The inequality provided is:

$x^2 > 5x + 14$

Let's solve it step by step.

Step 1: Rearrange the inequality

Move all terms to one side to set the inequality to 0:

$x^2 - 5x - 14 > 0$

Step 2: Solve the corresponding quadratic equation

To find critical points, solve the quadratic equation:

$x^2 - 5x - 14 = 0$

Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = -5$, and $c = -14$. Plug these values into the formula:

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}$ $x = \frac{5 \pm \sqrt{25 + 56}}{2}$ $x = \frac{5 \pm \sqrt{81}}{2}$ $x = \frac{5 \pm 9}{2}$

Thus, the two solutions are:

$x = \frac{5 + 9}{2} = 7 \quad \text{and} \quad x = \frac{5 - 9}{2} = -2$

Step 3: Analyze the sign of the quadratic expression

We have the critical points $x = -2$ and $x = 7$. Now we need to check the sign of the expression $x^2 - 5x - 14$ in the intervals defined by these points: $(- \infty, -2)$, $(-2, 7)$, and $(7, \infty)$.

• For $x \in (-\infty, -2)$, choose $x = -3$: $(-3)^2 - 5(-3) - 14 = 9 + 15 - 14 = 10 > 0$ The expression is positive in this interval.

• For $x \in (-2, 7)$, choose $x = 0$: $(0)^2 - 5(0) - 14 = -14 < 0$ The expression is negative in this interval.

• For $x \in (7, \infty)$, choose $x = 8$: $(8)^2 - 5(8) - 14 = 64 - 40 - 14 = 10 > 0$ The expression is positive in this interval.

Step 4: Write the solution

We are looking for where $x^2 - 5x - 14 > 0$. From the sign analysis, the expression is positive in the intervals $(- \infty, -2)$ and $(7, \infty)$.

Thus, the solution to the inequality is:

$x \in (-\infty, -2) \cup (7, \infty)$

Would you like a detailed explanation of any step or further details?

Here are 5 related questions you might find useful:

1. How do you solve quadratic inequalities in general?
2. What is the graphical interpretation of quadratic inequalities?
3. How do you apply the quadratic formula to different types of quadratic equations?
4. Can you explain the significance of the critical points in inequalities?
5. How do sign charts work in analyzing inequalities?

Tip: Always check the sign of the expression in each interval to correctly determine where the inequality holds.