Exercise 38 Page 643

We want to solve the given system of inequalities by graphing.

{x^{2} < y - 2 y \leq ∣ x + 8 ∣ - 7 (I) (II)

Let's graph each inequality one at a time.

Inequality (I)

Notice that Inequality (I) is a quadratic inequality. We can write the boundary curve for Inequality (I) by replacing the less than sign with an equals sign.

x^{2} = y - 2

Now, we can isolate the

y -

variable and identify

a,

b,

and

c .

x^{2} = y - 2

AddEqn

$LHS + 2 = RHS + 2$

x^{2} + 2 = y

RearrangeEqn

Rearrange equation

y = x^{2} + 2

IdPropMult

Identity Property of Multiplication

y = 1 x^{2} + 2

IdPropAdd

Identity Property of Addition

y = 1 x^{2} + 0 + 2

ZeroPropMult

Zero Property of Multiplication

y = 1 x^{2} + 0 x + 2

Knowing that

a = 1,

b = 0,

and

c = 2,

we can find the vertex. To do so, we will need to think of

y

as a function of

x,

y = f (x) .

Vertex of the Parabola : (- \frac{b}{2 a}, f (- \frac{b}{2 a}))