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} \leq 5 x - 2

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

Interval Notation

[\frac{7 - 2 11}{5}, 2) \cup [\frac{7 + 2 11}{5}, \infty)

Decimal Notation

0.07335 \dots \leq x < 2 or x \geq 2.72664 \dots

Solution steps

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

Rewrite in standard form

\frac{- 5 x ^{2} + 14 x - 1}{x - 2} \leq 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} \leq 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} \geq 0 \cdot (- 1)

Simplify

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

Identify the intervals

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

8 x + 2 > 348 x + 2 > 34

x^{2} - 25 = - x^{2} + 25

352 - 32 \cdot x = 0

280

\frac{g}{2} - 1 \geq 1

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