Determine Which Points Satisfy The System Of Inequalities:$ \begin{array}{l} y \ \textgreater \ -2x + 3 \ y \leq X - 2 \end{array} }$Options A. { (0,0)$ $ B. { (0,-1)$}$ C. { (1,1)$}$ D.

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Introduction

In mathematics, systems of inequalities are a set of two or more inequalities that involve multiple variables. Solving these systems requires a combination of algebraic and graphical techniques. In this article, we will focus on determining which points satisfy a system of two inequalities.

Understanding the System of Inequalities

The given system of inequalities is:

{ \begin{array}{l} y \ \textgreater \ -2x + 3 \\ y \leq x - 2 \end{array} \}

To solve this system, we need to find the points that satisfy both inequalities simultaneously.

Graphing the Inequalities

Let's start by graphing the two inequalities on a coordinate plane.

Graphing the First Inequality

The first inequality is:

y \textgreater −2x+3{ y \ \textgreater \ -2x + 3 }

To graph this inequality, we need to find the equation of the line that represents the boundary. The equation of the line is:

y=−2x+3{ y = -2x + 3 }

This is a linear equation in slope-intercept form, where the slope is -2 and the y-intercept is 3.

To graph the inequality, we need to shade the region above the line. This is because the inequality is greater than, which means that the solution set includes all points above the line.

Graphing the Second Inequality

The second inequality is:

y≤x−2{ y \leq x - 2 }

To graph this inequality, we need to find the equation of the line that represents the boundary. The equation of the line is:

y=x−2{ y = x - 2 }

This is a linear equation in slope-intercept form, where the slope is 1 and the y-intercept is -2.

To graph the inequality, we need to shade the region below the line. This is because the inequality is less than or equal to, which means that the solution set includes all points below the line.

Finding the Intersection Points

To find the points that satisfy both inequalities, we need to find the intersection points of the two lines.

The first line is:

y=−2x+3{ y = -2x + 3 }

The second line is:

y=x−2{ y = x - 2 }

To find the intersection point, we need to set the two equations equal to each other and solve for x.

−2x+3=x−2{ -2x + 3 = x - 2 }

Solving for x, we get:

−3x=−5{ -3x = -5 }

x=53{ x = \frac{5}{3} }

Now that we have the value of x, we can substitute it into one of the equations to find the value of y.

Substituting x into the first equation, we get:

y=−2(53)+3{ y = -2\left(\frac{5}{3}\right) + 3 }

y=−103+3{ y = -\frac{10}{3} + 3 }

y=−103+93{ y = -\frac{10}{3} + \frac{9}{3} }

y=−13{ y = -\frac{1}{3} }

So, the intersection point is:

(53,−13){ \left(\frac{5}{3}, -\frac{1}{3}\right) }

Analyzing the Options

Now that we have the intersection point, let's analyze the options.

Option A

Option A is:

(0,0){ (0,0) }

To determine if this point satisfies the system of inequalities, we need to check if it lies in the shaded region.

Substituting x = 0 and y = 0 into the first inequality, we get:

0 \textgreater −2(0)+3{ 0 \ \textgreater \ -2(0) + 3 }

0 \textgreater 3{ 0 \ \textgreater \ 3 }

This is not true, so the point (0,0) does not satisfy the first inequality.

Substituting x = 0 and y = 0 into the second inequality, we get:

0≤0−2{ 0 \leq 0 - 2 }

0≤−2{ 0 \leq -2 }

This is not true, so the point (0,0) does not satisfy the second inequality.

Therefore, option A is incorrect.

Option B

Option B is:

(0,−1){ (0,-1) }

To determine if this point satisfies the system of inequalities, we need to check if it lies in the shaded region.

Substituting x = 0 and y = -1 into the first inequality, we get:

−1 \textgreater −2(0)+3{ -1 \ \textgreater \ -2(0) + 3 }

−1 \textgreater 3{ -1 \ \textgreater \ 3 }

This is not true, so the point (0,-1) does not satisfy the first inequality.

Substituting x = 0 and y = -1 into the second inequality, we get:

−1≤0−2{ -1 \leq 0 - 2 }

−1≤−2{ -1 \leq -2 }

This is true, so the point (0,-1) satisfies the second inequality.

Therefore, option B is partially correct.

Option C

Option C is:

(1,1){ (1,1) }

To determine if this point satisfies the system of inequalities, we need to check if it lies in the shaded region.

Substituting x = 1 and y = 1 into the first inequality, we get:

1 \textgreater −2(1)+3{ 1 \ \textgreater \ -2(1) + 3 }

1 \textgreater 1{ 1 \ \textgreater \ 1 }

This is not true, so the point (1,1) does not satisfy the first inequality.

Substituting x = 1 and y = 1 into the second inequality, we get:

1≤1−2{ 1 \leq 1 - 2 }

1≤−1{ 1 \leq -1 }

This is not true, so the point (1,1) does not satisfy the second inequality.

Therefore, option C is incorrect.

Conclusion

In conclusion, the only point that satisfies both inequalities is the intersection point:

(53,−13){ \left(\frac{5}{3}, -\frac{1}{3}\right) }

Therefore, the correct answer is not among the options provided.

However, if we re-examine the options, we can see that option B is partially correct. The point (0,-1) satisfies the second inequality, but not the first inequality.

Therefore, the correct answer is option B, but with the understanding that it is partially correct.

Discussion

The system of inequalities is a classic example of a linear programming problem. The goal is to find the points that satisfy both inequalities simultaneously.

In this case, we used graphical techniques to visualize the solution set. We graphed the two inequalities on a coordinate plane and found the intersection point.

However, in real-world applications, we often encounter systems of inequalities with multiple variables. In such cases, we need to use more advanced techniques, such as linear programming algorithms or optimization methods.

Final Answer

The final answer is:

  • Option B is partially correct, but the correct answer is not among the options provided.
    Q&A: Systems of Inequalities =============================

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that involve multiple variables. It is a mathematical problem that requires finding the points that satisfy all the inequalities simultaneously.

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you need to graph each inequality separately on a coordinate plane. Then, you need to find the intersection points of the lines and shade the regions that satisfy the inequalities.

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 two or more equations that involve multiple variables. It is a mathematical problem that requires finding the points that satisfy all the equations simultaneously. A system of linear inequalities, on the other hand, is a set of two or more inequalities that involve multiple variables. It is a mathematical problem that requires finding the points that satisfy all the inequalities simultaneously.

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

A: To solve a system of linear inequalities, you need to follow these steps:

  1. Graph each inequality separately on a coordinate plane.
  2. Find the intersection points of the lines.
  3. Shade the regions that satisfy the inequalities.
  4. Find the points that lie in the shaded region.

Q: What is the intersection point of two lines?

A: The intersection point of two lines is the point where the two lines meet. It is the point that satisfies both equations simultaneously.

Q: How do I find the intersection point of two lines?

A: To find the intersection point of two lines, you need to follow these steps:

  1. Set the two equations equal to each other.
  2. Solve for x.
  3. Substitute x into one of the equations to find y.

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

A: A linear inequality is an inequality that involves a linear expression. It is an inequality that can be written in the form ax + by < c, where a, b, and c are constants. A nonlinear inequality, on the other hand, is an inequality that involves a nonlinear expression. It is an inequality that cannot be written in the form ax + by < c, where a, b, and c are constants.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you need to follow these steps:

  1. Graph the nonlinear expression on a coordinate plane.
  2. Find the points that satisfy the inequality.
  3. Shade the region that satisfies the inequality.

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

A: Systems of inequalities are important in real-world applications because they help us model and solve problems that involve multiple variables and constraints. They are used in fields such as economics, engineering, and computer science to optimize solutions and make decisions.

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

A: To use systems of inequalities in real-world applications, you need to follow these steps:

  1. Identify the variables and constraints involved in the problem.
  2. Write the inequalities that represent the constraints.
  3. Graph the inequalities on a coordinate plane.
  4. Find the points that satisfy the inequalities.
  5. Use the solution to make decisions or optimize solutions.

Conclusion

In conclusion, systems of inequalities are an important topic in mathematics that has many real-world applications. They help us model and solve problems that involve multiple variables and constraints. By understanding how to graph and solve systems of inequalities, we can make informed decisions and optimize solutions in a variety of fields.