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# problem - x is between the two zeros(think of the graph of...

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Some problem from class Andrew Berget October 3, 2011 Here’s the problem: Solve the inequality 1 + x 1 - x - 1 - x 1 + x < - 1 . Here’ the solution. We combine these into a single fraction. 1 + x 1 - x - 1 - x 1 + x = (1 + x ) 2 - (1 - x ) 2 (1 - x )(1 + x ) = 4 x (1 - x )(1 + x ) = 4 x 1 - x 2 . So we want to solve 4 x 1 - x 2 < - 1 Moving the - 1 to the other side of the inequality this becomes 4 x 1 - x 2 + 1 = 4 x + 1 - x 2 1 - x 2 < 0 ⇐⇒ x 2 - 4 x - 1 1 - x 2 > 0 We need to determine where the numerator and denominator have the same sign. The denominator is positive if - 1 < x < 1 and negative otherwise.
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Unformatted text preview: x is between the two zeros (think of the graph of this parabola). The zeros are x = 2 ± √ 5. Hence the numerator is negative if 2-√ 2 ≈ -. 2 < x < 2 + √ 5 ≈ 4 . 2. I drew a picture of the number line, plotting x = 1 ,-1 , 2+ √ 5 , 2-√ 5 to ﬁnd where the numerator and denominator have the same sign. I got the following answer: 1 < x < 2 + √ 5 or-1 < x < 2-√ 5 . 1...
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