Answer
(-infinity, $0)$ U $(5, 11/2)$
Work Step by Step
$z/(z-5) \geq 2z$
Denominator is zero when $z=5$
$z/(z-5) \geq 2z$
$z/(z-5) = 2z$
$z*(z-5)/(z-5) = 2z*(z-5)$
$z=2z^2-10z$
$0=2z^2-10z-z$
$0=2z^2-11z$
$0=z(2z-11)$
$z=0$
$2z-11=0$
$2z=11$
$2z/2=11/2$
$z= 11/2$
(-infinity, $0)$
$(0,5)$
$(5, 11/2)$
$(11/2$, infinity)
Let $z=-1$, $z=1$, $z=5.25$, $z=10$
$z=-1$
$z/(z-5) \geq 2z$
$-1/(-1-5) \geq 2*-1$
$-1/-6 \geq -2$
$1/6 \geq -2$ (true)
$z=1$
$z/(z-5) \geq 2z$
$1/(1-5) \geq 2*1$
$1/-4 \geq 2$
$-1/4 \geq 2$ (false)
$z=5.25$
$z/(z-5) \geq 2z$
$5.25/(5.25-5) \geq 2*5.25$
$5.25/.25 \geq 10.5$
$5.25*4/.25*4 \geq 10.5$
$21/1 \geq 10.5$
$21 \geq 10.5$ (true)
$z=10$
$z/(z-5) \geq 2z$
$10/(10-5) \geq 2*10$
$10/5 \geq 20$
$2 \geq 20$ (false)
