SOLVING TWO VARIABLE LINEAR
INEQUALITIES
INCLUDING ABSOLUTE VALUE
INEQUALITIES
Summary of Inequality Signs
> >
< <
Continuous
line
Dashed line
Shade
above
the line
Shade
below
the line
Graphing Linear Inequalities
The graph of a linear inequality is a region of the
coordinate plane that is bounded by a line. This
region represents the SOLUTION to the
inequality.
•A linear inequality is an inequality in two
variables whose graph is a region of the coordinate
plane that is bounded by a line.
1
y  x 1
2
2x  3y  6
Graph the following inequality:
x>2
Boundary is: x = 2
We shaded at the right of the line because x is more than 2. The line is
dashed because it is not equal or less than x, so the line which is the
boundary is not included in the solution.
Graph the following inequality: y < 6
Boundary: y = 0x + 6
m= 0
y- intercept = (0,6)
We shaded below the line because y is less than 6. The line is dashed
because it is not equal or less than y, so the line which is the
boundary is not included in the solution.
Example
3
Graph y > x  1.
2
3
Graph y > x  1.
2
1.
The boundary line is dashed.
2.
•
Substitute (0, 0) into the inequality to decide
where to shade.
3
y > x 1
2
3
0 > 0  1
2
•
•
0 > 0 1
0 > 1 False
So the graph is shaded away from (0, 0).
Graph the following inequality:
4x + 2y < 10
Solve for y  y < -2x + 5
Boundary is: y = -2x + 5
m= -2
y- intercept = (0,5)
We shaded below the line because y is less than the expression -2x + 5.
The line is dashed because it is not equal or less than y, so the line which
is the boundary is not included.
Graph the following inequality:
-9x + 3y< 3
Solve for y  y < 3x + 1
Boundary is: y = 3x + 1
m= 3
y- intercept = (0, 1)
We shaded below the line because y is less than the expression 3x +1.
The line is dashed because it is not equal or less than y, so the line which
is the boundary is not included.
Graph the following inequality:
5
y – 2 > 4 (x – 4)
5
4
Solve for y  y > x – 3
5
4
Boundary is: y = x – 3
m=
5
4
y-intercept = (0, -3)
We shaded above because y is
greater or equal than the
expression
and the line is continuous
because the word equal in
greater or equal indicates
that the boundary is included
in the solution.
EXAMPLE
EXAMPLE
EXAMPLE
EXAMPLE
Problem, con’t
Graph the following absolute value
equation: y = |x|
For x < 0
y = -x
For x > 0
y=x
Now let’s shift it two
units up:
y = |x| + 2
Now let’s shift it three
units to the right:
y = |x - 3| + 2
Now let’s graph it
upside down
y = – |x-3| + 2
Now let’s make it skinner
y = – 6|x-3| + 2
So, that’s how the different parameters in an
absolute value equation affect our graph.
Now let’s graph absolute value inequalities.
Absolute Value Inequalities
• Graph the absolute value function then shade
above OR below
Solid line…y <, y>
Shade above y>, y>
Shade below…y<, y<
Dashed line…y<, y>
Absolute Value Inequalities
Graph y < |x – 2| + 3
y < |x – 2| + 3
DASHED line
Shade BELOW
slope = 1
Vertex = (2, 3)
Absolute Value Inequalities
Graph y < |x – 2| + 3
Vertex = (2, 3)
Absolute Value Inequalities
Graph y < |x – 2| + 3
slope = 1
Absolute Value Inequalities
Graph y < |x – 2| + 3
DASHED line
Shade BELOW
Absolute Value Inequalities
Graph y < |x – 2| + 3
Shade BELOW
Absolute Value Inequalities
Graph –y + 1 < -2|x + 2|
-y < -2|x + 2| - 1
y > 2|x + 2| + 1
-y so CHANGE the
direction of the inequality
Absolute Value Inequalities
y
Solid line
Shade above
>
2|x
Slope = 2
+
2|
+
1
Vertex = (-2, 1)
Absolute Value Inequalities
y
>
2|x
+
2|
+
1
Absolute Value Inequalities
y
>
2|x
+
2|
+
1
Absolute Value Inequalities
y
>
2|x
+
2|
+
1
Absolute Value Inequalities
y
>
2|x
+
2|
+
1
Absolute Value Inequalities
Write an equation for the graph below.
Graph the following inequality:
y > |x|
Finding the boundary:
For x < 0
For x > 0
y = -x
y=x
There are two regions:
Testing point (0,2)
2 > | 0|
2 > 0 true
Therefore, the region
where (0,2) lies is the
solution region and we
shade it.
.
Graph the following inequality:
y < |x+1|– 3
Finding the boundary:
For x + 1 < 0 For x + 1 > 0
y = -(x+1) – 3 y = x+1 – 3
y = - x – 1 -3 y = - x – 4
y=x–2
.
There are two regions:
Testing point (0,0)
0 < | 0+1| – 3
0 < -2 false
So the region where (0,0) lies is not in the solution
region, therefore we shade the region below.
•Steps:
1. Decide if the boundary graph is solid or
dashed.
2. Graph the absolute value function as
the boundary.
3. Use the point (0, 0), if it is not on the
boundary graph, to decide how to
shade.
Graph y ≥ 2|x – 3| + 2
Graph y ≥ 2|x – 3| + 2
1. The boundary graph is solid.
2.
•
•
•
•
•
y ≥ 2|x – 3| + 2
0 ≥ 2|0 – 3| + 2
0 ≥ 2|-3| + 2
0≥6+2
0 ≥ 8 False
So shade away from (0, 0).
1
Graph y  x  1  4.
2
1
Graph y  x  1  4.
2
1.
The boundary graph is dashed.
1
2. y  x  1  4
2
1
0  0 1  4
2
1
0  4
2
1
0  3 False
2
So shade away from  0, 0 .
Your Turn!
8. Graph
y   x  2 3
9. Graph
2y  3   x 5
8.
9.
Example 10
Example 10
y ≤ |x+4| - 3
y ≥ 2x+5