Answer
The inequality is valid for values less than -1/3 and more than 5 (including them) i.e. $(-\infty,-\frac{1}{3}]\cap [5,\infty)$
Work Step by Step
First, we are going to set the right side to zero and factor to find the x-intercepts:
$3x^2=14x+5$
$3x^2-14x-5=14x+5-14x-5$
$3x^2-14x-5=0$
$(3x+1)(x-5)=0$
$x_1=-1/3$
$x_2=5$
These are the critical points. We are going to take three values: one less than -1/3, one between -1/3 and 5, and one more than 5 to test in the original equation and check if the inequality is true or not:
First test with a value less than -1/3:
$3(-1)^2\geq14(-1)+5$
$3(1)\geq-14+5$
$3\geq-9 \rightarrow \text{ TRUE}$
Second test with a value between -1/3 and 5:
$3(0)^2\geq14(0)+5$
$3(0)\geq0+5$
$0\geq5 \rightarrow \text{ FALSE}$
Third test with a value more than 5:
$3(6)^2\geq14(6)+5$
$3(36)\geq86+5$
$108\geq91 \rightarrow \text{ TRUE}$
These tests show that the inequality $3x^2\geq14x+5$ is valid for values less than -1/3 and more than 5 (including them) i.e. $(-\infty,-\frac{1}{3}]\cap [5,\infty)$
