# Find the range of values of x for which (x^2+x+1)/(x^2+2)<1/3, x being real.

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Find the range of values of x for which $\frac{x^2+x+1}{x^2+2}$ < $\frac13$, $x$ being real.

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$\frac{x^2+x+1}{x^2+2}$ - $\frac13$ < 0

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

Now we have to find the range of values in which 2x2 + 3x + 1 < 0 as 3(x2 + 2) is +ve for all real values of x.

Now 2x2 + 3x + 1 < 0 ⇒ 2x2 + 2x + x + 1 < 0

⇒ (2x + 1) (x + 1) < 0  ⇒ $\big(x+\frac12\big)(x + 1)<0.$

Critical points are –1 and $-\frac12$ . Plotting the critical points on the real number line and observing the sign of the expression (2x + 1) (x + 1) in the intervals formed, we see that the expression $\big(x+\frac12\big)$ (x + 1) is negative or less than zero in the interval $\big(-1,-\frac12\big)$, , i.e., –1 < x < $-\frac12.$

$x$ ∈ $\big(-1,-\frac12\big)$