Solve the following inequality:
x^2 - 2x >= 3
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To solve the inequality x^2 - 2x >= 3, we first move all terms to one side of the inequality to get:
x^2 - 2x - 3 >= 0
Next, we factor the quadratic to get:
(x - 3)(x + 1) >= 0
Setting each factor equal to zero gives the potential boundary points of x = 3 and x = -1.
To determine which intervals satisfy the inequality, we test a number from each interval in the inequality. The intervals are x < -1, -1 < x < 3, and x > 3.
For x < -1, we can test x = -2. Substituting -2 into the inequality gives (-2 - 3)(-2 + 1) = 5 > 0, so the interval x < -1 satisfies the inequality.
For -1 < x < 3, we can test x = 0. Substituting 0 into the inequality gives (0 - 3)(0 + 1) = -3 < 0, so the interval -1 < x < 3 does not satisfy the inequality.
For x > 3, we can test x = 4. Substituting 4 into the inequality gives (4 - 3)(4 + 1) = 5 > 0, so the interval x > 3 satisfies the inequality.
Therefore, the solution to the inequality x^2 - 2x >= 3 is x <= -1 or x >= 3.
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