What is (x-2)/(x+2)>= (2x-3)/(x-1) ?

The solution to (x-2)/(x+2)>= (2x-3)/(x-1) is -2(1+sqrt(3))<= x<-2\lor 1<x<= 2(sqrt(3)-1)

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(x-2)/(x+2)>= (2x-3)/(x-1)

\frac{x - 2}{x + 2} \geq \frac{2 x - 3}{x - 1}

- 2 (1 + 3) \leq x < - 2 or 1 < x \leq 2 (3 - 1)

Solution steps

\frac{x - 2}{x + 2} \geq \frac{2 x - 3}{x - 1}

Rewrite in standard form

\frac{- x ^{2} - 4 x + 8}{( x + 2 ) ( x - 1 )} \geq 0

Factor

- x^{2} - 4 x + 8 : - (x - 2 (3 - 1)) (x + 2 (1 + 3))

\frac{- ( x - 2 ( 3 - 1 ) ) ( x + 2 ( 1 + 3 ) )}{( x + 2 ) ( x - 1 )} \geq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x - 2 ( 3 - 1 ) ) ( x + 2 ( 1 + 3 ) ) ) ( - 1 )}{( x + 2 ) ( x - 1 )} \leq 0 \cdot (- 1)

Simplify

\frac{( x - 2 ( 3 - 1 ) ) ( x + 2 ( 1 + 3 ) )}{( x + 2 ) ( x - 1 )} \leq 0

Identify the intervals

- 2 (1 + 3) \leq x < - 2 or 1 < x \leq 2 (3 - 1)

1 - 3 x - 2 x = 3 - 2 (x + 3)

X^{2} - 4 = 0

\frac{5 x - 11}{7} = \frac{1}{13}

- 4 x - 2 \leq 6

25 6^{- \frac{1}{8}}

What is (x-2)/(x+2)>= (2x-3)/(x-1) ?
The solution to (x-2)/(x+2)>= (2x-3)/(x-1) is -2(1+sqrt(3))<= x<-2\lor 1<x<= 2(sqrt(3)-1)