What is (2x+3)/(x-2)<5x-2 ?

The solution to (2x+3)/(x-2) (7+2sqrt(11))/5

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(2x+3)/(x-2)<5x-2

\frac{2 x + 3}{x - 2} < 5 x - 2

\frac{7 - 2 11}{5} < x < 2 or x > \frac{7 + 2 11}{5}

Solution steps

\frac{2 x + 3}{x - 2} < 5 x - 2

Rewrite in standard form

\frac{- 5 x ^{2} + 14 x - 1}{x - 2} < 0

Factor

- 5 x^{2} + 14 x - 1 : - (x - \frac{7 + 2 11}{5}) (x - \frac{7 - 2 11}{5})

\frac{- ( x - \frac{7 + 2 11}{5} ) ( x - \frac{7 - 2 11}{5} )}{x - 2} < 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x - \frac{7 + 2 11}{5} ) ( x - \frac{7 - 2 11}{5} ) ) ( - 1 )}{x - 2} > 0 \cdot (- 1)

Simplify

\frac{( x - \frac{7 + 2 11}{5} ) ( x - \frac{7 - 2 11}{5} )}{x - 2} > 0

Identify the intervals

\frac{7 - 2 11}{5} < x < 2 or x > \frac{7 + 2 11}{5}

4 x^{2} + x y - 18 y^{2}

x^{2} + 8 x - 36 = 0

20 + 17 v = - 6 + 15 v

2 (x + 1) + 2 x = 4 x + 2

25 p^{2} + 30 p + 9

What is (2x+3)/(x-2)<5x-2 ?
The solution to (2x+3)/(x-2)<5x-2 is (7-2sqrt(11))/5 <x<2\lor x>(7+2sqrt(11))/5