Graph Linear Inequalities Worksheet

Graph Linear Inequalities Worksheet
  • Page 1
 1.  
Which of the graphs represents the inequality? y < x + 4


a.
Graph 1
b.
Graph 2
c.
Graph 3
d.
Graph 4


Solution:


Since the inequality involves less than (<), use dashed line to represent the boundary of y < x + 4.

y < x + 4
0 < 0 + 4
0 < 4 True
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute the values.]


Since the inequality is true for (0, 0), shade the region containing (0, 0).

The above graph matches with the graph 2.


Correct answer : (2)
 2.  
Which of the graphs, represents the inequality? yx - 4


a.
Graph 1
b.
Graph 2
c.
Graph 3
d.
Graph 4


Solution:


Since the inequality involves greater than or equal to (≥), the boundary line of the inequality y = x - 4 is a solid line.

yx - 4
0 ≥ 0 - 4
0 ≥ - 4
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute the values.]


Since the inequality is true for (0, 0), shade the region that contains (0, 0).

The above graph matches with the graph 4.


Correct answer : (4)
 3.  
Graph the system of linear inequalities.
x ≥ 0
y ≥ 0
4x + 6y ≤ 12

a.
Graph 1
b.
Graph 2
c.
Graph 3
d.
Graph 4


Solution:


The graph of x ≥ 0 is the half-plane on and to the right of the solid line x = 0.


The graph of y ≥ 0 is the half-plane on and above the solid line y = 0.


The graph of 4x + 6y ≤ 12 is the half-plane on and below the solid line 4x + 6y = 12.


The shaded portion shown below is the intersection of the three half-planes, which is the graph of the system.

So, Graph 1 represents the system of linear inequalities.


Correct answer : (1)
 4.  
Which of the graphs represents the inequality x - y > - 1?


a.
Graph 2
b.
Graph 1
c.
Both A and B
d.
None of the above


Solution:

x - y > - 1

x - y = - 1
[Write corresponding equation].

y = x + 1
[Subtracting x from the two sides of the equation].

The corresponding equation in slope-intercept form is y = x + 1.


The graph of the line has a slope of 1 and a y-intercept of 1. Since the inequality is > use a dashed line.

The point (0, 0) satisfies the inequality. So, the solution is the half of the plane that includes the point (0, 0).


Correct answer : (2)
 5.  
Find the two numbers whose difference is less than or equal to 5. Show the solution by graphing an inequality.

a.
Graph 1
b.
Graph 2
c.
Graph 3
d.
Graph 4


Solution:

Let one number be x and the other number be y.

x - y ≤ 5
[The difference of the two numbers is less than or equal to 5.]


Since the inequality involves less than or equal to(≤) the boundary line of the equation x - y ≤ 5 is a solid line as shown

Substitute any point in the equation that does not lie on the boundary line.

x - y ≤ 5

2 - 1 ≤ 5
[Test (2, 1).]

1 ≤ 5
[True.]


The inequality 1 ≤ 5 is true. So, shade the region that contains (2, 1) as shown

The above graph matches with the graph 2.


Correct answer : (2)
 6.  
Graph the linear inequality: |x| ≥ 3


a.
Graph C
b.
Graph A
c.
Graph B
d.
Graph D


Solution:

|x| ≥ 3

x ≤ - 3 or x ≥ 3


The graph of |x| ≥ 3 consists of the solid lines x = - 3 and x = 3 and the region between these lines.


Correct answer : (1)
 7.  
Check whether the ordered pair (0, -6 5 ) is a solution of the inequality 4x + 5y ≤ 4.
a.
No
b.
Yes


Solution:

4x + 5y ≤ 4

4x + 5y
[Take left hand side of the inequality.]

= 4(0) + 5(-65 )
[Substitute the values.]

= 0 + (-6)
[Simplify.]

-6 ≤ 4
[Compare with the right hand side of the orginal expression.]

The ordered pair is a solution of 4x + 5y ≤ 4.


Correct answer : (2)
 8.  
Which of the graphs represents the inequality? y ≤ 5x + 3


a.
Graph 1
b.
Graph 2
c.
Graph 3
d.
Graph 4


Solution:


Since the inequality involves less than or equal to (≤), the boundary line of the inequality y = 5x + 3 is a solid line as shown in the graph.

y ≤ 5x + 3
0 ≤ 0 + 3
0 ≤ 3
Test a point not on the boundary line.
Test (0, 0) in the inequality.
[Substitute the values.]


Since the inequality is true for (0, 0), shade the region that contains (0, 0).

The above graph matches graph 3.


Correct answer : (3)
 9.  
Graph the system of linear inequalities.
2x - 3y < 6
- x + y ≤ 4
2x + 4y < 8

a.
Graph 1
b.
Graph 2


Solution:

The graph of 2x - 3y < 6 is the half-plane above the dashed line 2x - 3y = 6.

The graph of - x + y ≤ 4 is the half-plane on and below the solid line - x + y = 4.

The graph of 2x + 4y < 8 is the half-plane below the dashed line 2x + 4y = 8.


Finally, the graph of the system is the intersection of the three half-plane as shown in the graph.


Correct answer : (2)
 10.  
Which of the graphs best represents the system of inequalities, x + y < 3, - 5x + 2y ≤ 10, y ≥ - 3?


a.
Graph 1
b.
Graph 3
c.
Graph 4
d.
Graph 2


Solution:


Graph all the three inequalitites in the same coordinate plane.

The graph of x + y < 3 is the half-plane below the dashed line x + y = 3.

The graph of - 5x + 2y ≤ 10 is the half-plane on and below the solid line - 5x + 2y = 10.

The graph of y ≥ - 3 is the half-plane on and above the solid line y = - 3.

The graph of the system is the intersection of the three half planes as shown in the graph.

Graph 1 is the correct answer.


Correct answer : (1)

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