Find the solution set for the inequality: x^2 - 6x + 8 > 0.

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Question: Find the solution set for the inequality: x^2 - 6x + 8 > 0.

Options:

Correct Answer: x < 2 or x > 4

Solution:

Step 1: Factor: (x - 2)(x - 4) > 0. Step 2: Critical points are x = 2 and x = 4. Step 3: Test intervals: valid for x < 2 or x > 4.

Find the solution set for the inequality: x^2 - 6x + 8 > 0.

Find the solution set for the inequality: x^2 - 6x + 8 > 0.

Step 1: Factor the quadratic expression x^2 - 6x + 8. This can be factored into (x - 2)(x - 4).
Step 2: Identify the critical points by setting each factor equal to zero. This gives us x = 2 and x = 4.
Step 3: Determine the intervals to test. The critical points divide the number line into three intervals: (-∞, 2), (2, 4), and (4, ∞).
Step 4: Test a point from each interval in the inequality (x - 2)(x - 4) > 0 to see if it holds true.
Step 5: For the interval (-∞, 2), test x = 0: (0 - 2)(0 - 4) = 8 > 0, so this interval works.
Step 6: For the interval (2, 4), test x = 3: (3 - 2)(3 - 4) = -1 < 0, so this interval does not work.
Step 7: For the interval (4, ∞), test x = 5: (5 - 2)(5 - 4) = 3 > 0, so this interval works.
Step 8: Combine the intervals that work. The solution set is x < 2 or x > 4.

Inequalities – Understanding how to solve quadratic inequalities by factoring and testing intervals.
Critical Points – Identifying points where the expression equals zero to determine intervals for testing.
Interval Testing – Using test points in the intervals created by critical points to determine where the inequality holds.