Is { (4,1)$}$ A Solution To This System Of Inequalities?${ \begin{array}{l} x+y \geq 5 \ 2x+7y \ \textless \ 20 \end{array} }$A. Yes B. No

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Introduction

In mathematics, a system of inequalities is a set of statements that involve one or more variables and are connected by the words "greater than," "less than," "greater than or equal to," or "less than or equal to." These statements are often represented graphically on a coordinate plane, where each point on the plane corresponds to a possible solution to the system. In this article, we will explore whether the point $(4,1)$ is a solution to the system of inequalities given by:

$$\begin{array}{l} x+y \geq 5 \\ 2x+7y \ \textless \ 20 \end{array}$$

Understanding the System of Inequalities

To determine whether the point $(4,1)$ is a solution to the system of inequalities, we need to understand what each inequality represents. The first inequality, $x+y \geq 5$, states that the sum of the $x$-coordinate and the $y$-coordinate of any point on the graph of this inequality must be greater than or equal to $5$. This means that any point on the graph of this inequality must lie on or above the line $x+y=5$.

The second inequality, $2x+7y \ \textless \ 20$, states that the sum of twice the $x$-coordinate and seven times the $y$-coordinate of any point on the graph of this inequality must be less than $20$. This means that any point on the graph of this inequality must lie below the line $2x+7y=20$.

Graphing the Inequalities

To visualize the system of inequalities, we can graph each inequality on a coordinate plane. The graph of the first inequality, $x+y \geq 5$, is a line with a slope of $-1$ and a $y$-intercept of $5$. Any point on or above this line satisfies the first inequality.

The graph of the second inequality, $2x+7y \ \textless \ 20$, is a line with a slope of $-\frac{2}{7}$ and a $y$-intercept of $\frac{20}{7}$. Any point below this line satisfies the second inequality.

Determining Whether $(4,1)$ is a Solution

To determine whether the point $(4,1)$ is a solution to the system of inequalities, we need to check whether it satisfies both inequalities. We can do this by substituting the values of $x$ and $y$ into each inequality and checking whether the resulting statement is true.

For the first inequality, $x+y \geq 5$, we have:

$$4+1 \geq 5$$

This statement is false, since $5$ is greater than $5$. Therefore, the point $(4,1)$ does not satisfy the first inequality.

For the second inequality, $2x+7y \ \textless \ 20$, we have:

$$2(4)+7(1) \ \textless \ 20$$

$$8+7 \ \textless \ 20$$

$$15 \ \textless \ 20$$

This statement is true, since $15$ is less than $20$. Therefore, the point $(4,1)$ satisfies the second inequality.

Conclusion

Since the point $(4,1)$ does not satisfy the first inequality, it is not a solution to the system of inequalities. Therefore, the correct answer is B. No.

Final Thoughts

In this article, we explored whether the point $(4,1)$ is a solution to the system of inequalities given by:

$$\begin{array}{l} x+y \geq 5 \\ 2x+7y \ \textless \ 20 \end{array}$$

We found that the point $(4,1)$ does not satisfy the first inequality, and therefore is not a solution to the system of inequalities. This example illustrates the importance of carefully checking each inequality in a system of inequalities to determine whether a given point is a solution.

References

• [1] "Systems of Inequalities" by Math Open Reference. Retrieved from https://www.mathopenref.com/systemsinequalities.html
• [2] "Graphing Inequalities" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7/graphing-inequalities/v/graphing-inequalities

Related Topics

• Systems of Equations
• Graphing Inequalities
• Linear Programming

Tags

• Systems of Inequalities
• Graphing Inequalities
• Linear Programming
• Mathematics
  

Introduction

In our previous article, we explored whether the point $(4,1)$ is a solution to the system of inequalities given by:

$$\begin{array}{l} x+y \geq 5 \\ 2x+7y \ \textless \ 20 \end{array}$$

We found that the point $(4,1)$ does not satisfy the first inequality, and therefore is not a solution to the system of inequalities. In this article, we will answer some frequently asked questions about systems of inequalities.

Q&A

Q: What is a system of inequalities?

A: A system of inequalities is a set of statements that involve one or more variables and are connected by the words "greater than," "less than," "greater than or equal to," or "less than or equal to."

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you can graph each inequality separately and then find the region where the two graphs overlap. This region represents the solution to the system of inequalities.

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

A: A system of equations is a set of statements that involve one or more variables and are connected by the words "equal to." A system of inequalities, on the other hand, is a set of statements that involve one or more variables and are connected by the words "greater than," "less than," "greater than or equal to," or "less than or equal to."

Q: How do I determine whether a point is a solution to a system of inequalities?

A: To determine whether a point is a solution to a system of inequalities, you need to check whether it satisfies each inequality in the system. If the point satisfies all the inequalities, then it is a solution to the system.

Q: What is the importance of systems of inequalities in real-life applications?

A: Systems of inequalities are used in a variety of real-life applications, including linear programming, optimization problems, and decision-making. They are also used in fields such as economics, finance, and engineering.

Q: Can a system of inequalities have multiple solutions?

A: Yes, a system of inequalities can have multiple solutions. This occurs when the system has multiple regions where the inequalities are satisfied.

Q: How do I solve a system of inequalities with multiple variables?

A: To solve a system of inequalities with multiple variables, you can use a variety of methods, including graphing, substitution, and elimination.

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

A: A linear inequality is an inequality that can be written in the form $ax+by \geq c$ or $ax+by \leq c$, where $a$, $b$, and $c$ are constants. A nonlinear inequality, on the other hand, is an inequality that cannot be written in this form.

Q: Can a system of inequalities have no solutions?

A: Yes, a system of inequalities can have no solutions. This occurs when the system has no regions where the inequalities are satisfied.

Conclusion

In this article, we answered some frequently asked questions about systems of inequalities. We hope that this article has provided you with a better understanding of systems of inequalities and how to solve them.

Final Thoughts

Systems of inequalities are an important topic in mathematics and have many real-life applications. By understanding how to solve systems of inequalities, you can apply this knowledge to a variety of fields, including economics, finance, and engineering.

References

• [1] "Systems of Inequalities" by Math Open Reference. Retrieved from https://www.mathopenref.com/systemsinequalities.html
• [2] "Graphing Inequalities" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7/graphing-inequalities/v/graphing-inequalities

Related Topics

• Systems of Equations
• Graphing Inequalities
• Linear Programming

Tags

• Systems of Inequalities
• Graphing Inequalities
• Linear Programming
• Mathematics