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Let a be the smallest integer satisfying the inequality x^2 - 15 < 2x, and let b be the largest integer satisfying the same inequality. What is b-a?

 May 23, 2022
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Let a be the smallest integer satisfying the inequality x^2 - 15 < 2x, and let b be the largest integer satisfying the same inequality. What is b-a?


\(x^2 - 15 < 2x\\ x^2 - 15 -2x <0\\ x^2 -2x - 15 <0\\\)

This is a concave up parabola

It will be less than 0 between the roots

the roots are

 

\(x = {2 \pm \sqrt{4+60} \over 2}\\ x = {2 \pm 8 \over 2}\\ x=1\pm4 \)

 

the equation is satified for all x values between -3 and 5  BUT  NOT inclusive of -3 and 5)

So the smallest integer is a=-2   and the largest is b=4

4 - - 2 = 6

 May 23, 2022
edited by Melody  May 23, 2022

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