Which Inequality Pairs With $y \leq -2x - 1$ To Complete The System Of Linear Inequalities?A. $y \ \textless \ -2x + 2$ B. $y \ \textgreater \ -2x + 2$ C. $y \ \textless \ 2x - 2$ D. $y \ \textgreater \

Feb 28, 2025 by ADMIN 213 views

$**Which Inequality Pairs with $y \leq -2x - 1$ to Complete the System of Linear Inequalities?**$

Understanding the Basics of Linear Inequalities

Linear inequalities are mathematical expressions that contain a variable and a constant, separated by an inequality sign. They are used to describe a set of values that a variable can take, and are often used in real-world applications such as finance, economics, and science. In this article, we will focus on finding the correct inequality that pairs with $y \leq -2x - 1$ to complete the system of linear inequalities.

The Concept of Systems of Linear Inequalities

A system of linear inequalities is a set of two or more linear inequalities that are combined to form a single mathematical expression. Each inequality in the system represents a constraint on the values of the variables, and the solution to the system is the set of values that satisfy all the inequalities simultaneously. In this case, we are looking for the inequality that pairs with $y \leq -2x - 1$ to form a system of linear inequalities.

Analyzing the Options

Let's analyze each of the options given:

Option A: $y \ \textless \ -2x + 2$

This inequality represents a line with a slope of -2 and a y-intercept of 2. However, when we compare it with $y \leq -2x - 1$ , we can see that the two lines are parallel, but the inequality in option A is strict (<), whereas the inequality in $y \leq -2x - 1$ is non-strict (≤). This means that the two inequalities are not equivalent, and option A is not the correct answer.

Option B: $y \ \textgreater \ -2x + 2$

This inequality represents a line with a slope of -2 and a y-intercept of 2, but with a strict inequality (>). When we compare it with $y \leq -2x - 1$ , we can see that the two lines are parallel, but the inequality in option B is strict (>), whereas the inequality in $y \leq -2x - 1$ is non-strict (≤). This means that the two inequalities are not equivalent, and option B is not the correct answer.

Option C: $y \ \textless \ 2x - 2$

This inequality represents a line with a slope of 2 and a y-intercept of -2. However, when we compare it with $y \leq -2x - 1$ , we can see that the two lines are not parallel, and the inequality in option C is strict (<), whereas the inequality in $y \leq -2x - 1$ is non-strict (≤). This means that the two inequalities are not equivalent, and option C is not the correct answer.

Option D: $y \ \textgreater \ 2x - 2$

This inequality represents a line with a slope of 2 and a y-intercept of -2, but with a strict inequality (>). When we compare it with $y \leq -2x - 1$ , we can see that the two lines are not parallel, and the inequality in option D is strict (>), whereas the inequality in $y \leq -2x - 1$ is non-strict (≤). This means that the two inequalities are not equivalent, and option D is not the correct answer.

Finding the Correct Inequality

To find the correct inequality, we need to find an inequality that is parallel to $y \leq -2x - 1$ and has a non-strict inequality (≤). Since the slope of $y \leq -2x - 1$ is -2, the correct inequality must also have a slope of -2. The only option that satisfies this condition is option A, but we need to modify it to make it non-strict (≤).

Modifying Option A

To modify option A, we need to change the strict inequality (<) to a non-strict inequality (≤). This can be done by changing the inequality sign from < to ≤. Therefore, the correct inequality is:

$y \leq -2x + 2$

This inequality is parallel to $y \leq -2x - 1$ and has a non-strict inequality (≤), making it the correct answer.

Conclusion

In conclusion, the correct inequality that pairs with $y \leq -2x - 1$ to complete the system of linear inequalities is $y \leq -2x + 2$ . This inequality is parallel to $y \leq -2x - 1$ and has a non-strict inequality (≤), making it the correct answer.

Final Answer

The final answer is $y \leq -2x + 2$ .

Q: What is a system of linear inequalities?

A: A system of linear inequalities is a set of two or more linear inequalities that are combined to form a single mathematical expression. Each inequality in the system represents a constraint on the values of the variables, and the solution to the system is the set of values that satisfy all the inequalities simultaneously.

Q: How do I determine if two linear inequalities are equivalent?

A: Two linear inequalities are equivalent if they have the same slope and y-intercept, and the same inequality sign (either <, >, ≤, or ≥). If the inequality signs are different, the inequalities are not equivalent.

Q: How do I find the correct inequality to pair with a given inequality?

A: To find the correct inequality, you need to find an inequality that is parallel to the given inequality and has the same inequality sign (either <, >, ≤, or ≥). If the given inequality has a non-strict inequality (≤ or ≥), you need to find an inequality that also has a non-strict inequality.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is represented by a strict inequality sign (< or >), while a non-strict inequality is an inequality that is represented by a non-strict inequality sign (≤ or ≥). Strict inequalities are used to describe a set of values that a variable can take, while non-strict inequalities are used to describe a set of values that a variable can take, including the boundary values.

Q: How do I graph a system of linear inequalities?

A: To graph a system of linear inequalities, you need to graph each inequality separately and then find the region that satisfies all the inequalities simultaneously. You can use a graphing calculator or a computer program to graph the inequalities.

Q: What is the solution to a system of linear inequalities?

A: The solution to a system of linear inequalities is the set of values that satisfy all the inequalities simultaneously. The solution can be a single point, a line, a region, or a set of points.

Q: How do I find the solution to a system of linear inequalities?

A: To find the solution to a system of linear inequalities, you need to find the intersection of the regions that satisfy each inequality. You can use a graphing calculator or a computer program to find the solution.

Q: What are some real-world applications of systems of linear inequalities?

A: Systems of linear inequalities have many real-world applications, including finance, economics, science, and engineering. They are used to model and solve problems in fields such as budgeting, resource allocation, and optimization.

Q: How do I use systems of linear inequalities in real-world applications?

A: To use systems of linear inequalities in real-world applications, you need to identify the constraints and the objective function, and then use the inequalities to model and solve the problem. You can use a graphing calculator or a computer program to solve the system of inequalities.

Q: What are some common mistakes to avoid when working with systems of linear inequalities?

A: Some common mistakes to avoid when working with systems of linear inequalities include:

Not checking if the inequalities are equivalent
Not checking if the inequalities have the same slope and y-intercept
Not checking if the inequalities have the same inequality sign
Not graphing the inequalities correctly
Not finding the intersection of the regions that satisfy each inequality

Q: How do I check if two linear inequalities are equivalent?

A: To check if two linear inequalities are equivalent, you need to check if they have the same slope and y-intercept, and the same inequality sign (either <, >, ≤, or ≥). If the inequality signs are different, the inequalities are not equivalent.

Q: How do I check if a linear inequality has a non-strict inequality?

A: To check if a linear inequality has a non-strict inequality, you need to check if the inequality sign is ≤ or ≥. If the inequality sign is < or >, the inequality is strict.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding equation and then shade the region that satisfies the inequality. If the inequality is strict, you need to shade the region on one side of the line, while if the inequality is non-strict, you need to shade the region on both sides of the line.

Q: How do I find the intersection of two linear inequalities?

A: To find the intersection of two linear inequalities, you need to find the point where the two lines intersect. You can use a graphing calculator or a computer program to find the intersection.

Q: What are some common real-world applications of linear inequalities?

A: Some common real-world applications of linear inequalities include:

Budgeting and financial planning
Resource allocation and optimization
Science and engineering
Economics and finance

Q: How do I use linear inequalities in real-world applications?

A: To use linear inequalities in real-world applications, you need to identify the constraints and the objective function, and then use the inequalities to model and solve the problem. You can use a graphing calculator or a computer program to solve the system of inequalities.

Q: What are some common mistakes to avoid when working with linear inequalities?

A: Some common mistakes to avoid when working with linear inequalities include:

Not checking if the inequalities are equivalent
Not checking if the inequalities have the same slope and y-intercept
Not checking if the inequalities have the same inequality sign
Not graphing the inequalities correctly
Not finding the intersection of the regions that satisfy each inequality