Which Graph Represents The Solution To The Following Inequality?$3x - 2y \leq 61$

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Introduction

In mathematics, inequalities are used to describe relationships between variables. Graphing inequalities is an essential skill in algebra and geometry, as it allows us to visualize the solution set of an inequality. In this article, we will explore how to graph the solution to the inequality $3x - 2y \leq 61$.

Understanding the Inequality

The given inequality is a linear inequality in two variables, $x$ and $y$. The inequality is in the form $ax + by \leq c$, where $a$, $b$, and $c$ are constants. In this case, $a = 3$, $b = -2$, and $c = 61$. The inequality states that the sum of $3x$ and $-2y$ is less than or equal to $61$.

Graphing the Inequality

To graph the inequality, we need to find the boundary line and the solution region. The boundary line is the line that divides the solution region from the non-solution region. In this case, the boundary line is the line $3x - 2y = 61$.

Finding the Boundary Line

To find the boundary line, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We can rewrite the boundary line equation as $y = \frac{3}{2}x - \frac{61}{2}$.

Graphing the Boundary Line

To graph the boundary line, we can use the slope-intercept form of the equation. The slope of the line is $\frac{3}{2}$, which means that for every 2 units we move to the right, we move up by 3 units. The y-intercept is $-\frac{61}{2}$, which means that the line intersects the y-axis at $-\frac{61}{2}$.

Finding the Solution Region

The solution region is the area on one side of the boundary line. Since the inequality is in the form $ax + by \leq c$, the solution region is the area on or below the boundary line.

Graphing the Solution Region

To graph the solution region, we can use a solid line to represent the boundary line and shade the area on or below the line. This represents the solution set of the inequality.

Graphing the Inequality on a Coordinate Plane

To graph the inequality on a coordinate plane, we need to plot the boundary line and the solution region. We can use a graphing calculator or a computer program to graph the inequality.

Plotting the Boundary Line

To plot the boundary line, we can use the slope-intercept form of the equation. We can plot the y-intercept at $-\frac{61}{2}$ and use the slope to plot additional points.

Plotting the Solution Region

To plot the solution region, we can use a solid line to represent the boundary line and shade the area on or below the line.

Conclusion

In conclusion, graphing the solution to the inequality $3x - 2y \leq 61$ involves finding the boundary line and the solution region. The boundary line is the line $3x - 2y = 61$, and the solution region is the area on or below the line. By graphing the inequality on a coordinate plane, we can visualize the solution set of the inequality.

Common Mistakes to Avoid

When graphing the inequality, there are several common mistakes to avoid. These include:

• Graphing the wrong boundary line: Make sure to graph the correct boundary line, which is the line $3x - 2y = 61$.
• Graphing the wrong solution region: Make sure to graph the correct solution region, which is the area on or below the boundary line.
• Not using a solid line for the boundary line: Make sure to use a solid line to represent the boundary line.
• Not shading the solution region: Make sure to shade the area on or below the boundary line to represent the solution region.

Tips for Graphing Inequalities

When graphing inequalities, there are several tips to keep in mind. These include:

• Use a graphing calculator or computer program: Graphing calculators and computer programs can help you graph the inequality quickly and accurately.
• Plot the boundary line first: Plot the boundary line first, and then plot the solution region.
• Use a solid line for the boundary line: Use a solid line to represent the boundary line.
• Shade the solution region: Shade the area on or below the boundary line to represent the solution region.

Final Thoughts

Graphing the solution to the inequality $3x - 2y \leq 61$ involves finding the boundary line and the solution region. By graphing the inequality on a coordinate plane, we can visualize the solution set of the inequality. Remember to avoid common mistakes and use tips for graphing inequalities to ensure that you graph the inequality correctly.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear inequality is an inequality that can be written in the form $ax + by \leq c$ or $ax + by \geq c$, where $a$, $b$, and $c$ are constants. A linear equation is an equation that can be written in the form $ax + by = c$, where $a$, $b$, and $c$ are constants.

Q: How do I graph a linear inequality on a coordinate plane?

A: To graph a linear inequality on a coordinate plane, you need to find the boundary line and the solution region. The boundary line is the line that divides the solution region from the non-solution region. The solution region is the area on or below the boundary line.

Q: What is the boundary line of a linear inequality?

A: The boundary line of a linear inequality is the line that divides the solution region from the non-solution region. It is the line that is represented by the equation $ax + by = c$, where $a$, $b$, and $c$ are constants.

Q: How do I find the boundary line of a linear inequality?

A: To find the boundary line of a linear inequality, you need to rewrite the inequality in the form $ax + by = c$. You can do this by adding or subtracting the same value to both sides of the inequality.

Q: What is the solution region of a linear inequality?

A: The solution region of a linear inequality is the area on or below the boundary line. It is the region that satisfies the inequality.

Q: How do I graph the solution region of a linear inequality?

A: To graph the solution region of a linear inequality, you need to use a solid line to represent the boundary line and shade the area on or below the line.

Q: Can I use a dashed line to represent the boundary line of a linear inequality?

A: No, you should use a solid line to represent the boundary line of a linear inequality. A dashed line is used to represent the boundary line of a linear equation.

Q: Can I use a dotted line to represent the boundary line of a linear inequality?

A: No, you should use a solid line to represent the boundary line of a linear inequality. A dotted line is not typically used to represent the boundary line of a linear inequality.

Q: How do I determine the direction of the solution region of a linear inequality?

A: To determine the direction of the solution region of a linear inequality, you need to look at the inequality sign. If the inequality sign is $\leq$, the solution region is on or below the boundary line. If the inequality sign is $\geq$, the solution region is on or above the boundary line.

Q: Can I use a graphing calculator or computer program to graph a linear inequality?

A: Yes, you can use a graphing calculator or computer program to graph a linear inequality. These tools can help you graph the inequality quickly and accurately.

Q: What are some common mistakes to avoid when graphing linear inequalities?

A: Some common mistakes to avoid when graphing linear inequalities include:

• Graphing the wrong boundary line
• Graphing the wrong solution region
• Not using a solid line for the boundary line
• Not shading the solution region

Q: How do I check my work when graphing a linear inequality?

A: To check your work when graphing a linear inequality, you need to make sure that you have graphed the correct boundary line and solution region. You can do this by checking your work against the original inequality.

Q: Can I graph a linear inequality with a negative slope?

A: Yes, you can graph a linear inequality with a negative slope. The boundary line will be a downward-sloping line, and the solution region will be on or below the line.

Q: Can I graph a linear inequality with a positive slope?

A: Yes, you can graph a linear inequality with a positive slope. The boundary line will be an upward-sloping line, and the solution region will be on or below the line.

Q: How do I graph a linear inequality with a zero slope?

A: To graph a linear inequality with a zero slope, you need to graph a horizontal line. The boundary line will be a horizontal line, and the solution region will be on or below the line.

Q: Can I graph a linear inequality with a vertical line?

A: Yes, you can graph a linear inequality with a vertical line. The boundary line will be a vertical line, and the solution region will be on or below the line.

Q: How do I graph a linear inequality with a fractional slope?

A: To graph a linear inequality with a fractional slope, you need to graph a line with a slope that is a fraction. The boundary line will be a line with a slope that is a fraction, and the solution region will be on or below the line.

Q: Can I graph a linear inequality with a negative y-intercept?

A: Yes, you can graph a linear inequality with a negative y-intercept. The boundary line will be a line with a negative y-intercept, and the solution region will be on or below the line.

Q: Can I graph a linear inequality with a positive y-intercept?

A: Yes, you can graph a linear inequality with a positive y-intercept. The boundary line will be a line with a positive y-intercept, and the solution region will be on or below the line.

Q: How do I graph a linear inequality with a zero y-intercept?

A: To graph a linear inequality with a zero y-intercept, you need to graph a line that passes through the origin. The boundary line will be a line that passes through the origin, and the solution region will be on or below the line.

Q: Can I graph a linear inequality with a vertical line and a horizontal line?

A: Yes, you can graph a linear inequality with a vertical line and a horizontal line. The boundary line will be a vertical line or a horizontal line, and the solution region will be on or below the line.

Q: How do I graph a linear inequality with a line that is parallel to the x-axis?

A: To graph a linear inequality with a line that is parallel to the x-axis, you need to graph a horizontal line. The boundary line will be a horizontal line, and the solution region will be on or below the line.

Q: Can I graph a linear inequality with a line that is parallel to the y-axis?

A: Yes, you can graph a linear inequality with a line that is parallel to the y-axis. The boundary line will be a vertical line, and the solution region will be on or below the line.

Q: How do I graph a linear inequality with a line that is perpendicular to the x-axis?

A: To graph a linear inequality with a line that is perpendicular to the x-axis, you need to graph a vertical line. The boundary line will be a vertical line, and the solution region will be on or below the line.

Q: Can I graph a linear inequality with a line that is perpendicular to the y-axis?

A: Yes, you can graph a linear inequality with a line that is perpendicular to the y-axis. The boundary line will be a horizontal line, and the solution region will be on or below the line.

Q: How do I graph a linear inequality with a line that is parallel to the x-axis and a line that is parallel to the y-axis?

A: To graph a linear inequality with a line that is parallel to the x-axis and a line that is parallel to the y-axis, you need to graph two lines that are parallel to the x-axis and y-axis. The boundary line will be two lines that are parallel to the x-axis and y-axis, and the solution region will be on or below the lines.

Q: Can I graph a linear inequality with a line that is perpendicular to the x-axis and a line that is perpendicular to the y-axis?

A: Yes, you can graph a linear inequality with a line that is perpendicular to the x-axis and a line that is perpendicular to the y-axis. The boundary line will be two lines that are perpendicular to the x-axis and y-axis, and the solution region will be on or below the lines.

Q: How do I graph a linear inequality with a line that is parallel to the x-axis and a line that is perpendicular to the y-axis?

A: To graph a linear inequality with a line that is parallel to the x-axis and a line that is perpendicular to the y-axis, you need to graph a horizontal line and a vertical line. The boundary line will be a horizontal line and a vertical line, and the solution region will be on or below the lines.

Q: Can I graph a linear inequality with a line that is perpendicular to the x-axis and a line that is parallel to the y-axis?

A: Yes, you can graph a linear inequality with a line that is perpendicular to the x-axis and a line that is parallel to the y-axis. The boundary line will be a vertical line and a horizontal line, and the solution region will be on or below the lines.

Q: How do I graph a linear inequality with a line that is parallel to the x-axis and a line that is parallel to the y-axis and a line that is perpendicular to the x-axis and a line that is perpendicular to the y-axis?

A: To graph a linear inequality with a line that is parallel to the x-axis and a line that is parallel to the y-axis and a line that is perpendicular to the x-axis and a line that is perpendicular to the y-axis, you need to graph four lines that are parallel to the x