What is (12)/(x^2+2x)< 3/(x^2+4x+4) ?

The solution to (12)/(x^2+2x)< 3/(x^2+4x+4) is -8/3 <x<-2\lor-2<x<0

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(12)/(x^2+2x)< 3/(x^2+4x+4)

\frac{12}{x ^{2} + 2 x} < \frac{3}{x ^{2} + 4 x + 4}

- \frac{8}{3} < x < - 2 or - 2 < x < 0

Interval Notation

(- \frac{8}{3}, - 2) \cup (- 2, 0)

Decimal Notation

- 2.66666 \dots < x < - 2 or - 2 < x < 0

Solution steps

\frac{12}{x ^{2} + 2 x} < \frac{3}{x ^{2} + 4 x + 4}

Rewrite in standard form

\frac{9 x + 24}{x ( x + 2 ) ^{2}} < 0

Factor

\frac{9 x + 24}{x ( x + 2 ) ^{2}} : \frac{3 ( 3 x + 8 )}{x ( x + 2 ) ^{2}}

\frac{3 ( 3 x + 8 )}{x ( x + 2 ) ^{2}} < 0

Divide both sides by

3

\frac{\frac{3 ( 3 x + 8 )}{x ( x + 2 ) ^{2}}}{3} < \frac{0}{3}

Simplify

\frac{3 x + 8}{x ( x + 2 ) ^{2}} < 0

Identify the intervals

- \frac{8}{3} < x < - 2 or - 2 < x < 0

4 (x^{2} + 5) = 3 x^{2} + 10 x - 4

- 3 - 49

3 x^{2} + 3 = 0

0 = x^{2} - 5 x - 24

\frac{1}{2} - \frac{5}{8}

What is (12)/(x^2+2x)< 3/(x^2+4x+4) ?
The solution to (12)/(x^2+2x)< 3/(x^2+4x+4) is -8/3 <x<-2\lor-2<x<0