Answer
$(−∞,-1)$ U $(0,6)$ U $(7,∞)$
Work Step by Step
$x(x-6)/(x-7)(x+1)) \ge 0$
$x-6=0$
$x-6+6=0+6$
$x=6$
$x-7=0$
$x-7+7=0+7$
$x=7$
$x+1=0$
$x+1-1=0-1$
$x=-1$
The denominator is zero when $x=7$ or $x=-1$, and the numerator is zero when $x=0$ or $x=6$.
We have five sections: $(−∞,-1)$, $(-1, 0)$, $(0,6)$, $(6,7)$, and $(7,∞)$. 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 $\ge$ sign, we include the end points and use brackets instead of parentheses.
Let $x=-3$, $x=-.5$, $x=2$, $x=6.1$, and $x=10$
$x=-3$
$x(x-6)/(x-7)(x+1)) \ge 0$
$-3(-3-6)/(-3-7)(-3+1)) \ge 0$
$-3*-9/(-10*-2) \ge 0$
$27/20 \ge 0$ (true)
$x=-.5$
$x(x-6)/(x-7)(x+1)) \ge 0$
$-.5(-.5-6)/(-.5-7)(-.5+1)) \ge 0$
$-.5 (-6.5)/(-7.5)(.5) \ge 0$
$3.25/-3.75 \ge 0$
$6.5/-7.5 \ge 0$
$-13/15 \ge 0$ (false)
$x=2$
$x(x-6)/(x-7)(x+1))\ge0$
$2(2-6)/(2-7)(2+1))\ge 0$
$2*-4/-5*3\ge 0$
$-8/-15 \ge 0$
$8/15 \ge 0$ (true)
$x=6.1$
$x(x-6)/(x-7)(x+1))$
$6.1(6.1-6)/(6.1-7)(6.1+1))\ge0$
$6.1*.1/-.9*7.1\ge0$
$.61/-.639 \ge0$
$-.61/.639 \ge 0$ (false)
$x=10$
$x(x-6)/(x-7)(x+1))\ge0$
$10(10-6)/(10-7)(10+1))\ge0$
$10*4/3*11\ge0$
$40/33 \ge0$ (true)
