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Solve the inequality

\(\displaystyle 1 + \frac{1}{1 + 1/x} < \frac{8}{5}\)

 Jun 17, 2020
 #1
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First simplify the left hand side \(\frac{2x+1}{x+1}<\frac{8}{5}\).

Subtract 8/5 from both sides and simplify with a common denominator 5(x+1). Now our inequality looks like this:\(\frac{2x-3}{5x+5}<0\)

This can only be true if one is positive and one is negative. (if both are positive, their quotient is obviously positive. If both are negative, their -1 sign cancels out, and their quotient is also positive)

 

So we have two cases to consider:

 

Case 1: 2x-3 > 0, 5x+5 < 0.

Solving this gets us x > 3/2 and x < -1 which doesn't make sense.

 

Case 2: 2x-3 < 0, 5x+5 > 0.

Solving this gets us -1 < x < 3/2, which is our answer.

 Jun 17, 2020
edited by thelizzybeth  Jun 17, 2020

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