Solve the inequality: x^2 - 4 > 0.

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Question: Solve the inequality: x^2 - 4 > 0.

Options:

Correct Answer: x < -2 or x > 2

Solution:

Step 1: Factor: (x - 2)(x + 2) > 0. Step 2: Test intervals: solution is x < -2 or x > 2.

Solve the inequality: x^2 - 4 > 0.

Solve the inequality: x^2 - 4 > 0.

Step 1: Rewrite the inequality x^2 - 4 > 0. Notice that x^2 - 4 can be factored.
Step 2: Factor the expression: x^2 - 4 = (x - 2)(x + 2). So, we rewrite the inequality as (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 product (x - 2)(x + 2) is positive or negative.
Step 6: For the interval (-∞, -2), test x = -3: (-3 - 2)(-3 + 2) = (-5)(-1) > 0, so this interval is part of the solution.
Step 7: For the interval (-2, 2), test x = 0: (0 - 2)(0 + 2) = (-2)(2) < 0, so this interval is NOT part of the solution.
Step 8: For the interval (2, ∞), test x = 3: (3 - 2)(3 + 2) = (1)(5) > 0, so this interval is part of the solution.
Step 9: Combine the results: The solution to the inequality is x < -2 or x > 2.

Inequalities – Understanding how to solve quadratic inequalities by factoring and testing intervals.
Factoring Quadratics – Recognizing how to factor quadratic expressions to simplify the inequality.
Interval Testing – Applying the method of testing intervals to determine where the inequality holds true.