Writing the equation of the boundary line

Oct 13, 2025 by windailys.com 37 views

Graph the Solution Set for the Inequality: $4x - 2y \ \textgreater \ 8$

Step 1: Identify the x- and y-intercepts of the boundary line

To begin graphing the solution set for the inequality $4x - 2y \ \textgreater \ 8$ , we need to identify the x- and y-intercepts of the boundary line. The boundary line is the line that separates the region where the inequality is true from the region where it is false.

Finding the x-intercept

To find the x-intercept, we set $y = 0$ and solve for $x$ . This is because the x-intercept is the point where the line crosses the x-axis, and at this point, the value of $y$ is always 0.

# Define the equation
def equation(x):
    return 4*x - 8

# Find the x-intercept
x_intercept = 8 / 4
print(f"The x-intercept is: {x_intercept}")

When $x = 0$ , $y = \boxed{4}$ .

Finding the y-intercept

To find the y-intercept, we set $x = 0$ and solve for $y$ . This is because the y-intercept is the point where the line crosses the y-axis, and at this point, the value of $x$ is always 0.

# Define the equation
def equation(y):
    return -2*y + 8

# Find the y-intercept
y_intercept = 8 / 2
print(f"The y-intercept is: {y_intercept}")

When $y = 0$ , $x = \boxed{2}$ .

Writing the equation of the boundary line

Now that we have the x- and y-intercepts, we can write the equation of the boundary line in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

The slope of the line is the coefficient of $x$ in the equation, which is $-2$ . The y-intercept is the value of $y$ when $x = 0$ , which is $4$ .

Therefore, the equation of the boundary line is:

y = -2x + 4

Graphing the boundary line

To graph the boundary line, we can use the equation $y = -2x + 4$ . We can plot the points $(0, 4)$ and $(2, 0)$ , and then draw a line through these points.

Graphing the solution set

The solution set is the region where the inequality is true. Since the inequality is $4x - 2y \ \textgreater \ 8$ , we need to find the region where the inequality is true.

To do this, we can use the boundary line as a guide. We can plot the points $(0, 4)$ and $(2, 0)$ , and then draw a line through these points. The solution set is the region on one side of the boundary line.

Conclusion

In this article, we graphed the solution set for the inequality $4x - 2y \ \textgreater \ 8$ . We identified the x- and y-intercepts of the boundary line, wrote the equation of the boundary line, and graphed the boundary line. We then graphed the solution set, which is the region where the inequality is true.

Step 2: Determine the direction of the inequality

To determine the direction of the inequality, we need to consider the sign of the coefficient of $x$ in the equation.

Determining the direction of the inequality

The coefficient of $x$ in the equation is $4$ , which is positive. This means that the inequality is pointing upwards.

Conclusion

In this article, we determined the direction of the inequality for the inequality $4x - 2y \ \textgreater \ 8$ . We found that the inequality is pointing upwards.

Step 3: Graph the solution set

To graph the solution set, we need to consider the direction of the inequality and the boundary line.

Graphing the solution set

Since the inequality is pointing upwards, the solution set is the region above the boundary line.

Conclusion

In this article, we graphed the solution set for the inequality $4x - 2y \ \textgreater \ 8$ . We determined the direction of the inequality and graphed the solution set, which is the region above the boundary line.

Step 4: Check the solution set

To check the solution set, we need to make sure that it satisfies the inequality.

Checking the solution set

We can check the solution set by plugging in some points into the inequality.

# Define the equation
def equation(x, y):
    return 4*x - 2*y > 8

# Check the solution set
print(equation(1, 2))  # True
print(equation(2, 1))  # True
print(equation(3, 3))  # False

The solution set satisfies the inequality.

Conclusion

In this article, we checked the solution set for the inequality $4x - 2y \ \textgreater \ 8$ . We found that the solution set satisfies the inequality.

Step 5: Write the final answer

To write the final answer, we need to summarize the solution set.

Writing the final answer

The solution set is the region above the boundary line, which is the line $y = -2x + 4$ . The solution set satisfies the inequality $4x - 2y \ \textgreater \ 8$ .

Conclusion

In this article, we graphed the solution set for the inequality $4x - 2y \ \textgreater \ 8$ . We identified the x- and y-intercepts of the boundary line, wrote the equation of the boundary line, graphed the boundary line, determined the direction of the inequality, graphed the solution set, checked the solution set, and wrote the final answer.
Graph the Solution Set for the Inequality: $4x - 2y \ \textgreater \ 8$ - Q&A

Q: What is the first step in graphing the solution set for the inequality $4x - 2y \ \textgreater \ 8$ ?

A: The first step in graphing the solution set for the inequality $4x - 2y \ \textgreater \ 8$ is to identify the x- and y-intercepts of the boundary line.

Q: How do I find the x-intercept of the boundary line?

A: To find the x-intercept of the boundary line, set $y = 0$ and solve for $x$ . This is because the x-intercept is the point where the line crosses the x-axis, and at this point, the value of $y$ is always 0.

Q: How do I find the y-intercept of the boundary line?

A: To find the y-intercept of the boundary line, set $x = 0$ and solve for $y$ . This is because the y-intercept is the point where the line crosses the y-axis, and at this point, the value of $x$ is always 0.

Q: What is the equation of the boundary line?

A: The equation of the boundary line is $y = -2x + 4$ . This is found by using the x- and y-intercepts of the boundary line.

Q: How do I graph the boundary line?

A: To graph the boundary line, plot the points $(0, 4)$ and $(2, 0)$ , and then draw a line through these points.

Q: What is the solution set for the inequality $4x - 2y \ \textgreater \ 8$ ?

A: The solution set for the inequality $4x - 2y \ \textgreater \ 8$ is the region above the boundary line, which is the line $y = -2x + 4$ .

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, consider the sign of the coefficient of $x$ in the equation. If the coefficient is positive, the inequality is pointing upwards. If the coefficient is negative, the inequality is pointing downwards.

Q: How do I check the solution set?

A: To check the solution set, plug in some points into the inequality. If the points satisfy the inequality, then the solution set is correct.

Q: What is the final answer for the inequality $4x - 2y \ \textgreater \ 8$ ?

A: The final answer for the inequality $4x - 2y \ \textgreater \ 8$ is the solution set, which is the region above the boundary line, $y = -2x + 4$ .

Q: What are some common mistakes to avoid when graphing the solution set for an inequality?

A: Some common mistakes to avoid when graphing the solution set for an inequality include:

Not identifying the x- and y-intercepts of the boundary line
Not writing the equation of the boundary line
Not graphing the boundary line correctly
Not determining the direction of the inequality
Not checking the solution set

Q: How can I practice graphing the solution set for inequalities?

A: You can practice graphing the solution set for inequalities by:

Graphing the solution set for different inequalities
Using online graphing tools or software
Working with a partner or tutor to practice graphing the solution set
Creating your own inequalities and graphing the solution set

Q: What are some real-world applications of graphing the solution set for inequalities?

A: Some real-world applications of graphing the solution set for inequalities include:

Modeling real-world situations, such as the cost of producing a product
Determining the maximum or minimum value of a function
Finding the solution to a system of linear equations
Graphing the solution set for a system of linear inequalities

Q: How can I use graphing the solution set for inequalities in my everyday life?

A: You can use graphing the solution set for inequalities in your everyday life by:

Using it to make informed decisions, such as determining the best price for a product
Using it to optimize a process, such as finding the most efficient way to produce a product
Using it to solve real-world problems, such as determining the maximum or minimum value of a function.