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

The solution to ((x+1))/(((x-1)^2)) = 3

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

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

x \leq 0 or x \geq 3

Interval Notation

(- \infty, 0] \cup [3, \infty)

Solution steps

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

Rewrite in standard form

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

Factor

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

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

Multiply both sides by

- 1

(reverse the inequality)

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

Simplify

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

Identify the intervals

x \leq 0 or x \geq 3

\frac{( 6 - 2 x )}{( x ^{2} + 5 )} \geq 0

\frac{( x ^{3} + 1 )}{( x ^{4} + x ^{3} - 10 x ^{2} - 13 x + 21 )} > 0

(x + 1) (x^{2} + x - 1) < (x + 1)

(x - 1)^{2} \cdot (x + 1)^{3} \cdot (x - 4) \geq 0

4 x^{3} - 3 x^{2} - 16 x + 12 < 0

What is ((x+1))/(((x-1)^2))<= 1 ?
The solution to ((x+1))/(((x-1)^2))<= 1 is x<= 0\lor x>= 3