Graphing Linear Inequalities
A linear inequality in two variables, x and y, is any inequality that can
be written in one of the forms below where A  0 and B  0
Ax + By ≥ 0
Ax + By > 0
Ax + By ≤ 0
Ax + By < 0
Steps for Graphing a Linear Inequality
1.
Isolate y.
2. Graph the boundary line. Use a solid line for
. Use a dashed line for < or >.
or
3. Shade above the boundary line for or >. Shade
below the boundary line for or <.
Step by Step Example
y  2x  4
Start at the y-intercept (0, -4) and then
move rise over run for the slope ( 2).
Notice that the boundary
line is SOLID because
there is a less than or
equal to symbol
Now we need to shade. We will shade below the line
because the symbol is a less than or equal to.
This is the
answer.
Try this one on your own. Move to the next slide
when you are ready to check your answer.
 y < -x + 3
Notice the line is dashed and we shaded below the
boundary line.
 y < -x + 3
Answer
Remember . . .
 When you have a greater or less than only you will have
a dashed boundary line.
 When you have greater than/equal to or less
than/equal to, then you have a solid boundary line.
 When you have a greater than or greater than or equal
to, you shade above the line or to the right if it is a
vertical line.
 When you have a less than or less than or equal to, you
shade below the line or to the left if it is a vertical line.
Here is a summary of what to do based on
the symbol in each inequality:
If y < (an expression)
• dashed line
• shade below
If y > (an expression)
• dashed line
• shade above
If y < (an expression)
• solid line
• shade below
If y > (an expression)
• solid line
• shade above
If your inequality isn’t already solved for y, do that first!
Example:
Graph: 3x + 4y > - 4
a)3x  4 y  4
4 y  3x  4
3
 y   x  1
4
Then, graph! The boundary line will be dashed because
we had a less than or equal to symbol. We shaded above
the boundary line for the same reason!
What happens when our inequality
contains absolute value?
 We will still use a dashed line for less than or greater
than symbols.
 We will still use a solid line for less than or equal to or
greater than or equal to symbols.
 We will still shade above the boundary line for
greater than or greater than or equal to symbols.
 We will still shade below the boundary line for less
than or less than or equal to symbols.
Remember How to Graph an
Absolute Value Equation:
f ( x)  a x  h  k
Makes graph wider,
more narrow, or flips
Makes graph move
Makes graph move
left or right opposite the sign up or down same as sign
Example:
y
5
4
3
2
1
vertex (2, 1)








x


-1
-2
-3
-4
-5
•The V is more narrow (because of the 3 in front), shifted right 2 units and up
1 unit.
•Notice that the V is solid because of the less than or equal to symbol. Also,
notice that we would shade below the V.
Try this one on your own. Move to the next slide
when you are ready to check your answer.
y
5
4
3
2
1
x










-1
-2
vertex (0, -3)
-3
-4
-5
•Your V should be wider, should open down, and should be shifted down 3 units.
•Your V should be dashed, and you should shade above.