Why d o es y = x - 2
represent the
boundary line?
Fo r a n y v a l u e o f x, the
correspondi ng val ue of y
is the boundary between
values of y t hat are
greater than x - 2 and
values of y t hat are less
than x - 2.
T h i n k
Have you gr aphed
inequalities like
these before?
Ye s. I n Le sso n 3 - 1 , y o u
graphed inequalities in
one variable on a number
line. Here you graph
them in the coordinate
plane.
The graph of a linear inequality in two variables consists of all points in the coordinate
plane that represent solutions. The graph is a region called a half-plane that is bounded
by a line. All points on one side of the boundary line are solutions, while all points on
the other side are not solutions.Ea c h p o i n t o n a solid
line is a solution. A
so lid li ne is used f o r
inequalities with > or <.Gr ap h in g an In eq u alit y in Tw o Var iab les
What is the graph ofy > x — 2?First, graph the boundary line y = x - 2. Since the inequality symbol
is >, the points on the boundary line are not solutions. Use a dashed
line to indicate that the points are not included in the solution.
To determine which side of the boundary line to shade, test a point that
is not on the line. For example, test the point (0, 0).
y>x~ 2
0 < 0 - 2 Su b s t i t u t e ( 0 , 0 ) f o r ( x , y ).
0 > - 2 (0, 0) is a solution.
Because the point (0, 0) is a solution of the inequality, so are all the points on the same
side of the boundary line as (0, 0). Shade the area above the boundary line.Go t It? 2. What is the graph ofy — |x + 1?|y
(^1) L
/
/ X
-2 0 /
(^11) f
:>
y>
1 y ft
3 L 1 /
2 f X
2 0 / 2
V
ft '
Ea c h p o i n t o n a dashed
line is not a solution. A
dashed line is used for
inequalit ies wit h > or <.
L. I....... jV -y^3
\ ) <
X
-2 O. -
'
rX +1
An inequality in one variable can be graphed on a number line or in the coordinate
plane. The boundary line will be a horizontal or vertical line.
Gr ap h in g a Lin ear In eq u alit y in On e Var iab le
What is the graph of each inequality in the coordinate plane?
Qx>-1 Qys 2
Graph x = — 1 using a dashed line.
Use (0, 0) as a test point.
x > —1
0 > -It/
Shade on the side
of the line that
contains ( 0 , 0 ).
4 ?y
i:
1 x
10
__LJ
t
Graph y = 2 using a solid line.
Use (0, 0) as a test point,
y — 2
0> 2 X
Shade on the side
of the line that does
not contain ( 0 , 0 ).
y lf |
4
\ * j
X
0 '
j
c
Lesso n 6- 5 Linear Inequalities^395