We want to solve the given system of inequalities by graphing. Note that both inequalities of the system are quadratic inequalities.

{y \geq x^{2} - 4 y \leq - 2 x^{2} + 7 x + 4 (I) (II)

Let's graph each of them, one at a time.

Inequality I

To graph the quadratic inequality, we will follow three steps.

Graph the related quadratic function.
Test a point not on the parabola.
Shade accordingly. If the point satisfies the inequality, we shade the region that contains the point. If not, we shade the opposite region.

Let's draw the graph of the related function, which is $y = x^{2} - 4 .$

Next, let's determine which region to shade by testing a point. For simplicity, we will use

(0, 0)

as our test point. Let's see if it satisfies the given inequality.

y \geq x^{2} - 4

SubstituteII

$x = 0$ , $y = 0$

0 \geq ? (0)^{2} - 4

CalcPow

Calculate power

0 \geq ? 0 - 4

SubTerms

Subtract terms

0 \geq - 4 ✓

Since

(0, 0)

produced a true statement, we will shade the region that contains the point. Also, note that the inequality is not strict. Therefore, the parabola will be solid.

Inequality II

We will follow the same three steps as before. Let's draw the graph of the related function, which is $y = - 2 x^{2} + 7 x + 4 .$

Next, let's determine which region to shade by testing a point. We will use $(0, 0)$ as our test point.

Inequality	Test Point	Statement	Is It the Good Region?
$y \leq - 2 x^{2} + 7 x + 4$	$(0, 0)$	$0 \leq 4 ✓$	Yes

Let's graph it!

Final Solution Set

The solution set is the overlapping region.

Exercise