Find the solution set for the inequality: x^2 + 2x - 8 > 0.

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Question: Find the solution set for the inequality: x^2 + 2x - 8 > 0.

Options:

Correct Answer: (-∞, -4) ∪ (2, ∞)

Solution:

Step 1: Factor the quadratic: (x - 2)(x + 4) > 0. Step 2: The solution is outside the roots: (-∞, -4) ∪ (2, ∞).

Find the solution set for the inequality: x^2 + 2x - 8 > 0.

Find the solution set for the inequality: x^2 + 2x - 8 > 0.

Step 1: Start with the inequality x^2 + 2x - 8 > 0.
Step 2: Factor the quadratic expression. We need to find two numbers that multiply to -8 and add to 2. The numbers are 4 and -2.
Step 3: Write the factored form: (x - 2)(x + 4) > 0.
Step 4: Identify the roots of the equation (where it equals 0): x - 2 = 0 gives x = 2, and x + 4 = 0 gives x = -4.
Step 5: Plot the roots on a number line: -4 and 2.
Step 6: Determine the intervals created by the roots: (-∞, -4), (-4, 2), and (2, ∞).
Step 7: Test a point from each interval in the inequality (x - 2)(x + 4) > 0.
Step 8: For (-∞, -4), test x = -5: (-5 - 2)(-5 + 4) = (-7)(-1) > 0, so this interval works.
Step 9: For (-4, 2), test x = 0: (0 - 2)(0 + 4) = (-2)(4) < 0, so this interval does not work.
Step 10: For (2, ∞), test x = 3: (3 - 2)(3 + 4) = (1)(7) > 0, so this interval works.
Step 11: Combine the intervals that work: The solution set is (-∞, -4) ∪ (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.
Sign Analysis – Determining the sign of the quadratic expression in different intervals based on its roots.