What is the solution to the inequality: x^2 - 9 > 0?

What is the solution to the inequality: x^2 - 9 > 0?

Step 1: Start with the inequality x^2 - 9 > 0.
Step 2: Factor the left side: x^2 - 9 can be factored into (x - 3)(x + 3).
Step 3: Rewrite the inequality: (x - 3)(x + 3) > 0.
Step 4: Identify the critical points by setting each factor to zero: x - 3 = 0 gives x = 3, and x + 3 = 0 gives x = -3.
Step 5: These critical points divide the number line into three intervals: (-∞, -3), (-3, 3), and (3, ∞).
Step 6: Test a point from each interval to see where the product (x - 3)(x + 3) is positive.
Step 7: For the interval (-∞, -3), test x = -4: (-4 - 3)(-4 + 3) = (-7)(-1) > 0, so this interval works.
Step 8: For the interval (-3, 3), test x = 0: (0 - 3)(0 + 3) = (-3)(3) < 0, so this interval does not work.
Step 9: For the interval (3, ∞), test x = 4: (4 - 3)(4 + 3) = (1)(7) > 0, so this interval works.
Step 10: Combine the results: the solution to the inequality is x < -3 or x > 3.

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