What is ((x^2+x+1))/((x^2+2))< 1/3 ?

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

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((x^2+x+1))/((x^2+2))< 1/3

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

- 1 < x < - \frac{1}{2}

Interval Notation

(- 1, - \frac{1}{2})

Decimal Notation

- 1 < x < - 0.5

Solution steps

\frac{x ^{2} + x + 1}{x ^{2} + 2} < \frac{1}{3}

Rewrite in standard form

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

Factor

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

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

Identify the intervals

- 1 < x < - \frac{1}{2}

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

\frac{0.5}{x - x ^{2} - 1} < 0

(x - 1)^{3} < 4 (x - 1)

(x + 2)^{2} > 9

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

What is ((x^2+x+1))/((x^2+2))< 1/3 ?
The solution to ((x^2+x+1))/((x^2+2))< 1/3 is -1<x<-1/2