Solve the Quadratic Inequality: x² - 16 > 0

To solve the inequality $x^2 - 16 > 0$, we'll first factor the quadratic expression.

• Step 1: Recognize that $x^2 - 16$ is a difference of squares, and factor it as $(x - 4)(x + 4) > 0$.
• Step 2: Identify the critical points where the expression equals zero: $x - 4 = 0$ gives $x = 4$, and $x + 4 = 0$ gives $x = -4$.
• Step 3: Test intervals determined by these critical points. We consider the intervals:
  a) $x < -4$
  b) $-4 < x < 4$
  c) $x > 4$
• Step 4: Determine the sign of the expression in each interval:
  - For $x < -4$, choose $x = -5$. Then $(x - 4)(x + 4) = (-5 - 4)(-5 + 4) = (-9)(-1) = 9 > 0$. So, this interval satisfies the inequality.
  - For $-4 < x < 4$, choose $x = 0$. Then $(x - 4)(x + 4) = (0 - 4)(0 + 4) = (-4)(4) = -16 < 0$. This interval does not satisfy the inequality.
  - For $x > 4$, choose $x = 5$. Then $(x - 4)(x + 4) = (5 - 4)(5 + 4) = (1)(9) = 9 > 0$. So, this interval satisfies the inequality.
• Step 5: Combine these results to state the solution as $x < -4$ or $x > 4$.

Hence, the solution to the inequality is: $x < -4, \, 4 < x$.