What is x/(x-9)>6x-2 ?

The solution to x/(x-9)>6x-2 is x<(19-sqrt(313))/4 \lor 9<x<(19+sqrt(313))/4

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x/(x-9)>6x-2

\frac{x}{x - 9} > 6 x - 2

x < \frac{19 - 313}{4} or 9 < x < \frac{19 + 313}{4}

Interval Notation

(- \infty, \frac{19 - 313}{4}) \cup (9, \frac{19 + 313}{4})

Decimal Notation

x < 0.32704 \dots or 9 < x < 9.17295 \dots

Solution steps

\frac{x}{x - 9} > 6 x - 2

Rewrite in standard form

\frac{- 6 x ^{2} + 57 x - 18}{x - 9} > 0

Factor

\frac{- 6 x ^{2} + 57 x - 18}{x - 9} : \frac{- 3 ( x - \frac{19 + 313}{4} ) ( x - \frac{19 - 313}{4} )}{x - 9}

\frac{- 3 ( x - \frac{19 + 313}{4} ) ( x - \frac{19 - 313}{4} )}{x - 9} > 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - 3 ( x - \frac{19 + 313}{4} ) ( x - \frac{19 - 313}{4} ) ) ( - 1 )}{x - 9} < 0 \cdot (- 1)

Simplify

\frac{3 ( x - \frac{19 + 313}{4} ) ( x - \frac{19 - 313}{4} )}{x - 9} < 0

Divide both sides by

3

\frac{\frac{3 ( x - \frac{19 + 313}{4} ) ( x - \frac{19 - 313}{4} )}{x - 9}}{3} < \frac{0}{3}

Simplify

\frac{( x - \frac{19 + 313}{4} ) ( x - \frac{19 - 313}{4} )}{x - 9} < 0

Identify the intervals

x < \frac{19 - 313}{4} or 9 < x < \frac{19 + 313}{4}

4 x - 3 = x + 9

- 16 t^{2} + 128 t

0 < 2 x + 2 \leq 6

6 x^{2} + 29 x - 5 = 0

\frac{- 4 ( r ^{4} t ^{- 2} ) ^{- 1}}{4 r ^{2} t ^{4}}

What is x/(x-9)>6x-2 ?
The solution to x/(x-9)>6x-2 is x<(19-sqrt(313))/4 \lor 9<x<(19+sqrt(313))/4