Find the solution set for the inequality: x^2 - 4 > 0.

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Question: Find the solution set for the inequality: x^2 - 4 > 0.

Options:

Correct Answer: (-∞, -2) ∪ (2, ∞)

Solution:

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

Find the solution set for the inequality: x^2 - 4 > 0.

Find the solution set for the inequality: x^2 - 4 > 0.

Step 1: Start with the inequality x^2 - 4 > 0.
Step 2: Factor the left side of the inequality. It becomes (x - 2)(x + 2) > 0.
Step 3: Identify the critical points by setting each factor to zero: x - 2 = 0 gives x = 2, and x + 2 = 0 gives x = -2.
Step 4: These critical points (-2 and 2) divide the number line into three intervals: (-∞, -2), (-2, 2), and (2, ∞).
Step 5: Test a point from each interval to see if the inequality (x - 2)(x + 2) > 0 holds true.
Step 6: For the interval (-∞, -2), test x = -3: (-3 - 2)(-3 + 2) = (-5)(-1) > 0, so this interval works.
Step 7: For the interval (-2, 2), test x = 0: (0 - 2)(0 + 2) = (-2)(2) < 0, so this interval does not work.
Step 8: For the interval (2, ∞), test x = 3: (3 - 2)(3 + 2) = (1)(5) > 0, so this interval works.
Step 9: Combine the intervals that work: The solution set is (-∞, -2) ∪ (2, ∞).

Quadratic Inequalities – Understanding how to solve inequalities involving quadratic expressions by factoring and analyzing sign changes.
Interval Notation – Using interval notation to express the solution set of inequalities.