Answer
$(-∞,-4)$ U $(4,6)$
Work Step by Step
$(2x-8)(x+4)(x-6) \leq 0$
$2x-8\leq 0$
$2x-8+8 \leq 0$
$2x \leq 8$
$2x/2 \leq 8/2$
$x \leq 4$
$x+4 \leq 0$
$x+4-4 \leq 0-4$
$x \leq -4$
$x-6\leq 0$
$x-6+6 \leq 0+6$
$x \leq 6$
We have four sections: $(-∞,-4)$, $(-4, 4)$, $(4,6)$, and $(6, ∞)$. 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 $x=-5$, $x=0$, $x=5$, and $x=10$
$x=-5$
$(2x-8)(x+4)(x-6) \leq 0$
$(2*-5-8)(-5+4)(-5-6) \leq 0$
$(-10-8)(-1)(-11) \leq 0$
$-18*11 \leq 0$
$-198 \leq 0$ (true)
$x=0$
$(2x-8)(x+4)(x-6) \leq 0$
$(2*0-8)(0+4)(0-6) \leq 0$
$(0-8)*4*-6 \leq 0$
$-8 *-24 \leq 0$
$192 \leq 0$ (false)
$x=5$
$(2x-8)(x+4)(x-6) \leq 0$
$(2*5-8)(5+4)(5-6) \leq 0$
$(10-8)(9)(-1)\leq 0$
$2*-9 \leq 0$
$-18 \leq 0$ (true)
$x=10$
$(2x-8)(x+4)(x-6) \leq 0$
$(2*10-8)(10+4)(10-6) \leq 0$
$(20-8)(14*4)\leq 0$
$12 *56 \leq 0$
$672 \leq 0$ (false)
