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

The solution to |(x^2-1)/(x^2+x+1)| 0

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|(x^2-1)/(x^2+x+1)|<1

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

- 2 < x < - \frac{1}{2} or x > 0

Solution steps

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

Apply absolute rule: If

∣ u ∣ < a, a > 0

then

- a < u < a

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

\frac{x ^{2} - 1}{x ^{2} + x + 1} > - 1 and \frac{x ^{2} - 1}{x ^{2} + x + 1} < 1

(x < - \frac{1}{2} or x > 0) and x > - 2

Merge Overlapping Intervals

- 2 < x < - \frac{1}{2} or x > 0

∣ 2 (x - 1) + 4 ∣ < 4

∣ x - 1 ∣ + ∣ x - 3 ∣ \leq 6

\frac{- 3}{x + \frac{1}{2}} < - 6

\frac{∣ x + 3 ∣ + x}{x + 2} > 1

∣ x - 3 ∣ > 2 x + 1

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