Understanding the Equation of the Boundary Line

Oct 13, 2025 by windailys.com 143 views

The Equation of Inequality: Understanding the Graph of $y \leq -2x+3$

In mathematics, inequalities are used to describe relationships between variables. When it comes to graphing inequalities, understanding the equation of the boundary line is crucial. In this article, we will explore the graph of the inequality $y \leq -2x+3$ and determine which sentence describes its graph.

Understanding the Equation of the Boundary Line

The equation $y=-2x+3$ is a linear equation in the slope-intercept form, where the slope is -2 and the y-intercept is 3. This equation represents a straight line that passes through the points (0, 3) and (-1.5, 0). The line has a negative slope, indicating that it slopes downward from left to right.

Graphing the Inequality

To graph the inequality $y \leq -2x+3$ , we need to consider the boundary line $y=-2x+3$ . Since the inequality is of the form $y \leq$ , we know that the region below the boundary line is shaded. However, we need to determine whether the boundary line is solid or dashed.

Solid or Dashed Boundary Line?

Since the inequality is of the form $y \leq$ , the boundary line is not included in the solution set. Therefore, the boundary line should be represented as a dashed line.

Shading the Region

Now that we have determined the type of boundary line, we can shade the region below the dashed boundary line. The region below the line $y=-2x+3$ is shaded, indicating that all points below this line satisfy the inequality $y \leq -2x+3$ .

In conclusion, the graph of the inequality $y \leq -2x+3$ is the region shaded below a dashed boundary line. This is because the inequality is of the form $y \leq$ , indicating that the region below the boundary line is shaded, and the boundary line is represented as a dashed line since it is not included in the solution set.

The Correct Answer

The correct answer is:

A. The region shaded below a dashed boundary line.

Additional Examples

To further reinforce our understanding of graphing inequalities, let's consider a few more examples.

Example 1: $y \geq 2x-1$

The equation $y=2x-1$ is a linear equation in the slope-intercept form, where the slope is 2 and the y-intercept is -1. This equation represents a straight line that passes through the points (0, -1) and (1, 1). The line has a positive slope, indicating that it slopes upward from left to right.

To graph the inequality $y \geq 2x-1$ , we need to consider the boundary line $y=2x-1$ . Since the inequality is of the form $y \geq$ , we know that the region above the boundary line is shaded. However, we need to determine whether the boundary line is solid or dashed.

Since the inequality is of the form $y \geq$ , the boundary line is included in the solution set. Therefore, the boundary line should be represented as a solid line.

Shading the region above the solid boundary line indicates that all points above this line satisfy the inequality $y \geq 2x-1$ .

Example 2: $y < -3x+2$

The equation $y=-3x+2$ is a linear equation in the slope-intercept form, where the slope is -3 and the y-intercept is 2. This equation represents a straight line that passes through the points (0, 2) and (-2/3, 0). The line has a negative slope, indicating that it slopes downward from left to right.

To graph the inequality $y < -3x+2$ , we need to consider the boundary line $y=-3x+2$ . Since the inequality is of the form $y <$ , we know that the region below the boundary line is shaded. However, we need to determine whether the boundary line is solid or dashed.

Since the inequality is of the form $y <$ , the boundary line is not included in the solution set. Therefore, the boundary line should be represented as a dashed line.

Shading the region below the dashed boundary line indicates that all points below this line satisfy the inequality $y < -3x+2$ .

In conclusion, graphing inequalities requires a thorough understanding of the equation of the boundary line and the type of boundary line (solid or dashed). By following the steps outlined in this article, you can accurately graph inequalities and determine which sentence describes their graph.
Frequently Asked Questions: Graphing Inequalities

Q: What is the difference between a solid and dashed boundary line in graphing inequalities?

A: In graphing inequalities, a solid boundary line represents an equation that is included in the solution set, while a dashed boundary line represents an equation that is not included in the solution set.

Q: How do I determine whether the boundary line is solid or dashed?

A: To determine whether the boundary line is solid or dashed, you need to look at the inequality symbol. If the inequality symbol is $\leq$ , $\geq$ , or $=$ , the boundary line is solid. If the inequality symbol is $<$ , $>$ , or $\neq$ , the boundary line is dashed.

Q: What is the significance of shading the region in graphing inequalities?

A: Shading the region in graphing inequalities indicates which side of the boundary line satisfies the inequality. If the inequality is of the form $y \leq$ , $y \geq$ , or $y =$ , the region below the boundary line is shaded. If the inequality is of the form $y <$ , $y >$ , or $y \neq$ , the region above the boundary line is shaded.

Q: Can I have a region shaded on both sides of the boundary line?

A: No, you cannot have a region shaded on both sides of the boundary line. In graphing inequalities, the region is shaded on only one side of the boundary line.

Q: How do I graph an inequality with a negative slope?

A: To graph an inequality with a negative slope, you need to draw a line with a negative slope and shade the region on the opposite side of the line. For example, if the inequality is $y < -2x+3$ , you would draw a line with a negative slope and shade the region below the line.

Q: Can I have a vertical boundary line in graphing inequalities?

A: Yes, you can have a vertical boundary line in graphing inequalities. A vertical boundary line is represented by an equation of the form $x=a$ , where $a$ is a constant. To graph a vertical boundary line, you need to draw a vertical line at the value of $x$ and shade the region on one side of the line.

Q: How do I graph an inequality with a horizontal boundary line?

A: To graph an inequality with a horizontal boundary line, you need to draw a horizontal line at the value of $y$ and shade the region on one side of the line. For example, if the inequality is $y \geq 2$ , you would draw a horizontal line at $y=2$ and shade the region above the line.

Q: Can I have a boundary line with a slope of zero in graphing inequalities?

A: Yes, you can have a boundary line with a slope of zero in graphing inequalities. A boundary line with a slope of zero is represented by an equation of the form $y=b$ , where $b$ is a constant. To graph a boundary line with a slope of zero, you need to draw a horizontal line at the value of $y$ and shade the region on one side of the line.

Q: How do I graph an inequality with a boundary line that is not a straight line?

A: To graph an inequality with a boundary line that is not a straight line, you need to draw the boundary line and shade the region on one side of the line. For example, if the inequality is $y \leq x^2+1$ , you would draw a parabola and shade the region below the parabola.

In conclusion, graphing inequalities requires a thorough understanding of the equation of the boundary line and the type of boundary line (solid or dashed). By following the steps outlined in this article, you can accurately graph inequalities and answer frequently asked questions about graphing inequalities.