Worksheet 4 Graphing Linear Inequalities and system of Linear Inequalities
Basic Vocabulary: When we want to say that a quantity is less than, less than equal to, greater than or greater than equal to, we do so by the following notation:
To denote x is greater than y: x
> y
To denote x is greater than or equal to y: x
³ y
To denote x is less than y: x
< y
To denote x is less than or equal to y: x
£ y
Note: The inequality sign always open towards the greater than quantity and closes towards the less than quantity.
Example 1:
“All points on the xy plane where x is greater than 2” is x
>
2
Example 2:
“All points on the xy plane where y is less than or equal to than the line 2x -
1” is the inequality y
£
2 x
-
1 (Discussion of this kind of graphs which are in two variables will be the focus of this activity)
Example 3: Graphs of linear inequalities in one variable.
a) “All numbers greater than 2” x > 2
(
2 b)
“All numbers less than and equal to -7”
]
-7
x  –7

Steps on how to Graph a Linear Inequality on two variables:
Take for example y
³
2 x
-
1
Step 1) Make sure your inequality has the form, isolated y on the left hand side and the x variable terms on the right hand side.
Step 2): Change the sign.
Change the inequality sign
<
,
£
,
³
, or > , to an equal sign, =
Step 3): Draw the line that corresponds to the resulting equation in Step 1.
 If the inequality is strict, < or >, use a dashed line.
 If the inequality is not strict, £
or
³
,
use a solid line.
Step 4): Looking at the inequality sign you can how you want to shade the solution region. If the inequality has >
, or
³ sign, shade the top of the equation. On the other hand if the inequality has <
,
£ sign, shade the bottom of the equation.
Step 5) : Test a point to check your answer.
 Select a point a point in one half-plane and test to see whether it satisfies the inequality. If it does, then so do all the points in that region (half-plane). It should match your shading from step 4). If not, then none of the points in that half-plane satisfy the inequality.
 Repeat this step for the other half-plane.
Example 3:
The second statement from the example is the inequality y > 2x -1 . Just like before, graph the boundary line first by replacing the inequality symbol with an equal sign (y =
2x – 1). This time the line will be solid because y is greater than or equal to the line 2x –
1 and is included in the solution. x
-3 -2 -1 y > 2x -1
3
2
1
0
-1
-2
0 1 2 3
-3 y

When shading, once again ask yourself, “ Where are the points on the graph with y coordinates greater than the line y= 2x – 1
?” Looking this time on the y-axis, it is clear that the y values become more positive above (greater than) the line. So we shade (in red) above the line. See the graph below.
Once again, to check that we shaded the correct side of the line, choose the origin (0, 0) as a test point. Replace the x and y in the inequality with zero and see if it reads as a true statement when simplified.
. y
³
2 x
-
1
0
³
2(0)
-
1
0
³
0
-
1
0
³ -
1
Zero is greater than -1, a true statement. Therefore the origin must be included within the solution space. y > 2 x -1

Exercise 1:
Write the inequality statement for all the points on the xy plane that symbolizes the statements below. a.
x is less than and equal to 3 b.
x is greater than 8 c.
y is less than – ¼ d.
y is greater than and equal to the line 2 x + 4 e.
y is less than and equal to the line 5 x – 1
2.
Consider the linear inequality y < 5. a.
Describe in words the inequality statement.
10
8
6
4 b.
Graph the inequality on the axes to the right. 2
-10 -8 -6 -4 -2
0
0
-2 c.
Substitute a point into the inequality as a test point to verify you shaded the correct area.
Show your work.
-4
-6
-8
-10
3.
Consider the linear inequality y > x - 4. a.
Describe in words the inequality statement.
10
8
6 b.
Graph the inequality on the axes to the right. c.
Substitute a point into the inequality as a test point to verify you shaded the correct area.
Show your work.
-10 -8 -6 -4 -2
4
2
0
0
-2
-4
-6
-8
-10
4.
Consider the linear inequality x + y < -5. a.
Describe in words the inequality statement.
10
8
6
4
2
0
0 -10 -8 -6 -4 -2 2
2
2
4
4
4
6
6
6
8
8
8
10
10
10
-2
-4
-6
-8
-10

b.
Graph the inequality on the axes to the right. c.
Substitute a point into the inequality as a test point to verify you shaded the correct area.
Describe the Regions in a Plane Using Inequalities
Example:
Describe in words the shaded region of this graph. The line is solid.
Step 1: Find the equation of the line.
In our example slope, m=0 and y-intercept, b=3,
Hence, the equation of the line is y = 3.
Step 2: Test with a point: Pick a point in the shaded area
and find the correct inequality.
We pick the point (1, 2) and put in the equation
to see 1 < 3. Since the line is solid we use 1

3
instead.
The inequality that represents the shaded area is y

3.
Exercise 5 : a. Describe in words the shaded region of this graph. The line is dashed. b. Describe in words the shaded region of this graph. The line is dashed.

Exercise 6:
Graph the solution set for each inequality on graph paper. Make a separate graph for each letter below. (Remember to shade the solution set and use either a solid line or a broken line for each equation). a.
x + y > – 2 c.
y  2x – 3 e.
y < x b. x – y d. 2x – y < 0 f. y 
 – 1
– 4x

g.
y > – 2 h. x  3
System of Linear Inequalities
Systems of linear inequalities describe regions on a plane using two or more linear inequalities simultaneously. The solution to the system is the region where all inequalities are true at the same time.
Consider the system x > 4 and y < -2. The graphs for each inequality are below. If we superimpose (lay on top) both inequalities, the area of the graph where both inequalities are shaded would be the solution on the graph. x >4 y < -2

x > 4 superimposed on y < - 2
Final solution of the system of inequalities:

Exercise 7: (a) Graph the system: y < 5 and y > - 2 (b) Graph the system: y < x + 3 and y < -3 x
14
14
12
12
10
10
8
8
6
6
4
4
2
-14 -12 -10 -8 -6 -4 -2
-2
0
0 2 4 6 8 10 12 14
-14 -12 -10 -8 -6 -4
2
-2
0
0
-2
2 4 6 8 10 12 14
-4
-4
-6 -6
-8 -8
-10 -10
-12 -12
-14 -14
Exercise 8: State whether the origin (0, 0) is included in the solution space for the following system of linear inequalities. Show your work! a.
y > -2 and x < 1 b. y > 4x -2 and x + y < -1

Exercise 9 : Write the system of inequalities for the shaded region in the graph to the right.