#### Solved Examples and Worksheet for Solving and Graphing Linear Inequalities on a Number Line

Q1Pick an appropriate graph for the compound inequality. -5 ≤ x - 3 ≤ 2 A. Figure 2
B. Figure 3
C. Figure 1
D. All the figures

Step: 1
-5 ≤ x - 3 ≤ 2
[Original inequality.]
Step: 2
-5 + 3 ≤ x - 3 + 3 ≤ 2 + 3
Step: 3
-2 ≤ x ≤ 5
[Simplify.]
Step: 4
The solution is all real numbers greater than or equal to -2 and less than or equal to 5. The graph for the solution is Correct Answer is :   Figure 2
Q2Solve -4 ≤ x - 6 ≤ 3 and graph the inequality. A. Figure 1
B. Figure 2
C. Figure 3
D. None of the above

Step: 1
-4 ≤ x -6 and x - 6 ≤ 3
[Write the inequality as two inequalities.]
Step: 2
-4 + 6 ≤ x - 6 + 6 and x - 6 + 6 ≤ 3 + 6
Step: 3
2 ≤ x and x ≤ 9
[Simplify.]
Step: 4
2 ≤ x ≤ 9
[Write compound inequality]
Step: 5
The solution is all real numbers greater than 2 and less than or equal to 9. The graph of the solution can be represented as shown below.
[The red colored region is the solution of the problem.] Correct Answer is :   Figure 1
Q3Which of the graphs represents the inequality?
- 5.5 + x < - 1.6 A. Figure 1
B. Figure 2
C. Figure 3
D. Figure 4

Step: 1
- 5.5 + x < - 1.6
[Original inequality.]
Step: 2
- 5.5 + x + 5.5 < - 1.6 + 5.5
Step: 3
x < 3.9
[Simplify.]
Step: 4
The solution for the inequality is the set of all real numbers less than 3.9.
Step: 5
So, among the choices, Figure 2 is the appropriate graph for the inequality.
[The set of numbers to the left of 3.9 is the solution for the inequality.]
Correct Answer is :   Figure 2
Q4Which of the graphs represents the inequality?
- 2.4 + x ≤ - 5.8 A. Figure 1
B. Figure 2
C. Figure 3
D. Figure 4

Step: 1
- 2.4 + x ≤ - 5.8
[Original inequality.]
Step: 2
- 2.4 + x + 2.4 ≤ - 5.8 + 2.4
Step: 3
x ≤ - 3.4
[Simplify.]
Step: 4
The solution for the inequality is set of all real numbers less than or equal to - 3.4.
Step: 5
Among the choices, Figure 4 satisfies the inequality.
Correct Answer is :   Figure 4
Q5Which of the graphs represents the inequality x6 < - 3? A. Figure 1
B. Figure 2
C. Figure 3
D. Figure 4

Step: 1
x6 < - 3
[Original inequality.]
Step: 2
(6) (x6) < (- 3) (6)
[Multiply each side by 6.]
Step: 3
x < -18
[Simplify.]
Step: 4
The solution of the inequality includes the set of all the integers less than - 18.
Step: 5
- 18 is not included in the solution, which is represented by an open dot.
Step: 6
The region to the left of - 18 is the solution of the inequality.
[All the numbers to the left of a number are less than the number.]
Step: 7
So, the graph in figure 4 represents the inequality.
Correct Answer is :   Figure 4
Q6Which of the graphs represents the solution of the inequality n + 6 ≥ 11? A. Graph 1
B. Graph 2
C. Graph 3
D. Graph 4

Step: 1
n + 6 ≥ 11
[Original inequality.]
Step: 2
n + 6 - 6 ≥ 11 - 6
[Subtract 6 from each side.]
Step: 3
n ≥ 5
[Simplify.]
Step: 4
The solution of the inequality is all real numbers greater than or equal to 5.
Step: 5
From the choice, graph 2 represents the inequality n ≥ 5.
Correct Answer is :   Graph 2
Q7Which of the graphs represents the solution of the inequality, x9 > - 2? A. Figure 1
B. Figure 2
C. Figure 3
D. Figure 4

Step: 1
x9 > - 2
[Original inequality.]
Step: 2
(9) (x9) > (- 2) (9)
[Multiply each side by 9.]
Step: 3
x > - 18
[Simplify.]
Step: 4
The solution of the inequality includes the set of all the integers greater than - 18.
Step: 5
- 18 is not included in the solution, which is represented by an open dot.
Step: 6
The region to the right of - 18 is the solution of the inequality.
[All the numbers to the right of a number are greater than the number.]
Step: 7
So, the graph in Figure 1 represents the inequality.
Correct Answer is :   Figure 1
Q8Which of the graphs represents the solution of - 3 ≤ x < 5? A. Figure 1
B. Figure 2
C. Figure 4
D. Figure 3

Step: 1
The solution is all real numbers greater than or equal to - 3 and less than 5. The graph for the solution is: Step: 2
So, among the choices Figure 1 is the correct choice.
Correct Answer is :   Figure 1
Q9Which of the graphs represents the inequality?
- 6.8 + x > - 2.9 A. Figure 1
B. Figure 2
C. Figure 3
D. Figure 4

Step: 1
- 6.8 + x > - 2.9
[Original inequality.]
Step: 2
- 6.8 + x + 6.8 > - 2.9 + 6.8
Step: 3
x > 3.9
[Simplify.]
Step: 4
The solution for the inequality is the set of all real numbers greater than 3.9.
Step: 5
So, among the choices, Figure 3 is the appropriate graph for the inequality.
[The set of numbers to the right of 3.9 is the solution for the inequality.]
Correct Answer is :   Figure 3
Q10Which of the graphs represents the solution for the inequality, x6 < - 3? A. Figure 3
B. Figure 1
C. Figure 4
D. Figure 2

Step: 1
The solution of the inequality includes the set of all the integers less than -18.
Step: 2
The region to the left of -18 is the solution of the inequality.
[All the numbers to the right of a number are greater than the number.]
Step: 3
So, the number line in Figure 4 represents the inequality.
Correct Answer is :   Figure 4
Q11Which graph represents the solution of the inequality 2x ≥ 12? A. Figure 3
B. Figure 4
C. Figure 1
D. Figure 2

Step: 1
2x ≥ 12
[Original inequality.]
Step: 2
2x2 122
[Divide each side by 2.]
Step: 3
x ≥ 6
[Simplify.]
Step: 4
The inequality x ≥ 6 takes all the values greater than 6 including 6.
Step: 5
The inequalities, ≥ or ≤ are represented with closed dot on number line.
Step: 6
The following figure represents the solutions of the inequality. Correct Answer is :   Figure 2
Q12Which of the graphs represents the solution for the inequality?
3.5 + x + 5.6 ≥ 4.3 A. Figure 3
B. Figure 1
C. Figure 4
D. Figure 2

Step: 1
3.5 + x + 5.6 ≥ 4.3
[Original inequality.]
Step: 2
x + 9.1 ≥ 4.3
[Combine like terms.]
Step: 3
x + 9.1 - 9.1 ≥ 4.3 - 9.1
[Subtract 9.1 from each side.]
Step: 4
x ≥ -4.8
[Simplify.]
Step: 5
Among the choices, Figure 2 satisfies the inequality.
Correct Answer is :   Figure 2
Q13Which of the graphs represents the inequality?
- 7.7 + x > - 3.8 A. Figure 1
B. Figure 3
C. Figure 4
D. Figure 2

Step: 1
- 7.7 + x > - 3.8

Step: 2
- 7.7 + x + 7.7 > - 3.8 + 7.7
[Add 7.7 to both sides of the equation.]
Step: 3
x > 3.9
[Simplify.]
Step: 4
The solution for the inequality is the set of all real numbers greater than 3.9.
Step: 5
As x cannot be equal to 3.9, it should be represented by a open dot.
Step: 6
So, among the choices, Figure 3 is the appropriate graph for the inequality.
[The solution for the inequality is the set of numbers to the right of 3.9.]
Correct Answer is :   Figure 3