Find integer integer x

Level 2

{ ( x − 1 ) ( x 2 − 7 x + 10 ) = 10 x ( x − 1 ) 2 − 2 < x ( x − 5 ) 2 + 6 \begin{cases} (x-1)(x^2-7x+10) = 10 \\ \dfrac {x(x-1)}2 - 2 < \dfrac {x(x-5)}2+6 \end{cases}

Find the number of real x x satisfying the system of equation and inequality above.

2 4 3 5

Manolis Arseniou by Manolis Arseniou , 31, Greece

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1 solution

Joshua Lowrance
Mar 15, 2019

( x − 1 ) ( x 2 − 7 x + 10 ) = ( x − 1 ) ( x − 2 ) ( x − 5 ) = > x = 1 , 2 , 5 (x-1)(x^2-7x+10) = (x-1)(x-2)(x-5) => x=1,2,5

x ( x − 1 ) 2 − 2 < x ( x − 5 2 + 6 \frac{x(x-1)}{2}-2<\frac{x(x-5}{2}+6

x 2 − x − 4 < x 2 − 5 x + 12 x^2-x-4<x^2-5x+12

4 x < 16 4x<16

x < 4 = > x ≠ 5 x<4 => x\neq 5

x = 1 , 2 (but as 1 is not a possible answer) x = 2 x=1,2 \text{ (but as 1 is not a possible answer) } x=2

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