Intermediate Algebra (11th edition)

SECTION 11.5 Second-Degree Inequalities and Systems of Inequalities 665

Graphing a Second-Degree Inequality

Graph

Subtract

Divide by 144.

This form shows that the boundary is the hyperbola

given by

Since the graph is a vertical hyperbola, the desired

region will be either the region between the branches

or the regions above the top branch and below the

bottom branch. Choose as a test point. Substi-

tuting into the original inequality leads to

a true statement, so the region between the branches

containing 1 0, 0 2 is shaded, as shown in FIGURE 35.

0 ...144,

1 0, 0 2

y^2

9

-

x^2

16

=1.

y^2

9

-

x^2

16

... 1

16 y^2 - 9 x^2 ... 144 9 x^2.

16 y^2 ... 144 + 9 x^2.

EXAMPLE 3

x

y

16 y^2 ≤ 144 + 9x^2

0

3

–3

–4 4

FIGURE 35

OBJECTIVE 2 Graph the solution set of a system of inequalities. If two or

more inequalities are considered at the same time, we have a system of inequalities.

To find the solution set of the system, we find the intersection of the graphs (solution

sets) of the inequalities in the system.

Graphing a System of Two Inequalities

Graph the solution set of the system.

Begin by graphing the solution set of The boundary line is the

graph of and is a dashed line because of the symbol The test point

leads to a false statement in the inequality so shade the region

above the line, as shown in FIGURE 36.

The graph of is the interior of a dashed circle centered at the ori-

gin with radius 4. This is shown in FIGURE 37.

x^2 +y^2616

1 0, 0 2 2 x+ 3 y 7 6,

2 x+ 3 y= 6 7.

2 x+ 3 y 7 6.

x^2 +y^2616

2 x+ 3 y 7 6

EXAMPLE 4

x

y

0

2

3

2 x + 3y > 6

FIGURE 36

x

y

04

4

x^2 + y^2 < 16

–4

–4

FIGURE 37

x

y

03

2

4

x^2 + y^2 < 16

2 x + 3y > 6

FIGURE 38

NOW TRY

NOW TRY
EXERCISE 3
Graph 25x^2 - 16 y^27 400.

NOW TRY ANSWERS

3. 
   
   
   
   4. 
      
   
   
   
   

x

y

–5

–4 4

5

0

25 x^2 – 16y^2 > 400

The graph of the solution set of the system is the intersection of the graphs of the

two inequalities. The overlapping region in FIGURE 38is the solution set. NOW TRY

NOW TRY
EXERCISE 4
Graph the solution set of the
system.

y 7 x^2 - 1

x^2 +y^279

x

y

–3–1 3

3

0

x^2 + y^2 > 9

y > x^2 – 1