Answer
The inequality is valid for values less than -4 and values more than 0.5 (including them) i.e. $(-\infty,-4]\cap [0.5,\infty)$.
Work Step by Step
First, we are going to set everything to the left side and factor:
$7x-4\geq-2x^2$
$2x^2+7x-4\geq0$
$(2x-1)(x+4)\geq0$
Now, we find critical points by equating the numerator and denominator to zero:
$(2x-1)(x+4)=0$
There are two critical points:
$x_1+4=0\rightarrow x_1=-4$
$2x_2-1=0\rightarrow x_2=\frac{1}{2}=0.5$
Next, we are going to take three values: one less than -4; one between -4 and 0.5; and one more than 0.5 to test in the original equation and check if the inequality is true or not:
First test with a value less than -4:
$7(-5)-4\geq-2(-5)^2$
$-35-4\geq-2(25)$
$-39\geq-50 \rightarrow \text{ TRUE}$
Second test with a value between -4 and 0.5:
$7(0)-4\geq-2(0)^2$
$-4\geq0 \rightarrow \text{ FALSE}$
Third test with a value more than 0.5:
$7(1)-4\geq-2(1)^2$
$7-4\geq-2$
$3\geq-2 \rightarrow \text{ TRUE}$
These tests show that the inequality $7x-4\geq-2x^2$ is valid for values less than -4 and values more than 0.5 (including them) i.e. $(-\infty,-4]\cap [0.5,\infty)$
