#### Solved Examples and Worksheet for Graphing Linear Equations and Inequalities

Q1Which of the following inequalities when graphed will have a dashed boundary line?
A. 4x + 5y ≤ 6
B. 4x - y ≥ 6
C. x + 4y ≤ 6
D. 4x + y > 6

Step: 1
The boundary line of the half-plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.
Step: 2
For the inequalities 4x - y ≥ 6, 4x + 5y ≤ 6 and x + 4y ≤ 6 the boundary line of the half-plane is solid as they contain ≥ and ≤ symbols.
Step: 3
For the inequality 4x + y > 6 the boundary line of the half-plane is dashed as it contains > symbol.
Correct Answer is :   4x + y > 6
Q2Which of the following inequalities when graphed will have a solid boundary line?
A. 6x + 5y < 9
B. 6x - 4y > 9
C. 2x + 5y ≤ 4
D. 2x + 5y < 4

Step: 1
The boundary line of the half-plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.
Step: 2
For the inequalities 6x - 4y > 9, 2x + 5y < 4, and 6x + 5y < 9 the boundary line of the half-plane is dashed as they contain > and < symbols.
Step: 3
For the inequality 2x + 5y ≤ 4 the boundary line of the half-plane is solid as it contains ≤ symbol.
Correct Answer is :   2x + 5y ≤ 4
Q3Which graph represents the equation x = - 2? A. Graph 3
B. Graph 1
C. Graph 4
D. Graph 2

Step: 1
The x-coordinate is always - 2, regardless of the value of y.
Step: 2
The graph of the equation x = - 2 is a vertical line 2 units to the left of the y-axis as shown in the following graph. Step: 3
The above graph matches with the graph 3.
Correct Answer is :   Graph 3
Q4Which graph represents the equation - y = x - 4? A. Graph 3
B. Graph 4
C. Graph 1
D. Graph 2

Step: 1
- y = x - 4
[Original equation.]
Step: 2
y = - x + 4
[Apply the slope-intercept form, y = mx + c.]
Step: 3
Choose values for x. Step: 4
Plot the points on a graph. Step: 5
The above graph matches with the Graph 4.
Step: 6
So, Graph 4 is the correct choice.
Correct Answer is :   Graph 4
Q5Which of the graphs best suits the inequality y ≥3x - 4 ? A. Graph 1
B. Graph 4
C. Graph 2
D. Graph 3

Step: 1
y = 3x - 4
[Write the corresponding equation of the given inequality.]
Step: 2
The equation is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 3
Graph the corresponding equation using the slope and y-intercept. As the inequality involves '≥', use a solid line. Step: 4
Test a point, which is not on the boundary line. Test (0, 0) in the inequality.
y ≥ 3x - 4 ⇒ 0 ≥ 3(0) - 4 ⇒ 0 ≥ - 4
[True.]
Step: 5
Since the inequality is true for (0, 0), shade the region that contains (0, 0). Step: 6
Therefore, graph 4 represents the inequality y ≥3x - 4.
Correct Answer is :   Graph 4
Q6Which of the graphs best suits the inequality y < x + 1? A. Graph 1
B. Graph 3
C. Graph 2
D. Graph 4

Step: 1
y = x + 1
[Write the corresponding equation of the given inequality.]
Step: 2
The equation is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 3
Graph the corresponding equation using the slope and y-intercept. As the inequality involves '<', use a dashed line. Step: 4
Test a point, which is not on the boundary line. Test (0, 0) in the inequality.
Step: 5
y < x + 1 ⇒ 0 < 0 + 1 ⇒ 0 < 1
[True.]
Step: 6
Since the inequality is true for (0, 0), shade the region that contains (0, 0). Step: 7
Therefore, graph 2 represents the inequality y < x + 1.
Correct Answer is :   Graph 2
Q7Which of the graphs best suits the inequality y ≥ 2x - 3? A. Graph 1
B. Graph 4
C. Graph 3
D. Graph 2

Step: 1
y = 2x - 3
[Write the corresponding equation of the given inequality.]
Step: 2
The equation is in slope-intercept form y = mx + b, where m is the slope and b is the y - intercept.
Step: 3
Graph the corresponding equation using the slope and y - intercept. As the inequality involves '≥', use a solid line. Step: 4
Test a point, which is not on the boundary line. Test (0, 0) in the inequality.
Step: 5
y ≥2x - 3 ⇒ 0 ≥ 2(0) - 3 ⇒ 0 ≥ - 3
[True.]
Step: 6
Since the inequality is true for (0, 0), shade the region that contains (0, 0). Step: 7
Therefore, graph 3 represents the inequality y ≥ 2x - 3
Correct Answer is :   Graph 3
Q8Which of the graphs represents the equation y2 = x3 - 1 A. Graph 3
B. Graph 1
C. Graph 2
D. Graph 4

Step: 1
y2 = x3 - 1 ⇒ y = 23x - 2
[Multiply by 2 on both sides of the equation.]
Step: 2
When x = 0, y = 23 (0) - 2 = - 2
[Substitute x = 0 in the equation.]
Step: 3
When x = 3, y = 23(3) - 2 = 0
[Substitute x = 3 in the equation.]
Step: 4
Thus, the points (0, -2) and (3, 0) are the solutions of the equation y2 = x3 - 1.
Step: 5
Draw a line passing through these points. Step: 6
Therefore, graph 1, represents the equation y2 = x3 - 1.
Correct Answer is :   Graph 1
Q9Which of the graphs represents the equation y = - 34x + 3? A. Graph 3
B. Graph 4
C. Graph 1
D. Graph 2

Step: 1
y = - 34x + 3
Step: 2
When x = 0, y = - 34 (0) + 3 = 3
[Substitute x = 0 in the equation.]
Step: 3
When x = 4, y = - 34(4) + 3 = 0
[Substitute x = 4 in the equation..]
Step: 4
Thus, the points (0, 3) and (4, 0) are the solutions of the equation y = - 34 x + 3.
Step: 5
Draw a line passing through these points. Step: 6
Therefore, graph 1 represents the equation y = - 34x + 3.
Correct Answer is :   Graph 1
Q10Which of the graphs represents the equation y = 43x - 1? A. Graph 1
B. Graph 2
C. Graph 3
D. Graph 4

Step: 1
y = 43x - 1
Step: 2
When x = 3, y = 43(3) - 1 = 3
[Substitute x = 3 in the equation.]
Step: 3
When x = 0, y = 43(0) - 1 = - 1
[Substitute x = 0 in the equation.]
Step: 4
Thus, the points (0, - 1) and (3, 3) are the solution of the equation y = 43x - 1.
Step: 5
Draw a line passing through these points. Step: 6
Therefore, graph 4, represents the equation y = 43x - 1.
Correct Answer is :   Graph 4
Q11Which of the graphs represents the equation y = - 45 ? A. Graph 4
B. Graph 2
C. Graph 3
D. Graph 1

Step: 1
y = - 45 = - 0.8
[Simplify the equation.]
Step: 2
The y - coordinate is always - 0.8, regardless of the value of x.
Step: 3
Thus, the graph of the equation y = - 0.8 is a horizontal line which is 0.8 units below the x-axis as shown in the graph. Step: 4
Therefore, graph 2 represents the equation y = - 45
Correct Answer is :   Graph 2
Q12Choose a graph that represents an equation whose x - intercept is 1 and y - intercept is - 2. A. Graph 2
B. Graph 3
C. Graph 4
D. Graph 1

Step: 1
The x - intercept is 1. Therefore, the line crosses the x-axis at a point (1, 0).
Step: 2
The y-intercept is -2. Therefore, the line crosses the y-axis at a point (0, -2).
Step: 3
Plot the points (1, 0) and (0, -2) and draw a line through them. Step: 4
Therefore, graph 4 represents the graph of the equation whose x-intercept is 1 and y-intercept is -2.
Correct Answer is :   Graph 4
Q13Choose a graph that represents an equation whose x-intercept is -3 and y-intercept is 1. A. Graph 4
B. Graph 3
C. Graph 1
D. Graph 2

Step: 1
The x - intercept is -3. Therefore, the line crosses the x-axis at a point (-3, 0).
Step: 2
The y-intercept is 1. Therefore, the line crosses the y-axis at a point (0, 1).
Step: 3
Plot the points (-3, 0) and (0, 1) and draw a line through them. Step: 4
Therefore, graph 1 represents the graph of the equation whose x-intercept is -3 and y-intercept is 1.
Correct Answer is :   Graph 1