Transcript 6.5 Graphing Linear Inequalities in Two Variables

6.5 Graphing Linear Inequalities in Two
Variables
7.6 Graphing Linear Inequalities System
Objectives
1. Graph linear inequalities.
2. Graph systems of linear inequalities.
Linear Inequalities
A linear inequality in two variables is an inequality
that can be written in the form
Ax + By < C,
where A, B, and C are real numbers and A and B are not
both zero. The symbol < may be replaced with , >, or .
The solution set of an inequality is the set of all ordered
pairs that make it true. The graph of an inequality
represents its solution set.
To Graph a Linear Inequality
Step
(1) Solve for y, convert the inequalities to Slope-Intercept Form.
If one variable is missing, solve it and go to step (2).
(2) Graph the related equation.
** If the inequality symbol is < or >, draw the line dashed.
** If the inequality symbol is  or , draw the line solid.
(3) If y < or y  , shade the region BELOW the line
If y > or y  , shade the region ABOVE the line
If x < or x  , shade the region LEFT to the line
If x > or x  , shade the region RIGHT to the line
Example
Graph y > x  4.
(1) Already in S-I form.
(2) The related equation is y = x
 4. Use a dashed line because
the inequality symbol is >. This
indicates that the line itself is
not in the solution set.
(3) Determine which half-plane
satisfies the inequality.
y>x4
“>” shade above
Example
Graph: 4x + 2y  8
1. Convert to S-I form:
y  -2x + 4
2. Graph the related
equation.
3. Determine which halfplane satisfies the
inequality.
“y ” shade below
Example
Graph 2x – 4 > 0 .
1. One variable is missing.
Solve it:
x>2
2. Graph the related equation
x=2
3. Determine which halfplane satisfies the
inequality.
“x >” shade the right.
Example
Graph 8 - 3y  2 .
1. One variable is missing.
Solve it:
y2
2. Graph the related equation
y=2
3. Determine the region to be
shaded.
“y ” shaded below
7.6 Systems of Linear Inequalities
Graph the solution set of
the system.
x y3
x y 1
First, we graph x + y  3
using a solid line.
y -x+3
“above”
Next, we graph x  y > 1
using a dashed line.
y< x–1
“below”
The solution set of the system of
equations is the region shaded both
red and green, including part of the
line x + y = 3.
Example
Graph the following system
of inequalities and find the
coordinates of any vertices
formed:
y20
x  y  2
x y0
We graph the related
equations using solid lines.
We shade the region common
to all three solution sets.
Example continued
y20
(1 )
The system of equations from
inequalities (1) and (3):
x  y  2
(2 )
y+2=0
x y0
(3 )
x+y=0
To find the vertices, we solve The vertex is (2, 2).
three systems of equations. The system of equations from
inequalities (2) and (3):
The system of equations
from inequalities (1) and (2): x + y = 2
x+y=0
y+2=0
The vertex is (1, 1).
x + y = 2
The vertex is (4, 2).
Summary
1. To graph a two-variable linear inequality, graph the
related equation first (variable y must be solved) with
appropriate boundary line:
“<” and “>” use dash line
“≤” and “≥” use solid line
2. “y < …” and “y ≤ …” shade the region below
3. “y > …” and “y ≥ …” shade the region above
4. “x < …” and “x ≤ …” shade the region left
5. “x > …” and “x ≥ …” shade the region right
6. When graphing the linear inequality system, follow
the step 1 ~ 3 and choose the region shaded most.
Assignment
6.5 P 363 #’s 10 - 22 (even), 32 - 40 (even), 45 - 60
(even)
7.6 P 435 #’s 9 - 26