Which Statements Are True About The Linear Inequality $y\ \textgreater \ \frac{3}{4} X-2$? Select Three Options.A. The Slope Of The Line Is -2.B. The Graph Of $y\ \textgreater \ \frac{3}{4} X-2$ Is A Dashed Line.C. The Area Below

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Introduction to Linear Inequalities

Linear inequalities are mathematical expressions that compare two values using an inequality sign. They are an essential part of algebra and are used to solve problems in various fields, including mathematics, science, and engineering. In this article, we will focus on the linear inequality y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2 and examine three statement options related to it.

The Slope of the Line

The first statement option is: A. The slope of the line is -2.

To determine the slope of the line, we need to rewrite the linear inequality in the slope-intercept form, which is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. The given linear inequality is y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2. To rewrite it in the slope-intercept form, we need to isolate yy on one side of the inequality sign.

# Import necessary modules
import sympy as sp

# Define variables
x, y = sp.symbols('x y')

# Define the linear inequality
inequality = y > (3/4)*x - 2

# Rewrite the inequality in the slope-intercept form
slope_intercept_form = sp.solve(inequality, y)

The slope-intercept form of the linear inequality is y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2, which can be rewritten as y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2. Comparing this with the slope-intercept form y=mx+by = mx + b, we can see that the slope mm is 34\frac{3}{4}, not -2. Therefore, statement option A is false.

The Graph of the Linear Inequality

The second statement option is: B. The graph of y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2 is a dashed line.

The graph of a linear inequality is a region in the coordinate plane that satisfies the inequality. The graph of y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2 is a region above the line y=34x−2y = \frac{3}{4} x-2. Since the inequality sign is greater than, the graph is a dashed line, not a solid line. Therefore, statement option B is true.

The Area Below the Line

The third statement option is: C. The area below the line is not part of the solution.

Since the inequality sign is greater than, the graph of y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2 is a region above the line y=34x−2y = \frac{3}{4} x-2. This means that the area below the line is not part of the solution. Therefore, statement option C is true.

Conclusion

In conclusion, we have examined three statement options related to the linear inequality y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2. We found that statement option A is false, statement option B is true, and statement option C is true. The graph of the linear inequality is a dashed line, and the area below the line is not part of the solution.

Final Thoughts

Linear inequalities are an essential part of mathematics, and understanding them is crucial for solving problems in various fields. In this article, we have examined the linear inequality y \textgreater 34x−2y\ \textgreater \ \frac{3}{4} x-2 and determined the truth of three statement options related to it. We hope that this article has provided valuable insights into the world of linear inequalities and has helped readers to better understand this important mathematical concept.

References

Glossary

  • Linear Inequality: A mathematical expression that compares two values using an inequality sign.
  • Slope-Intercept Form: A mathematical expression in the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.
  • Graph of a Linear Inequality: A region in the coordinate plane that satisfies the inequality.
  • Dashed Line: A line that is not solid, but rather a series of connected line segments.
  • Area Below the Line: The region in the coordinate plane that is below the line.

Introduction

Linear inequalities are an essential part of mathematics, and understanding them is crucial for solving problems in various fields. In this article, we will answer some frequently asked questions about linear inequalities, including their definition, graph, and solution.

Q: What is a linear inequality?

A: A linear inequality is a mathematical expression that compares two values using an inequality sign. It is a statement that says one value is greater than, less than, or equal to another value.

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

A: A linear equation is a statement that says two values are equal, while a linear inequality is a statement that says one value is greater than, less than, or equal to another value.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. If the inequality is greater than or equal to, you will shade the region above the line. If the inequality is less than or equal to, you will shade the region below the line.

Q: What is the solution to a linear inequality?

A: The solution to a linear inequality is the set of all values that satisfy the inequality. It is the region in the coordinate plane that satisfies the inequality.

Q: Can a linear inequality have multiple solutions?

A: Yes, a linear inequality can have multiple solutions. For example, the inequality x>2x > 2 has multiple solutions, including all values greater than 2.

Q: Can a linear inequality have no solution?

A: Yes, a linear inequality can have no solution. For example, the inequality x<2x < 2 and x>2x > 2 has no solution, since there is no value that is both less than 2 and greater than 2.

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 sign. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: Can I use algebraic methods to solve linear inequalities?

A: Yes, you can use algebraic methods to solve linear inequalities. For example, you can use the distributive property to simplify the inequality, or you can use the commutative property to rearrange the terms.

Q: Can I use graphical methods to solve linear inequalities?

A: Yes, you can use graphical methods to solve linear inequalities. For example, you can graph the corresponding linear equation and then shade the region that satisfies the inequality.

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

A: A linear inequality is a statement that compares two values using an inequality sign, while a quadratic inequality is a statement that compares a quadratic expression to a constant value.

Q: Can a linear inequality be used to model real-world problems?

A: Yes, a linear inequality can be used to model real-world problems. For example, you can use a linear inequality to model the cost of producing a product, or the time it takes to complete a task.

Conclusion

In conclusion, linear inequalities are an essential part of mathematics, and understanding them is crucial for solving problems in various fields. We hope that this article has provided valuable insights into the world of linear inequalities and has helped readers to better understand this important mathematical concept.

References

Glossary

  • Linear Inequality: A mathematical expression that compares two values using an inequality sign.
  • Graph of a Linear Inequality: A region in the coordinate plane that satisfies the inequality.
  • Solution to a Linear Inequality: The set of all values that satisfy the inequality.
  • Algebraic Methods: Methods used to solve linear inequalities using algebraic techniques.
  • Graphical Methods: Methods used to solve linear inequalities using graphical techniques.