Answer
$[-5, 0]$ U $(3/4, ∞)$
Work Step by Step
$x*(x+5)/(4x-3) \ge 0$
$4x-3=0$
$4x-3+3=0+3$
$4x=3$
$4x/4 =3/4$
$x=3/4$
The denominator is zero when $x=3/4$.
$x+5=0$
$x+5-5=0-5$
$x=-5$
$x=0$
The numerator is zero when $x=0$ and $x=-5$.
Four regions to test: $(-∞, -5]$, $[-5, 0]$, $[0, 3/4)$, $(3/4, ∞)$
Let $x=-6$, $x=-1$, $x=1/2$, $x=1$
$x=-6$
$x*(x+5)/(4x-3) \ge 0$
$-6*(-6+5)/(4*-6-3) \ge 0$
$-6*-1/(-24-3) \ge 0$
$6/-27 \ge 0$
$-2/9 \ge 0$ (false)
$x=-1$
$x*(x+5)/(4x-3) \ge 0$
$-1*(-1+5)/(4*-1-3) \ge 0$
$-1*4/(-4-3) \ge 0$
$-4 /-7 \ge 0$
$4/7 \ge 0$ (true)
$x=1/2$
$x*(x+5)/(4x-3) \ge 0$
$1/2*(1/2+5)/(4*1/2-3) \ge 0$
$1/2*11/2 / 2-3 \ge 0$
$11/4 / -1 \ge 0$
$-11/4 \ge 0$ (false)
$x=1$
$x*(x+5)/(4x-3) \ge 0$
$1*(1+5)/(4*1-3) \ge 0$
$1*6/(4-3) \ge 0$
$6/1 \ge 0$
$6 \ge 0$ (true)
