Is ( 1 , 5 (1,5 ( 1 , 5 ] A Solution To This System Of Inequalities?${ \begin{align*} 10x + Y &\ \textgreater \ 5 \ 8x + 2y &\geq 18 \end{align*} }$A. YesB. No

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Introduction

When dealing with systems of inequalities, it's essential to determine whether a given point satisfies all the conditions. In this case, we're given a system of two linear inequalities and asked to determine if the point $(1,5)$ is a solution. To do this, we'll substitute the coordinates of the point into each inequality and check if the resulting statement is true.

Understanding the Inequalities

Before we proceed, let's take a closer look at the two inequalities:

1. $10x + y > 5$
2. $8x + 2y \geq 18$

We need to understand the meaning of these inequalities. The first inequality states that the sum of $10x$ and $y$ is greater than $5$. This means that if we substitute any value of $x$ and $y$ into the equation, the result should be greater than $5$. The second inequality states that the sum of $8x$ and $2y$ is greater than or equal to $18$. This means that if we substitute any value of $x$ and $y$ into the equation, the result should be greater than or equal to $18$.

Substituting the Point into the Inequalities

Now that we understand the inequalities, let's substitute the point $(1,5)$ into each one:

1. $10(1) + 5 > 5$
2. $8(1) + 2(5) \geq 18$

Evaluating the First Inequality

Let's evaluate the first inequality:

$10(1) + 5 > 5$

This simplifies to:

$15 > 5$

Since $15$ is indeed greater than $5$, the first inequality is satisfied.

Evaluating the Second Inequality

Now let's evaluate the second inequality:

$8(1) + 2(5) \geq 18$

This simplifies to:

$8 + 10 \geq 18$

Which further simplifies to:

$18 \geq 18$

Since $18$ is equal to $18$, the second inequality is also satisfied.

Conclusion

Since both inequalities are satisfied, the point $(1,5)$ is a solution to the system of inequalities.

Importance of Systems of Inequalities

Systems of inequalities are used to model real-world problems that involve constraints. For example, a company may want to maximize its profits while minimizing its costs. In this case, the system of inequalities would represent the constraints on the company's resources and the objective function would represent the profit.

Real-World Applications

Systems of inequalities have many real-world applications, including:

• Optimization problems: Systems of inequalities can be used to model optimization problems, such as maximizing profits or minimizing costs.
• Resource allocation: Systems of inequalities can be used to model resource allocation problems, such as allocating resources to different projects or departments.
• Scheduling: Systems of inequalities can be used to model scheduling problems, such as scheduling tasks or meetings.

Tips for Solving Systems of Inequalities

Here are some tips for solving systems of inequalities:

• Understand the inequalities: Before solving the system, make sure you understand the meaning of each inequality.
• Substitute values: Substitute the values of the variables into each inequality and check if the resulting statement is true.
• Use algebraic manipulations: Use algebraic manipulations to simplify the inequalities and make them easier to solve.
• Graph the inequalities: Graph the inequalities on a coordinate plane to visualize the solution set.

Conclusion

In conclusion, the point $(1,5)$ is a solution to the system of inequalities. Systems of inequalities are used to model real-world problems that involve constraints, and have many real-world applications, including optimization problems, resource allocation, and scheduling. By understanding the inequalities and using algebraic manipulations, we can solve systems of inequalities and find the solution set.

Final Answer

The final answer is: A. Yes

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Introduction

In our previous article, we determined that the point $(1,5)$ is a solution to the system of inequalities:

1. $10x + y > 5$
2. $8x + 2y \geq 18$

However, we received many questions from readers asking for clarification on the solution. In this article, we'll address some of the most frequently asked questions.

Q&A

Q: What if the point $(1,5)$ doesn't satisfy one of the inequalities?

A: If the point $(1,5)$ doesn't satisfy one of the inequalities, then it's not a solution to the system of inequalities. However, in this case, we found that the point $(1,5)$ satisfies both inequalities.

Q: Can you explain why the second inequality is satisfied?

A: The second inequality is satisfied because $8(1) + 2(5) = 8 + 10 = 18$. Since $18$ is equal to $18$, the second inequality is satisfied.

Q: What if the inequalities are not linear?

A: If the inequalities are not linear, then the solution process would be more complex. However, the basic idea remains the same: substitute the values of the variables into each inequality and check if the resulting statement is true.

Q: Can you provide more examples of systems of inequalities?

A: Here are a few examples of systems of inequalities:

1. $x + 2y > 3$

2. $2x - 3y \geq -6$

3. $x - 2y < 1$

4. $3x + 4y \geq 12$

5. $2x + 3y > 5$

6. $x - 2y \leq -3$

Q: How do I graph the inequalities on a coordinate plane?

A: To graph the inequalities on a coordinate plane, follow these steps:

1. Plot the boundary line for each inequality.
2. Test a point on each side of the boundary line to determine which side of the line satisfies the inequality.
3. Shade the region that satisfies the inequality.

Q: What if I'm having trouble graphing the inequalities?

A: If you're having trouble graphing the inequalities, try the following:

1. Use a graphing calculator or software to graph the inequalities.
2. Plot the boundary line and test a point on each side of the line.
3. Shade the region that satisfies the inequality.

Q: Can you provide more tips for solving systems of inequalities?

A: Here are some additional tips for solving systems of inequalities:

1. Understand the inequalities: Before solving the system, make sure you understand the meaning of each inequality.
2. Substitute values: Substitute the values of the variables into each inequality and check if the resulting statement is true.
3. Use algebraic manipulations: Use algebraic manipulations to simplify the inequalities and make them easier to solve.
4. Graph the inequalities: Graph the inequalities on a coordinate plane to visualize the solution set.

Conclusion

In conclusion, the point $(1,5)$ is a solution to the system of inequalities. We've addressed some of the most frequently asked questions and provided additional tips for solving systems of inequalities. By understanding the inequalities and using algebraic manipulations, we can solve systems of inequalities and find the solution set.

Final Answer

The final answer is: A. Yes