Intermediate Algebra (11th edition)

664 CHAPTER 11 Nonlinear Functions, Conic Sections, and Nonlinear Systems

OBJECTIVES

Second-Degree Inequalities and Systems of Inequalities

11.5

1 Graph second-

degree inequalities.

2 Graph the solution

set of a system of

inequalities.

OBJECTIVE 1 Graph second-degree inequalities. A second-degree inequal-

ityis an inequality with at least one variable of degree 2 and no variable with degree

greater than 2.

Graphing a Second-Degree Inequality

Graph

The boundary of the inequality is the graph of the equation

a circle with radius 6 and center at the origin, as shown in FIGURE 33.

The inequality will include either the points outside the circle or

the points inside the circle, as well as the boundary. To

decide which region to shade, we substitute any test

point not on the circle into the original inequality.

Original inequality

Use as a test point.

✓ True

Since a true statement results, the original inequality

includes the points insidethe circle, the shaded region

in FIGURE 33, and the boundary.

0 ... 36

02 + 02 ... 1 0, 0 2

?

36

x^2 + y^2 ... 36

x^2 +y^2 ... 36

x^2 + y^2 = 36,

x^2 +y^2 ... 36

x^2 +y^2 ... 36.

EXAMPLE 1

xx

y

0

(0, 0)

Test

point

• 6


• 6

6

6

x^22 y^22 36

FIGURE 33

NOTE Since the substitution is easy, the origin is the test point of choice unless the

graph actually passes through 1 0, 0 2.

Graphing a Second-Degree Inequality

Graph

The boundary, is a parabola that opens down with vertex

at

Original inequality

Use as a test point.

Simplify.

False

Because the final inequality is a false statement, the

points in the region containing do not satisfy

the inequality. In FIGURE 34the parabola is drawn as

a dashed curve since the points of the parabola itself

do not satisfy the inequality, and the region inside

(or below) the parabola is shaded.

1 0, 0 2

0 6- 35

06

?

- 32 - 3

06 1 0, 0 2

?

- 210 - 422 - 3

y6- 21 x- 422 - 3

1 4, - 32.

y=- 21 x- 422 - 3,

y6- 21 x- 422 - 3.

EXAMPLE 2

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yyyyyyyyyyyyyyyyyyyyy

(( 44 ,,,,,,,,–– 3 ))

((((((((( 33 ,,,,,,– 5555 ))))))))) ((((((((( 5555 ,,–– 55 )))))))))

0

y<<–2(2( 22 ((x– 4444 ))^222 – 33

FIGURE 34

NOW TRY
EXERCISE 1
Graph x^2 +y^2 Ú9.

NOW TRY ANSWERS

1. 
   

x

y

0 3

–3

x^2 + y^2 êê 9

NOW TRY
EXERCISE 2
Graph yÚ- 1 x+ 222 +1.

2.

x

y

0

–3

–3

(–2, 1) 1

y êê –(x + 2)^2 + 1

NOW TRY

NOW TRY