LINEAR INEQUALITIES
IN TWO VARIABLES
4-4
OBJECTIVES
To graph linear inequalities in two
variables.
To use linear inequalities when modeling
real-world situations.
LINEAR INEQUALITY
A linear inequality in two variables, such as
𝑦 > 𝑥 − 3, can be formed by replacing the
equal sign in a linear equation with an
inequality sign.
A solution of an inequality in two variables is
an ordered pair that makes the inequality true.
LINEAR INEQUALITIES
A linear inequality in two variables has an
infinite number of solutions.
These solutions can be represented in the
coordinate plans as the set of all points on one
side of a boundary line.
IDENTIFYING SOLUTIONS OF A LINEAR
INEQUALITY
Is the ordered pair a solution of 𝑦 > 𝑥 − 3?
A. (1, 2)
B. (-3, -7)
2 > 1– 3
−7 > −3 – 3
2 > −2
−7 > – 6
True, so yes, it is a
Not true, so no, it is
solution.
not a solution.
IS THE ORDERED PAIR (3,6) A
2
SOLUTION OF 𝑦 ≤ 𝑥 + 4?
3
2
6 ≤
3 +4
3
6 ≤2+4
6 ≤6
True, so yes, it is a solution.
SUPPOSE AN ORDERED PAIR IS NOT A
SOLUTION OF 𝑦 > 𝑥 + 10. MUST IT BE
A SOLUTION OF 𝑦 < 𝑥 + 10?
No, any ordered pair that makes the inequality equal on both
sides is not a solution to either inequality.
Ex: (10, 20)
20 > 10 + 10
20 < 10 + 10
20 > 20
20 < 20
Neither are true so it is not a solution of either.
GRAPHING
The graph of a linear inequality in two
variables consists of all the 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.
GRAPHING – BOUNDARY LINE
Each point on a dashed line is
not a solution. A dashed line is
used for inequalities with > or <.
Each point on a solid line is a
solution. A solid line is used for
inequalities with ≥ or ≤.
GRAPHING – SHADING
All points in the shaded region are solutions to the inequality.
Shade above the line if the
inequality symbols is > or ≥.
Shade below the line if the
inequality symbols is < or ≤.
GRAPHING AN INEQUALITY
𝑦 ≤ 3𝑥 − 1
First graph the boundary line.
•If it’s > or <, draw a dashed line.
•If it’s ≥ or ≤, draw a solid line.
𝑦 ≤ 3𝑥 − 1
To determine which side of the boundary
line to shade, test a point that is not on the
line. (Try the origin.)
•If the point is a solution, shade the side that the
point is on.
•If it is not a solution, shade the other side.
HINT: If the sign is > or ≥, shade above the line.
If the sign is < or ≤, shade below the line.
𝑦 ≤ 3𝑥 − 1
0≤3 0 −1
0≤0−1
0 ≤ −1
DRAW IT
Boundary line:
> or <, dashed line
≥ or ≤, solid line
Test point: (0,0)
true: shade this side
false: share opposite
HINT:
> or ≥, shade above
< or ≤, shade below
What is the graph
of 𝑦 > 𝑥 − 2?
DRAW IT
Boundary line:
> or <, dashed line
≥ or ≤, solid line
Test point: (0,0)
true: shade this side
false: share opposite
HINT:
> or ≥, shade above
< or ≤, shade below
What is the graph
1
of 𝑦 ≤ 𝑥 + 1?
2
WRITING AN INEQUALITY FROM A GRAPH
Y-intercept: 𝑏 = −3
Slope: 𝑚 = 2
Equation: 𝑦 = 2𝑥 − 3
Above
Below
Dashed:
>
<
Solid:
≥
≤
Inequality: 𝑦 > 2𝑥 − 3
WRITING AN INEQUALITY FROM A GRAPH
Y-intercept: 𝑏 = 3
Slope: 𝑚 = −
5
4
Equation: 𝑦 = − 𝑥 + 3
Above
Below
Dashed:
>
<
Solid:
≥
≤
Inequality: 𝑦 ≥
5
− 𝑥
4
+3
5
4
WHICH INEQUALITY REPRESENTS THE GRAPH BELOW?
A. 𝑦 < −2𝑥 + 4
B. 𝑦 > −2𝑥 + 4
C. 𝑦 ≤ −2𝑥 + 4
D. 𝑦 ≥ −2𝑥 + 4