Chapter 8 - Section 8.4 - Nonlinear Inequalities in One Variable - Exercise Set - Page 511: 51

$(−∞,0]$ U $(5, 11/2]$

$z/(z-5) \geq 2z$ $z/(z-5)-2z \geq 2z-2z$ $z/(z-5)-2z \geq 0$ $z/(z-5)-2z*(z-5)/(z-5)\geq 0$ $z/(z-5)-2z^2+10z/(z-5)\geq 0$ $(-2z^2+11z)/(z-5) \geq 0$ $-2z^2+11z=0$ $z(-2z+11)=0$ $-2z+11=0$ $-2z+11+2z=0+2z$ $11=2z$ $11/2 =2z/2$ $11/2 =z$ $z=0$ The denominator is zero when $z=5$, and the numerator is zero when $z=0$ or $z=11/2$. We have four sections: $(−∞,0)$, $(0,5)$, $(5, 11/2)$, and $(11/2,∞)$. We need to test one value for x in each section to determine if the section would be a solution set. Since we have the $\geq$ sign, we include the end points and use brackets instead of parentheses. Let $z=-1$, $z=1$, $z=5.1$, and $z=10$ $z=-1$ $z/(z-5) \geq 2z$ $-1/(-1-5) \geq 2*-1$ $-1/-6 \geq -2$ $1/6 \geq -2$ (true) $z=1$ $z/(z-5) \geq 2z$ $1/(1-5) \geq 2*1$ $1/-4 \geq 2$ $-1/4 \geq 2$ (false) $z=5.1$ $z/(z-5) \geq 2z$ $5.1/(5.1-5) \geq 2*5.1$ $5.1/.1 \geq 10.2$ $51 \geq 10.2$ (true) $z=10$ $z/(z-5) \geq 2z$ $10/(10-5) \geq 2*10$ $10/5 \geq 20$ $2 \geq 20$ (false)