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Introduction

In mathematics, linear inequalities are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A linear inequality is an inequality involving a linear expression, and it can be written in the form of $ax + by \leq c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. In this article, we will focus on solving linear inequalities using a graphical approach, and we will determine which three points are solutions to the inequality $3x - 4y \leq 0$.

Understanding Linear Inequalities

A linear inequality is a statement that two linear expressions are not equal, but one is either greater than or less than the other. For example, the inequality $3x - 4y \leq 0$ means that the expression $3x - 4y$ is less than or equal to zero. To solve this inequality, we need to find the values of $x$ and $y$ that satisfy the inequality.

Graphing Linear Inequalities

To graph a linear inequality, we can use the same method as graphing a linear equation. However, when graphing a linear inequality, we need to consider the direction of the inequality sign. If the inequality sign is $\leq$, we need to include the boundary line, and if the inequality sign is $>$ or $<$, we need to exclude the boundary line.

Graphing the Inequality $3x - 4y \leq 0$

To graph the inequality $3x - 4y \leq 0$, we can start by graphing the boundary line $3x - 4y = 0$. The boundary line has a slope of $-\frac{3}{4}$ and a y-intercept of $\frac{3}{4}$. To graph the inequality, we need to shade the region below the boundary line, since the inequality sign is $\leq$.

Determining the Solutions

To determine which three points are solutions to the inequality $3x - 4y \leq 0$, we need to check if each point lies in the shaded region. Let's consider the five points given in the problem:

• $(-2, -8)$
• $(2, 8)$
• $(-8, 0)$
• $(2, -2)$
• $(8, 0)$

We can substitute each point into the inequality and check if it is true. If the inequality is true, then the point is a solution to the inequality.

Checking the Point $(-2, -8)$

To check if the point $(-2, -8)$ is a solution to the inequality, we can substitute $x = -2$ and $y = -8$ into the inequality:

$3(-2) - 4(-8) \leq 0$

$-6 + 32 \leq 0$

$26 \leq 0$

Since the inequality is not true, the point $(-2, -8)$ is not a solution to the inequality.

Checking the Point $(2, 8)$

To check if the point $(2, 8)$ is a solution to the inequality, we can substitute $x = 2$ and $y = 8$ into the inequality:

$3(2) - 4(8) \leq 0$

$6 - 32 \leq 0$

$-26 \leq 0$

Since the inequality is not true, the point $(2, 8)$ is not a solution to the inequality.

Checking the Point $(-8, 0)$

To check if the point $(-8, 0)$ is a solution to the inequality, we can substitute $x = -8$ and $y = 0$ into the inequality:

$3(-8) - 4(0) \leq 0$

$-24 - 0 \leq 0$

$-24 \leq 0$

Since the inequality is not true, the point $(-8, 0)$ is not a solution to the inequality.

Checking the Point $(2, -2)$

To check if the point $(2, -2)$ is a solution to the inequality, we can substitute $x = 2$ and $y = -2$ into the inequality:

$3(2) - 4(-2) \leq 0$

$6 + 8 \leq 0$

$14 \leq 0$

Since the inequality is not true, the point $(2, -2)$ is not a solution to the inequality.

Checking the Point $(8, 0)$

To check if the point $(8, 0)$ is a solution to the inequality, we can substitute $x = 8$ and $y = 0$ into the inequality:

$3(8) - 4(0) \leq 0$

$24 - 0 \leq 0$

$24 \leq 0$

Since the inequality is not true, the point $(8, 0)$ is not a solution to the inequality.

Conclusion

In this article, we have used a graphical approach to solve the linear inequality $3x - 4y \leq 0$. We have graphed the boundary line and shaded the region below the boundary line. We have then checked five points to determine which three points are solutions to the inequality. The points that are solutions to the inequality are:

• $(-8, 0)$
• $(2, -2)$
• $(8, 0)$

Q&A: Solving Linear Inequalities

In the previous article, we discussed how to solve linear inequalities using a graphical approach. In this article, we will answer some frequently asked questions about solving linear inequalities.

Q: What is a linear inequality?

A linear inequality is an inequality involving a linear expression. It can be written in the form of $ax + by \leq c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

Q: How do I graph a linear inequality?

To graph a linear inequality, you can use the same method as graphing a linear equation. However, when graphing a linear inequality, you need to consider the direction of the inequality sign. If the inequality sign is $\leq$, you need to include the boundary line, and if the inequality sign is $>$ or $<$, you need to exclude the boundary line.

Q: What is the boundary line?

The boundary line is the line that separates the region where the inequality is true from the region where the inequality is false. It is the line that is defined by the equation $ax + by = c$.

Q: How do I determine which points are solutions to the inequality?

To determine which points are solutions to the inequality, you need to check if each point lies in the shaded region. You can substitute each point into the inequality and check if it is true. If the inequality is true, then the point is a solution to the inequality.

Q: What if the inequality is not true for any point?

If the inequality is not true for any point, then the inequality has no solution. This means that there are no values of $x$ and $y$ that satisfy the inequality.

Q: Can I use a graphical approach to solve a system of linear inequalities?

Yes, you can use a graphical approach to solve a system of linear inequalities. You can graph each inequality separately and then find the intersection of the shaded regions.

Q: How do I find the intersection of the shaded regions?

To find the intersection of the shaded regions, you need to find the points that are common to both inequalities. You can do this by graphing each inequality separately and then finding the points where the two shaded regions intersect.

Q: What if the intersection of the shaded regions is empty?

If the intersection of the shaded regions is empty, then the system of linear inequalities has no solution. This means that there are no values of $x$ and $y$ that satisfy both inequalities.

Q: Can I use a graphical approach to solve a linear inequality with multiple variables?

Yes, you can use a graphical approach to solve a linear inequality with multiple variables. You can graph the inequality in a three-dimensional space and then find the points that satisfy the inequality.

Q: How do I graph a linear inequality in a three-dimensional space?

To graph a linear inequality in a three-dimensional space, you need to use a three-dimensional coordinate system. You can graph the inequality by plotting the points that satisfy the inequality and then connecting the points to form a surface.

Conclusion

In this article, we have answered some frequently asked questions about solving linear inequalities using a graphical approach. We have discussed how to graph a linear inequality, how to determine which points are solutions to the inequality, and how to find the intersection of the shaded regions. We have also discussed how to solve a system of linear inequalities and how to graph a linear inequality in a three-dimensional space.