Answer
$(-∞, -6)$ U $(-3/4,0)$ U $(5,∞)$
Work Step by Step
$(4x+3)(x-5)/x*(x+6) > 0$
$x+6=0$
$x+6-6=0-6$
$x=-6$
The denominator is zero when $x=0$ and $x=-6$.
$4x+3=0$
$4x+3-3=0-3$
$4x=-3$ $4x/4=-3/4$
$x=-3/4$
$x-5=0$
$x-5+5=0+5$
$x=5$
The numerator is zero when $x=-3/4$ and $x=5$.
Five regions to test: $(-∞, -6)$, $(-6, -3/4)$, $(-3/4,0)$, $(0,5)$, $(5,∞)$
Let $x=-10$, $x=-1$, $x=-1/2$, $x=1$, $x=6$
$x=-10$
$(4x+3)(x-5)/x*(x+6) > 0$
$(4*-10x+3)(-10-5)/-10*(-10+6) > 0$
$(-40+3)*-15/-10*(-4) > 0$
$-37*-15/40 >0$
$555/40 > 0$
$13.875 > 0$ (true, so this region is part of the solution set)
$x=-1$
$(4x+3)(x-5)/x*(x+6) > 0$
$(4*-1+3)(-1-5)/-1*(-1+6) > 0$
$(-4+3)*-6/-1*5 >0$
$-1*-6 /-5 >0$
$6/-5 >0$
$-6/5 > 0$ (false, so this region is not part of the solution set)
$x=-1/2$
$(4x+3)(x-5)/x*(x+6) > 0$
$(4*-1/2+3)(-1/2-5)/-1/2*(-1/2+6) > 0$
$(-2+3)(-11/2)/-1/2*(5.5) >0$
$1*-5.5 /-1/2 *(5.5) >0$
$-1/ -1/2 > 0$
$1/ .5 >0$
$2 > 0$ (true, so this region is part of the solution set)
$x=1$
$(4x+3)(x-5)/x*(x+6) > 0$
$(4*1+3)(1-5)/1*(1+6) > 0$
$(4+3)*-4/1*7 >0$
$7*-4/7 >0$
$-4 >0$ (false, so this region is not part of the solution set)
$x=6$
$(4x+3)(x-5)/x*(x+6) > 0$
$(4*6+3)(6-5)/6*(6+6) > 0$
$(24+3)*1/6*12 > 0$
$27*1/72 >0$
$27/72 > 0$
$9/24 >0$
$3/8 >0$ (true, so this region is part of the solution set)
