What is (x-1)/(x+1)<= (2x-5)/(x-1) ?

The solution to (x-1)/(x+1) = 3

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(x-1)/(x+1)<= (2x-5)/(x-1)

\frac{x - 1}{x + 1} \leq \frac{2 x - 5}{x - 1}

x \leq - 2 or - 1 < x < 1 or x \geq 3

Interval Notation

(- \infty, - 2] \cup (- 1, 1) \cup [3, \infty)

Solution steps

\frac{x - 1}{x + 1} \leq \frac{2 x - 5}{x - 1}

Rewrite in standard form

\frac{- x ^{2} + x + 6}{( x + 1 ) ( x - 1 )} \leq 0

Factor

\frac{- x ^{2} + x + 6}{( x + 1 ) ( x - 1 )} : \frac{- ( x + 2 ) ( x - 3 )}{( x + 1 ) ( x - 1 )}

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

Multiply both sides by

- 1

(reverse the inequality)

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

Simplify

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

Identify the intervals

x \leq - 2 or - 1 < x < 1 or x \geq 3

7 x^{2} + 2 x + 3 = 0

25 x^{4} - 1

a^{2} b x a^{4} b^{6}

- 2 (i)^{9}

- \frac{1}{2} (2 n + 4) + 6 = - 9 + 4 (2 n + 1)

What is (x-1)/(x+1)<= (2x-5)/(x-1) ?
The solution to (x-1)/(x+1)<= (2x-5)/(x-1) is x<=-2\lor-1<x<1\lor x>= 3