SOLUTION: The solution of the inequality: -x < x^2 < 2x + 1, is { ?? < x < ??} Thanks in advance!

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Question 860535: The solution of the inequality: -x < x^2 < 2x + 1, is { ?? < x < ??}
Thanks in advance!
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Analyze two separate inequalities.
-x%3Cx%5E2 AND x%5E2%3C2x%2B1; look for what is true for both at the same time.

-x%5E2-x%3C0
x%5E2%2Bx%3E0
x%28x%2B1%29%3E0
Critical points 0 and -1.
Solutions: x%3C-1 OR x%3E0.

x%5E2-2x-1%3C0
discriminant, 4-4(-1)=8
roots or horiz intercepts will be the critical points:
x=%282-2sqrt%282%29%29%2F2=1-sqrt%282%29
OR
x=1%2Bsqrt%282%29.
The parabola opens upward, and the points below the level y=0 occur BETWEEN the horizontal intercepts.

The x values common for both inequalities are highlight%280%3Cx%3C1%2Bsqrt%282%29%29, which is easier to see if you make a number line.