Answer
$[-5, 0]$ U $(3/4, ∞)$
Work Step by Step
$(x)(x+5)/(4x-3) \geq 0$
Numerator is zero when $x=0$ and $x+5=0$. Denominator is zero when $4x-3=0$
$x+5=0$
$x+5-5=0-5$
$x=-5$
$4x-3=0$
$4x-3+3=0+3$
$4x=3$
$4x/4=3/4$
$x=3/4$
Four regions to test: $(-∞, -5]$, $[-5, 0]$, $[0, 3/4)$, $(3/4, ∞)$
Let $x=-10$, $x=-2$, $x=1/4$, $x=1$
$x=-10$
$(x)(x+5)/(4x-3) \geq 0$
$(-10)(-10+5)/(4*-10-3) \geq 0$
$(-10)(-5)/(-40-3) \geq 0$
$50/-43 \geq 0$
$-50/43 \geq 0$ (false)
$x=-2$
$(x)(x+5)/(4x-3) \geq 0$
$(-2)(-2+5)/(4*-2-3) \geq 0$
$(-2)(3)/(-8-3) \geq 0$
$-6/-11 \geq 0$
$6/11 \geq 0$ (true)
$x=1/4$
$(x)(x+5)/(4x-3) \geq 0$
$(1/4)(1/4+5)/(4*1/4-3) \geq 0$
$(1/4)(21/4)/(1-3) \geq 0$
$(21/16)/-2 \geq 0$
$-21/32 \geq 0$ (false)
$x=1$
$(x)(x+5)/(4x-3) \geq 0$
$(1)(1+5)/(4*1-3) \geq 0$
$(1)(6)/(4-3) \geq 0$
$6/1 \geq 0$
$6 \geq 0$ (true)
