What is 2x^2+2x+1<(15)/((x^2+x+1)) ?

The solution to 2x^2+2x+1<(15)/((x^2+x+1)) is -2<x<1

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2x^2+2x+1<(15)/((x^2+x+1))

2 x^{2} + 2 x + 1 < \frac{15}{( x ^{2} + x + 1 )}

- 2 < x < 1

Interval Notation

(- 2, 1)

Solution steps

2 x^{2} + 2 x + 1 < \frac{15}{x ^{2} + x + 1}

Rewrite in standard form

\frac{2 x ^{4} + 4 x ^{3} + 5 x ^{2} + 3 x - 14}{x ^{2} + x + 1} < 0

Factor

\frac{2 x ^{4} + 4 x ^{3} + 5 x ^{2} + 3 x - 14}{x ^{2} + x + 1} : \frac{( x - 1 ) ( x + 2 ) ( 2 x ^{2} + 2 x + 7 )}{x ^{2} + x + 1}

\frac{( x - 1 ) ( x + 2 ) ( 2 x ^{2} + 2 x + 7 )}{x ^{2} + x + 1} < 0

Identify the intervals

- 2 < x < 1

\frac{2 + 3}{( x + 1 )} > \frac{2}{x}

\frac{x}{x + 2} > \frac{1}{x - 3}

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

\frac{\frac{x}{- ( x + 2 )}}{x} \cdot (x - 2) < 0

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

What is 2x^2+2x+1<(15)/((x^2+x+1)) ?
The solution to 2x^2+2x+1<(15)/((x^2+x+1)) is -2<x<1