Determine the solution set for the inequality: 2x^2 - 8 < 0.

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Question: Determine the solution set for the inequality: 2x^2 - 8 < 0.

Options:

Correct Answer: (-2, 2)

Solution:

Step 1: Factor the inequality: 2(x^2 - 4) < 0. Step 2: Roots are x = -2 and x = 2. Step 3: The solution is between the roots: (-2, 2).

Determine the solution set for the inequality: 2x^2 - 8 < 0.

Determine the solution set for the inequality: 2x^2 - 8 < 0.

Step 1: Start with the inequality 2x^2 - 8 < 0.
Step 2: Factor out the common factor of 2 from the left side: 2(x^2 - 4) < 0.
Step 3: Now, simplify the inequality: x^2 - 4 < 0.
Step 4: Factor x^2 - 4 as (x - 2)(x + 2) < 0.
Step 5: Identify the roots of the equation (x - 2)(x + 2) = 0. The roots are x = -2 and x = 2.
Step 6: The roots divide the number line into three intervals: (-∞, -2), (-2, 2), and (2, ∞).
Step 7: Test a point from each interval to see where the inequality (x - 2)(x + 2) < 0 holds true.
Step 8: Choose a test point from (-∞, -2), like x = -3: (-3 - 2)(-3 + 2) = (-5)(-1) > 0 (not a solution).
Step 9: Choose a test point from (-2, 2), like x = 0: (0 - 2)(0 + 2) = (-2)(2) < 0 (this is a solution).
Step 10: Choose a test point from (2, ∞), like x = 3: (3 - 2)(3 + 2) = (1)(5) > 0 (not a solution).
Step 11: The solution set is the interval where the inequality is satisfied: (-2, 2).

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