Solving of system of linear inequalities

A linear inequality in two variables has an
infinite number of solutions. These solutions can
be represented in the coordinate plane as the set
of all points on one side of a boundary line.
A solution of an inequality in two
variables is an ordered pair that makes the
inequality true.

a. (1,2) b. (-3,-7)
(1,2) is a solution (-3,-7) is not a solution
Write the inequality
Substitute
Simplify

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.
-2 2
Each point on right side
of the line is not a
solution.A dashed line
used for inequalities with
> or <.
-2 2
Each point on solid line is
a solution.A solid line
used for inequalities with
≤ or ≥.

Used the dashed line to
indicate that the points are not
included in the solution.
-2 2
2
-2
y
x
To determine which side of the boundary line is a solution or
not, test a point that is not the line. For example, test the point
(0,0).
Substitute (0,0) for (x,y).
(0,0) is a solution.

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.

Example:
What is the graph of each inequality in the coordinate
plane?
Graph x= -1 using a dashed line. Use
(0,0) as a test point.
-2 2
2
-2
y
x
The side of the line contains (0,0).

Graph y=2 using a solid line. Use
(0,0) as a test point.
-2 2
2
-2
y
x
The side of the line does not contain (0,0).

When a linear inequality is solved for y, the direction of
the inequality symbol determines which side of the
boundary line. If the symbol is < or ≤, below is the
boundary line. If the symbol is > or ≥, above is the
boundary line.
-2 2
2
-2
y
x

-2 2
2
-2
y
x
Now, we put the two
graphs together on the
same grid to determine
the solution set for the
system.A solution to a
system of inequalities is
an ordered pair that
makes every inequality in
the system true.
-2 2
2
-2
y
x
Solution
region for
the
system.

-2 2
2
-2
y
x
Solution region
for the system.

To check, we can select a point in the solution region such as (3,0)
and verify that it makes both inequalities true.
First Inequality:
True
Second Inequality:
True
Since (3,0) makes both inequalities true, it is a solution to the
system.Though we selected only one ordered pair in the solution
region, remember that every ordered pair in that region is a
solution.

To solve a system of linear inequalities,
graph all of the inequalities on the
same grid. The solution set for the
system contains all ordered pairs in the
region where the inequalities’ solution
sets overlap along with ordered pairs
on the portion of any solid line that
touches the region of overlap

Example:
Graph the solution set for the system of inequalities.
-2 2
2
-2
y
x
Solution region
for the system.
Solution:
Graph the inequalities on the
same grid. Because both lines
are dashed, the solution set
for the system contains only
does ordered pairs in the
region of overlap (written in
color blue).
Note: Ordered pairs on the
dashed lines are not part of
the solution region.

-2 2
2
-2
y
x
Solution
region for
the system.
Solution:
Graph the inequalities on the
same grid.The solution set for
this system contains all ordered
pairs in the region of overlap
(written on color blue) together
with all ordered pairs on the
portion of the solid line that
touches the solution region for
the system.

Inconsistent Systems
Some systems of linear inequalities have no solution.
We say these systems are consistent
-2 2
2
-2
y
x
Solution:

The slopes are equal, so the
lines are in fact parallel. Since
the lines are parallel and the
region do not overlap, there is
no solution region for this
system. The system is
inconsistent.

Seatwork
In problem 1-6, determine whether the ordered pair is a
solution of the linear inequality or not.
In problem 7-10, graph each linear inequality.

A.Answer the remaining activity
in page 279 (11-6).
B. Advance study about
deduction and proving triangles
congruent.

Thanks for
Listening 
Prepared by:
Ricie Anne V. Palisoc

Solving of system of linear inequalities

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