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

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Introduction

Linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will delve into the world of linear inequalities, focusing on the statement y \textgreater 34x−2y \ \textgreater \ \frac{3}{4} x - 2. We will examine three options and determine which ones are true.

Option A: The Slope of the Line is −2-2

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

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

By examining the slope-intercept form, we can see that the slope of the line is 34\frac{3}{4}, not −2-2. Therefore, Option A is false.

Option 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 dashed line if the inequality is strict, i.e., it is written as y \textgreater mx+by \ \textgreater \ mx + b or y \textless mx+by \ \textless \ mx + b. In this case, the inequality is y \textgreater 34x−2y \ \textgreater \ \frac{3}{4} x - 2, which is a strict inequality.

# Import necessary modules
import matplotlib.pyplot as plt
import numpy as np

# Define variables
x = np.linspace(-10, 10, 400)
y = (3/4)*x - 2

# Create a plot
plt.plot(x, y, label='y = (3/4)x - 2')
plt.plot(x, y + 0.1, label='y > (3/4)x - 2')
plt.legend()
plt.show()

As we can see from the plot, the graph of y \textgreater 34x−2y \ \textgreater \ \frac{3}{4} x - 2 is a dashed line. Therefore, Option B is true.

Option C: The y-Intercept of the Line is −2-2

The y-intercept of a line is the point where the line intersects the y-axis. To find the y-intercept, we need to set x=0x = 0 in the equation of the line.

# 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

# Find the y-intercept
y_intercept = sp.solve(inequality.subs(x, 0), y)

By examining the y-intercept, we can see that the y-intercept of the line is indeed −2-2. Therefore, Option C is true.

Conclusion

In conclusion, we have analyzed the linear inequality y \textgreater 34x−2y \ \textgreater \ \frac{3}{4} x - 2 and determined the truth of three options. We found that Option A is false, Option B is true, and Option C is true. This comprehensive analysis has provided a deeper understanding of linear inequalities and their graphical representations.

References

Further Reading

Introduction

Linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In our previous article, we delved into the world of linear inequalities, focusing on the statement y \textgreater 34x−2y \ \textgreater \ \frac{3}{4} x - 2. We examined three options and determined which ones were true. In this article, we will provide a comprehensive Q&A guide to help you better understand linear inequalities.

Q: What is a linear inequality?

A linear inequality is an inequality that can be written in the form ax+by \textgreater cax + by \ \textgreater \ c or ax+by \textless cax + by \ \textless \ c, where aa, bb, and cc are constants, and xx and yy are variables.

Q: How do I graph a linear inequality?

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 strict (i.e., y \textgreater mx+by \ \textgreater \ mx + b or y \textless mx+by \ \textless \ mx + b), you should use a dashed line. If the inequality is non-strict (i.e., y ≥ mx+by \ \geq \ mx + b or y ≤ mx+by \ \leq \ mx + b), you should use a solid line.

Q: How do I solve a linear inequality?

To solve a linear inequality, you need to isolate the variable on one side of the inequality. 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: What is the difference between a linear inequality and a linear equation?

A linear equation is an equation that can be written in the form ax+by=cax + by = c, where aa, bb, and cc are constants, and xx and yy are variables. A linear inequality, on the other hand, is an inequality that can be written in the form ax+by \textgreater cax + by \ \textgreater \ c or ax+by \textless cax + by \ \textless \ c.

Q: Can I use the same methods to solve linear inequalities as I do to solve linear equations?

No, you cannot use the same methods to solve linear inequalities as you do to solve linear equations. When solving linear inequalities, you need to consider the direction of the inequality and the signs of the coefficients.

Q: How do I determine the direction of the inequality?

To determine the direction of the inequality, you need to examine the sign of the coefficient of the variable. If the coefficient is positive, the inequality is pointing upwards. If the coefficient is negative, the inequality is pointing downwards.

Q: Can I use the same methods to graph linear inequalities as I do to graph linear equations?

No, you cannot use the same methods to graph linear inequalities as you do to graph linear equations. When graphing linear inequalities, you need to use a dashed line if the inequality is strict and a solid line if the inequality is non-strict.

Q: How do I determine the type of line to use when graphing a linear inequality?

To determine the type of line to use when graphing a linear inequality, you need to examine the inequality. If the inequality is strict (i.e., y \textgreater mx+by \ \textgreater \ mx + b or y \textless mx+by \ \textless \ mx + b), you should use a dashed line. If the inequality is non-strict (i.e., y ≥ mx+by \ \geq \ mx + b or y ≤ mx+by \ \leq \ mx + b), you should use a solid line.

Conclusion

In conclusion, linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. By understanding the basics of linear inequalities, you can better solve problems and make informed decisions. We hope this comprehensive Q&A guide has helped you better understand linear inequalities.

References

Further Reading