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avatar+766 

Find all real x where
2 \cdot \frac{x - 5}{x - 3} > \frac{2x - 5}{x + 2} + 20.
Give your answer in interval notation.

 
 Mar 15, 2025
 #1
avatar+15054 
+1

Find all real x for

 

        I                II

2x5x3>2x5x+2+20.f(x)=2x5x32x5x+220>02(x5)(x+2)(2x5)(x3)20(x3)(x+2)>0(2x10)(x+2)(2x26x5x+15)20(x2+2x3x6)>0

2x2+4x10x202x2+11x1520x240x+60x+120>0f(x)=20x2+25x+85>0

x1,2=b±b24ac2ax1,2=25±252+4208540x0{1.5292,2.7792}

The extreme points of the hyperbolas I and II are xI=3 and xII=2.

In the calculated range is(2x5x3) < (2x5x+2+20)xR (1.52921099245<x<2.77921099245)

The extreme points of the hyperbolas I and II are xI=3 and xII=2.This gives:(2x5x3) > (2x5x+2+20)xR(2<1.5292)andx=R(2.7792<3)

 

laugh !

 Mar 15, 2025
edited by asinus  Mar 16, 2025
edited by asinus  Mar 16, 2025, 11:34 PM
edited by asinus  Mar 16, 2025, 11:37 PM
edited by asinus  Mar 16, 2025, 11:40 PM
edited by asinus  Mar 16, 2025, 11:42 PM
edited by asinus  Mar 17, 2025, 5:05:27 AM
 #2
avatar+15054 
0

This gives:(2x5x3) > (2x5x+2+20)

xR(-2 < x < -1.5292)

and

xR(2.7792 < x < 3)

 

Sorry. It didn't display correctly in answer 1#.

laugh !

 Mar 16, 2025, 11:47 PM
edited by asinus  Mar 16, 2025, 11:54 PM

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