Substituting the Point into the Inequality

Oct 13, 2025 by windailys.com 29 views

Introduction

In mathematics, inequalities are used to describe relationships between variables. They are an essential part of algebra and are used to solve equations, graph functions, and model real-world problems. In this article, we will explore whether a given point satisfies a specific inequality.

The Inequality

The given inequality is $10x + 14y \leq 10$ . This is a linear inequality in two variables, $x$ and $y$ . The inequality states that the sum of $10x$ and $14y$ is less than or equal to $10$ .

The Point

The point in question is $(8, -5)$ . This is a coordinate point in the Cartesian plane, where $x = 8$ and $y = -5$ .

Substituting the Point into the Inequality

To determine whether the point satisfies the inequality, we need to substitute the values of $x$ and $y$ into the inequality. This means replacing $x$ with $8$ and $y$ with $-5$ in the inequality.

Calculating the Result

Substituting the values of $x$ and $y$ into the inequality, we get:

$10(8) + 14(-5) \leq 10$

Expanding the equation, we get:

$80 - 70 \leq 10$

Simplifying the equation, we get:

$10 \leq 10$

Conclusion

The result of the calculation is that the left-hand side of the inequality is equal to the right-hand side. This means that the point $(8, -5)$ satisfies the inequality.

Understanding the Concept

In mathematics, an inequality is satisfied when the left-hand side is less than or equal to the right-hand side. In this case, the left-hand side is equal to the right-hand side, which means that the point satisfies the inequality.

Graphical Representation

To visualize the inequality, we can graph the line $10x + 14y = 10$ on the Cartesian plane. The line will divide the plane into two regions: one where the inequality is satisfied and one where it is not.

Graphing the Line

To graph the line, we can use the slope-intercept form of a linear equation, which is $y = mx + b$ . In this case, the slope is $-\frac{10}{14} = -\frac{5}{7}$ and the y-intercept is $\frac{10}{14} = \frac{5}{7}$ .

Plotting the Line

Plotting the line on the Cartesian plane, we get a line with a slope of $-\frac{5}{7}$ and a y-intercept of $\frac{5}{7}$ . The line divides the plane into two regions: one where the inequality is satisfied and one where it is not.

Conclusion

In conclusion, the point $(8, -5)$ satisfies the inequality $10x + 14y \leq 10$ . This means that the point lies on or below the line $10x + 14y = 10$ .

Final Answer

The final answer is A. Yes. The point $(8, -5)$ satisfies the inequality $10x + 14y \leq 10$ .

Frequently Asked Questions

Q: What is the inequality in question?

A: The inequality in question is $10x + 14y \leq 10$ .

Q: What is the point in question?

A: The point in question is $(8, -5)$ .

Q: Does the point satisfy the inequality?

A: Yes, the point $(8, -5)$ satisfies the inequality $10x + 14y \leq 10$ .

Q: How can we graph the inequality?

A: We can graph the line $10x + 14y = 10$ on the Cartesian plane to visualize the inequality.

Q: What is the significance of the line in the inequality?

A: The line $10x + 14y = 10$ divides the plane into two regions: one where the inequality is satisfied and one where it is not.

Conclusion

In conclusion, the point $(8, -5)$ satisfies the inequality $10x + 14y \leq 10$ . This means that the point lies on or below the line $10x + 14y = 10$ . The line divides the plane into two regions: one where the inequality is satisfied and one where it is not.

Introduction

Inequalities are an essential part of mathematics, and they are used to describe relationships between variables. In this article, we will answer some frequently asked questions (FAQs) on inequalities.

Q: What is an inequality?

A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and nonlinear inequalities. Linear inequalities are of the form $ax + by \leq c$ , where $a$ , $b$ , and $c$ are constants. Nonlinear inequalities are of the form $f(x) \leq g(x)$ , where $f(x)$ and $g(x)$ are functions.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

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

A: A linear inequality is of the form $ax + by \leq c$ , where $a$ , $b$ , and $c$ are constants. A nonlinear inequality is of the form $f(x) \leq g(x)$ , where $f(x)$ and $g(x)$ are functions.

Q: Can I graph a linear inequality?

A: Yes, you can graph a linear inequality by graphing the corresponding linear equation and then shading the region that satisfies the inequality.

Q: How do I graph a nonlinear inequality?

A: To graph a nonlinear inequality, you need to graph the corresponding nonlinear equation and then shade the region that satisfies the inequality.

Q: What is the significance of the line in an inequality?

A: The line in an inequality divides the plane into two regions: one where the inequality is satisfied and one where it is not.

Q: Can I use inequalities to solve real-world problems?

A: Yes, inequalities are used to solve many real-world problems, such as optimizing costs, maximizing profits, and minimizing losses.

Q: What are some common applications of inequalities?

A: Inequalities are used in many fields, including economics, finance, engineering, and computer science.

Q: Can I use inequalities to solve systems of equations?

A: Yes, inequalities can be used to solve systems of equations by finding the intersection of the solution sets of the individual equations.

Q: What is the difference between a system of linear equations and a system of linear inequalities?

A: A system of linear equations is a set of equations of the form $ax + by = c$ , where $a$ , $b$ , and $c$ are constants. A system of linear inequalities is a set of inequalities of the form $ax + by \leq c$ , where $a$ , $b$ , and $c$ are constants.

Q: Can I use inequalities to solve optimization problems?

A: Yes, inequalities can be used to solve optimization problems by finding the maximum or minimum value of a function subject to certain constraints.

Q: What are some common optimization problems that can be solved using inequalities?

A: Some common optimization problems that can be solved using inequalities include maximizing profits, minimizing costs, and optimizing resource allocation.

Conclusion

In conclusion, inequalities are an essential part of mathematics, and they are used to describe relationships between variables. In this article, we have answered some frequently asked questions (FAQs) on inequalities, including the definition of an inequality, the different types of inequalities, and how to solve and graph inequalities.

Final Answer

The final answer is that inequalities are a powerful tool for solving mathematical problems and modeling real-world situations.