Answer
$[-6/5, 0)$ U $(5/6, 3]$
Work Step by Step
$(5x+6)(x-3)/(x)(6x-5) < 0$
Numerator is zero when either $5x+6 =0$ or $x-3=0$. Denominator is zero when $x=0$ or $6x-5=0$
$5x+6=0$
$5x+6-6=0-6$
$5x=-6$
$5x/5=-6/5$
$x=-6/5$
$x-3=0$
$x-3+3=0+3$
$x=3$
$x=0$
$6x-5=0$
$6x-5+5=0+5$
$6x=5$
$6x/6 =5/6$
Five regions to test: $(-∞, -6/5]$, $[-6/5, 0)$, $(0, 5/6)$, $(5/6, 3]$, $[3, ∞)$
Let $x=-2$, $x=-1$, $x=1/2$, $x=1$, $x=5$
$x=-2$
$(5x+6)(x-3)/(x)(6x-5) < 0$
$(5*-2+6)(-2-3)/(-2)(6*-2-5) < 0$
$(-10+6)(-5)/(-2)(-12-5) <0$
$(-4)(-5) /(-2)(-17) <0$
$20/34 < 0$ (false)
$x=-1$
$(5x+6)(x-3)/(x)(6x-5) < 0$
$(5*-1+6)(-1-3)/(-1)(6*-1-5) < 0$
$(-5+6)(-4)/(-1)(-6-5) < 0$
$(1)(-4)/(-1)(-11) < 0$
$-4/11 < 0$ (true)
$x=1/2$
$(5x+6)(x-3)/(x)(6x-5) < 0$
$(5*1/2+6)(1/2-3)/(1/2)(6*1/2-5) < 0$
$(5/2+6)(-5/2)/(1/2)(3-5) < 0$
$(17/2)(-5/2)/(1/2)(-2) < 0$
$-85/4/-1 < 0$
$85/4 <0$ (false)
$x=1$
$(5x+6)(x-3)/(x)(6x-5) < 0$
$(5*1+6)(1-3)/(1)(6*1-5) < 0$
$(5+6)(-2)/(1)(6-5) < 0$
$(11)(-2)/(1)(1) <0$
$-22 /1 <0$
$-22 <0 $ (true)
$x=5$
$(5x+6)(x-3)/(x)(6x-5) < 0$
$(5*5+6)(5-3)/(5)(6*5-5) < 0$
$(25+6)(2)/(5)(30-5) < 0$
$(31)(2)/(5)(25) <0$
$62/125 <0$ (false)
