Answer
$(-∞,-2/3)$ U $(3/2, ∞)$
Work Step by Step
$6x^2-5x \geq 6$
$6x^2-5x-6 \geq 6-6$
$6x^2-5x-6 \geq 0$
$(3x+2)(2x-3) \geq 0$
$3x+2\geq 0$
$3x+2-2 \geq 0-2$
$3x \geq -2$
$3x/3 \geq -2/3$
$x \geq -2/3$
$2x-3\geq 0$
$2x-3+3 \geq 0+3$
$2x \geq 3$
$2x/2 \geq 3/2$
$x \geq 3/2$
We have three sections: $(-∞,-2/3)$, $(-2/3, 3/2)$, and $(3/2, ∞)$. 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 $x=-3$, $x=0$, and $x=2$
$x=-3$
$6x^2-5x \geq 6$
$6(-3)^2-5*(-3) \geq 6$
$6*9+15 \geq 6$
$54+15 \geq 6$
$69 \geq 6$ (true)
$x=0$
$6x^2-5x \geq 6$
$6*0^2-5*0 \geq 6$
$6*0-0 \geq 6$
$0-0 \geq 6$
$0 \geq 6$ (false)
$x=2$
$6x^2-5x \geq 6$
$6*2^2-5*2 \geq 6$
$6*4-10\geq 6$
$24-10 \geq 6$
$14 \geq 6$ (true)
