Intermediate Algebra (11th edition)

Graphing a Linear System of Three Inequalities

Graph the solution set of the system.

Graph each inequality separately, on the same

axes. The graph of consists of all points

that lie below the dashed line The

graph of is the region that lies below

the solid line Finally, the graph of

is the region above the solid horizontal

line

The graph of the system, the intersection of

these three graphs, is the triangular region enclosed

by the three boundary lines in FIGURE 39, including

two of its boundaries.

y=-2.

yÚ- 2

y= 2 x+3.

y... 2 x+ 3

x+y=1.

x+ y 61

yÚ- 2

y... 2 x+ 3

x+y 61

EXAMPLE 5

x

y

0

1

3

–2

y ≥ –2

y ≤ 2 x + 3

x + y < 1

FIGURE 39

Graphing a System of Three Inequalities

Graph the solution set of the system.

The graph of is a parabola with vertex at Those points

above (or in the interior of ) the parabola satisfy the condition

Thus, the solution set of includes points on the parabola or in the

interior.

The graph of the equation is an ellipse. We draw it as a dashed

curve. To satisfy the inequality a point must lie outside the ellipse.

The graph of includes all points below the dashed line

The graph of the system is the shaded region in FIGURE 40, which lies outside the

ellipse, inside or on the boundary of the parabola, and below the line y= 4.

y 64 y=4.

2 x^2 +y^27 4,

2 x^2 +y^2 = 4

yÚx^2 - 2 x+ 1

y 7 x^2 - 2 x+ 1.

y= x^2 - 2 x+ 1 1 1, 0 2.

y 64

2 x^2 + y^274

yÚx^2 - 2 x+ 1

EXAMPLE 6

x

y

0

2

–2

4

y < 4

2 x^2 + y^2 > 4

y ≥ x^2 – 2x + 1

FIGURE 40

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5.

NOW TRY
EXERCISE 6
Graph the solution set of the
system.

y+ 370

y...x^2 - 2

x^2

4

+

y^2

16

... 1

6. 
   

NOW TRY
EXERCISE 5
Graph the solution set of the
system.

xÚ 0

yÚ

1

2

x- 2

3 x+ 2 y 76

x

y

–2

3

0 24

3 x + 2y > 6

2

y êê^1 x – 2

x êê 0

x

y

–4

–2

4

2

0

+ 16 yÄÄ 1

2

4

x^2

y ÄÄ x^2 – 2

y + 3 > 0

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

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