Question

Solve the inequality 2x>30+5/4x

Answer 1

Explanation:

`2x > 30+(5)/(4x) \\2x-(5)/(4x) > 30\\(8x^2-5)/(4x) > 30\\case~1\\if~x > 0\\8x^2-5 > 120x\\8x^2-120x > 5\\x^2-15x > (5)/(8) \\adding~(-(15)/(2) )^2~to~both~sides\\(x-(15)/(2) )^2 > (5)/(8)+(225)/(4) \\(x-(15)/(2) )^2 > (455)/(8) \\x-(15)/(2) < -\sqrt{(455)/(8) } \\x < (15)/(2)-\sqrt{(455)/(8) } \\or~x < 0\\rejected~as~x > 0`

`x-(15)/(2) > \sqrt{(455)/(8) } \\x > (15)/(2) +\sqrt{(455)/(8) }`

case~2

`if~x < 0\\8x^2-5 < 120x\\8x^2-120x < 5\\x^2-15x < (5)/(8) \\adding~(-(15)/(2) )^2\\(x-(15)/(2) )^2 < (5)/(8) +(-(15)/(2) )^2\\|x-(15)/(2) | < (5+450)/(8) \\-\sqrt{(455)/(8) } < x-(15)/(2) < \sqrt{(455)/(8) } \\(15)/(2) -\sqrt{(455)/(8) } < x < (15)/(2) +\sqrt{(455)/(8) } \\but~x < 0\\7.5-\sqrt{(455)/(8) } < x < 0`