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 and

Ax + By ≥ 0 Ax + By > 0 Ax + By ≤ 0 Ax + By < 0

0A 0B

Steps for Graphing a Linear Inequality1. Isolate y.

2. Graph the boundary line. Use a solid line for or

. Use a dashed line for < or >.

3. Shade above the boundary line for or >. Shade below the boundary line for or <.

42 xy Start at the y-intercept (0, -4) and then move rise over run for the slope ( 2).

Step by Step Example

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

Example:Graph: 3x + 4y > - 4

) 3 4 44 3 4

3 14

a x yy x

y x

If your inequality isn’t already solved for y, do that first!

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.

Makes graph wider, Makes graph move Makes graph move

more narrow, or flips left or right opposite the sign up or down same as sign

( )f x a x h k

Example:

x

y

-5

-4

-3

-2

-1

1

2

3

4

5

vertex (2, 1)

•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.

x

y

-5

-4

-3

-2

-1

1

2

3

4

5

vertex (0, -3)

•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.