Determine the solution set for the inequality: x^2 - 6x + 8 ≤ 0.

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

Options:

Correct Answer: [2, 4]

Solution:

Step 1: Factor: (x - 2)(x - 4) ≤ 0. Step 2: The solution is between the roots: [2, 4].

Determine the solution set for the inequality: x^2 - 6x + 8 ≤ 0.

Determine the solution set for the inequality: x^2 - 6x + 8 ≤ 0.

Step 1: Start with the inequality x^2 - 6x + 8 ≤ 0.
Step 2: Factor the quadratic expression. Look for two numbers that multiply to 8 and add to -6. The numbers are -2 and -4.
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 and x = 4.
Step 5: Determine the intervals to test: (-∞, 2), (2, 4), and (4, ∞).
Step 6: Choose a test point from each interval and substitute it into the factored inequality (x - 2)(x - 4).
Step 7: For the interval (-∞, 2), choose x = 0: (0 - 2)(0 - 4) = 8 > 0 (not part of the solution).
Step 8: For the interval (2, 4), choose x = 3: (3 - 2)(3 - 4) = -1 < 0 (part of the solution).
Step 9: For the interval (4, ∞), choose x = 5: (5 - 2)(5 - 4) = 3 > 0 (not part of the solution).
Step 10: The solution includes the points where the expression equals 0 (x = 2 and x = 4) and the interval where it is negative.
Step 11: Combine the results: The solution set is [2, 4].

Quadratic Inequalities – Understanding how to solve inequalities involving quadratic expressions by factoring and analyzing intervals.
Factoring Quadratics – The ability to factor quadratic expressions to find roots and analyze their behavior.
Interval Notation – Using interval notation to express the solution set of inequalities.