What is the solution set for the inequality x^2 - 4 > 0?

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Question: What is the solution set for the inequality x^2 - 4 > 0?

Options:

Correct Answer: (-∞, -2) ∪ (2, ∞)

Solution:

Factor the inequality: (x - 2)(x + 2) > 0. The solution set is x < -2 or x > 2.

What is the solution set for the inequality x^2 - 4 > 0?

What is the solution set for the inequality x^2 - 4 > 0?

Step 1: Start with the inequality x^2 - 4 > 0.
Step 2: Recognize that x^2 - 4 can be factored as (x - 2)(x + 2).
Step 3: Rewrite the inequality as (x - 2)(x + 2) > 0.
Step 4: Identify the critical points by setting each factor to zero: x - 2 = 0 gives x = 2, and x + 2 = 0 gives x = -2.
Step 5: Plot the critical points on a number line: -2 and 2.
Step 6: Test intervals around the critical points to see where the product (x - 2)(x + 2) is positive.
Step 7: Choose a test point in the interval (-∞, -2), for example, x = -3: (-3 - 2)(-3 + 2) = (-5)(-1) > 0, so this interval works.
Step 8: Choose a test point in the interval (-2, 2), for example, x = 0: (0 - 2)(0 + 2) = (-2)(2) < 0, so this interval does not work.
Step 9: Choose a test point in the interval (2, ∞), for example, x = 3: (3 - 2)(3 + 2) = (1)(5) > 0, so this interval works.
Step 10: Combine the intervals where the product is positive: the solution set is x < -2 or x > 2.

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