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

The solution to ((5x+1))/(((x+1)^2)) 3

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((5x+1))/(((x+1)^2))<1

\frac{( 5 x + 1 )}{( ( x + 1 ) ^{2} )} < 1

x < - 1 or - 1 < x < 0 or x > 3

Interval Notation

(- \infty, - 1) \cup (- 1, 0) \cup (3, \infty)

Solution steps

\frac{5 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 < - 1 or - 1 < x < 0 or x > 3

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

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

\frac{( 6 x - 5 )}{( 4 x + 1 )} < 0

\frac{( x - 3 )}{5} \geq \frac{( 3 - x )}{( 8 x )}

\frac{( x ^{2} - 5 x + 6 )}{( x ^{2} + x + 1 )} < 0

What is ((5x+1))/(((x+1)^2))<1 ?
The solution to ((5x+1))/(((x+1)^2))<1 is x<-1\lor-1<x<0\lor x>3