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Solve the inequality over the reals: (x² + 1)/(4x - 3) > 2.

1. x in (3/4, 1) U (7, infinity)
2. x in (1, 7)
3. x in (3/4, 1) U (1, 7)
4. x in (-infinity, 1) U (7, infinity)

Correct answer: x in (3/4, 1) U (7, infinity)

Solution

(x² + 1)/(4x - 3) - 2 > 0 => (x² + 1 - 2(4x - 3))/(4x - 3) > 0 => (x² - 8x + 7)/(4x - 3) > 0 => ((x - 1)(x - 7))/(4x - 3) > 0. Critical points: x = 3/4, 1, 7. Sign chart: for x < 3/4 the expression is negative; for 3/4 < x < 1 it is positive; for 1 < x < 7 negative; for x > 7 positive. So solution: (3/4, 1) U (7, infinity).

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