Mathematics

Mathematics

Problem

Consider the inequality arctan⁡(3x−2)>π3\arctan (3x-2) > \dfrac{\pi}{3}, where xx is a real number. a. (i) Convert the given inequality into an equivalent inequality in terms of 3x−23x-2 without any inverse trigonometric function. b. (ii) Hence determine the complete solution set of xx satisfying the inequality. c. (iii) Express your answer in interval notation on the real line.
Intuitive explanation

The key is recognizing that arctan⁡\arctan is a strictly increasing function, so you can "unwrap" the inequality by applying the tangent function to both sides. Once you do that, you'll get a simple algebraic inequality for 3x−23x - 2. Just remember that tan⁡(π/3)\tan(\pi/3) is a standard value you should know.

Intuitive explanation

The key is recognizing that arctan⁡\arctan is a strictly increasing function, so you can "unwrap" the inequality by applying the tangent function to both sides. Once you do that, you'll get a simple algebraic inequality for 3x−23x - 2. Just remember that tan⁡(π/3)\tan(\pi/3) is a standard value you should know.

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