Find-solution-set-the-inequality-2x-5-25-3x-gt-x-

Question Number 135424 by benjo_mathlover last updated on 13/Mar/21

F i n d s o l u t i o n s e t t h e i n e q u a l i t y

\sqrt{2 x - 5} + \sqrt{25 - 3 x} > x

Answered by EDWIN88 last updated on 13/Mar/21

(1) 2 x - 5 ⩾ 0 \to x ⩾ \frac{5}{2}

(2) 3 x - 25 ⩽ 0 \to x ⩽ \frac{25}{3}

(3) \sqrt{25 - 3 x} > x - \sqrt{2 x - 5}

\to 25 - 3 x > x^{2} - 2 x \sqrt{2 x - 5} + 2 x - 5

\to 20 - 5 x > x^{2} - 2 x \sqrt{2 x - 5}

\to 2 x \sqrt{2 x - 5} > x^{2} + 5 x - 20

\Rightarrow {4 x}^{2} - 2 x - 5) > x^{4} + {25 x}^{2} + 400 + {10 x}^{3} - {40 x}^{2} - 200 x

\to x^{4} + {10 x}^{3} - {19 x}^{2} - 198 x + 405 < 0

\to (x + 9.21) (x + 6.18) (x - 2.28) (x - 3.11) < 0

\to - 9.21 < x < - 6.18 \cup 2.28 < x < 3.11

solution set we get from (1) \cap (2) \cap (3)

\Rightarrow \frac{5}{2} ⩽ x < 3.11 or [\frac{5}{2}; 3.11)

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