SYSTEMS OF LINEAR INEQUALI

TIES

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.

Example:

Is the given ordered pair a solution of ?

a. (1,2)

b. (-3,-7)

𝑦>𝑥−3

2>1−3

2>−2

(1,2) is a solution

𝑦>𝑥−3

−7>−3−3

−7>−6

(-3,-7) is not a solution

Write the inequality

SubstituteSimplify

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

𝑦>32x−1

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

-2 2

𝑦 ≤32x+1

Each point on solid line is a solution. A solid line used for inequalities with ≤ or ≥.

Example:What is the graph of

First, graph the boundary line Since the inequality symbol is >, the points on the boundary line are not solution.

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).

𝑦>𝑥−2

0>0−2 Substitute (0,0) for (x,y).

0>−2 (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?

a.

Graph x= -1 using a dashed line. Use (0,0) as a test point.

-2 2

2

-2

y

x𝑥>−1

a.

The side of the line contains (0,0).

b.

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.

Consider the system of linear inequalities are First, let’s graph each inequality separately.

In the graph of , below of the dashed line is the region contains all ordered pairs that make true.

-2 2

2

-2

y

x

𝑥+2 𝑦<6

In the graph of , ordered pairs in the left side is the region and on the line itself make true.

-2 2

2

-2

y

x

2 𝑥− 𝑦≥2

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

2 𝑥− 𝑦≥2𝑥+2 𝑦<6

Solution region for the system.

On our graph containing both and the region where the solution of the system overlaps contains ordered pairs that make both inequalities true, so all ordered pairs in this region are in the solution set for the system. Also, ordered pairs on the solid line for where it touches the region of overlap are in the solution set for the system, whereas ordered pairs on the dashed line for are not. This implies the following procedure.

-2 2

2

-2

y

x

2 𝑥− 𝑦≥2𝑥+2 𝑦<6

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:

𝑥+2 𝑦<6

(3 )+2 (0 )<6

3<6 True

Second Inequality:

2 𝑥− 𝑦≥2

c

6≥2 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.a. and

-2 2

2

-2

y

x

𝑦 ≤2 𝑥−1

𝑥+3 𝑦>6Solution 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.

b. and

-2 2

2

-2

y

x

𝑦<2

5 𝑥− 𝑦 ≤4

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

Example:Graph the solution set of the system of

inequalities. and

-2 2

2

-2

y

x

𝑦 ≥14𝑥+3

𝑥−4 𝑦 ≥8

Solution:Graph the inequalities on the same grid. The lines appear to be parallel. We can verify by writing in slope – intercept form and comparing the slopes.𝑥−4 𝑦 ≥8−4 𝑦 ≥−𝑥+8

𝑦 ≤14𝑥−2

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.

1. (2,2)

2. (-1,0)

6. (-6,1)3. (0,0)

5. (0,0)

4. (0,1)

In problem 7-10, graph each linear inequality.

10.

9.

8.

7.

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