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}} < 1

x < 0 or x > 3

Interval Notation

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

Solution steps

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

Rewrite in standard form

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

Factor

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

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

Multiply both sides by

- 1

(reverse the inequality)

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

Simplify

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

Identify the intervals

x < 0 or x > 3

\frac{( 5 - 2 x )}{( 3 x ^{2} - 2 x - 16 )} < 1

(x^{2} - 4 x + 4) (1 - 2 x) (x^{2} + 3 x - 1) x > 0

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

\frac{( 3 x - 2 )}{( 2 x - 3 )} < 3

\frac{( x ^{2} - 5 ) ( x ^{2} - 4 )}{( x - 1 )} < 0