Answer
$(-5, 3/2)$ U $(4, ∞)$
Work Step by Step
$(3x-12)(x+5)(2x-3) \geq 0$
$3x-12 \geq 0$
$3x-12+12 \geq 0 +12$
$3x \geq 12$
$3x/3 \geq 12/3$
$x \geq 4$
$x+5\geq 0$
$x+5-5 \geq 0-5$
$x \geq -5$
$2x-3\geq0$
$2x-3+3 \geq 0+3$
$2x \geq 3$
$2x/2 \geq 3/2$
$x \geq 3/2$
We have four sections: $(-∞,-5)$, $(-5, 3/2)$, $(3/2,4)$, and $(4, ∞)$. 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 $\leq$ sign, we exclude the end points and use parentheses instead of brackets.
Let $x=-10$, $x=0$, $x=2$, and $x=10$
$x=-10$
$(3x-12)(x+5)(2x-3) \geq 0$
$(3*-10-12)(-10+5)(2*-10-3) \geq 0$
$(-30-12)(-5)(-20-3) \geq 0$
$-42*-5*-23 \geq 0$
$-4830 \geq 0$ (false)
$x=0$
$(3x-12)(x+5)(2x-3) \geq 0$
$(3*0-12)(0+5)(2*0-3) \geq 0$
$(0-12)(5)(0-3) \geq 0$
$-12*5*-3 \geq 0$
$180 \geq 0$ (true)
$x=2$
$(3x-12)(x+5)(2x-3) \geq 0$
$(3*2-12)(2+5)(2*2-3) \geq 0$
$(6-12)*7*(4-3)\geq 0$
$-6*7*1\geq 0$
$-42 \geq 0$ (false)
$x=10$
$(3x-12)(x+5)(2x-3) \geq 0$
$(3*10-12)(10+5)(2*10-3) \geq 0$
$(30-12)(15)(20-3)\geq 0$
$18*15*17 \geq 0$
$4590 \geq 0$ (true)
