-4(1)(-6)=49+24=73$

   $73$ is prime, hence square-free, so $\sqrt{73}$ cannot be simplified and the boundaries will be irrational. $\Delta>0$ guarantees two distinct real roots.

3. 3

   Find the roots.

   $x=\frac{7\pm\sqrt{73}}{2}$

   with $\sqrt{73}\approx 8.5440$, so $x_1\approx -0.7720$ and $x_2\approx 7.7720$.

4. 4

   Use the direction of opening. The coefficient of $x^2$ is $+1>0$, so the parabola opens upward and the expression is positive outside the roots, negative between them:

   $x<\frac{7-\sqrt{73}}{2}\qquad\text{or}\qquad x>\frac{7+\sqrt{73}}{2}$

5. 5

   Test and settle the endpoints. At $x=-1$: $1+7-6=2>0$ ✓ (left region). At $x=0$: $-6$, which fails ✓ (middle region excluded). At $x=8$: $64-56-6=2>0$ ✓ (right region). At each root the expression equals $0$, and the inequality is strict, so neither boundary is included.