Which of the following represents the solution to the inequality: x^2 - 9 > 0

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Question: Which of the following represents the solution to the inequality: x^2 - 9 > 0?

Options:

Correct Answer: (-∞, -3) ∪ (3, ∞)

Solution:

Step 1: Factor the inequality: (x - 3)(x + 3) > 0. Step 2: The solution is outside the roots: (-∞, -3) ∪ (3, ∞).

Which of the following represents the solution to the inequality: x^2 - 9 > 0?

Which of the following represents 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 of the inequality. It becomes (x - 3)(x + 3) > 0.
Step 3: Identify the roots of the equation (x - 3)(x + 3) = 0. The roots are x = 3 and x = -3.
Step 4: Determine the intervals created by the roots: (-∞, -3), (-3, 3), and (3, ∞).
Step 5: Test a number from each interval to see if it makes the inequality true.
Step 6: For the interval (-∞, -3), test x = -4: (-4 - 3)(-4 + 3) = (-7)(-1) > 0, so this interval works.
Step 7: For the interval (-3, 3), test x = 0: (0 - 3)(0 + 3) = (-3)(3) < 0, so this interval does not work.
Step 8: For the interval (3, ∞), test x = 4: (4 - 3)(4 + 3) = (1)(7) > 0, so this interval works.
Step 9: Combine the intervals that work: The solution is (-∞, -3) ∪ (3, ∞).

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