Solve The Inequality: $\[ 4y - 7x \ \textless \ 0 \\]
Introduction
In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. They are used to describe the relationship between variables and constants, and are essential in solving problems in various fields such as algebra, geometry, and calculus. In this article, we will focus on solving the inequality 4y - 7x < 0, which is a linear inequality in two variables.
Understanding the Inequality
The given inequality is 4y - 7x < 0, where x and y are the variables, and 4 and -7 are the coefficients. To solve this inequality, we need to isolate the variables on one side of the inequality sign. The goal is to find the values of x and y that satisfy the inequality.
Graphical Representation
To visualize the inequality, we can graph the related equation 4y - 7x = 0. This equation represents a line in the coordinate plane. The inequality 4y - 7x < 0 represents the region below the line.
Solving the Inequality
To solve the inequality, we can use the following steps:
- Isolate the variable y: We can rewrite the inequality as y < (7/4)x.
- Graph the related equation: The related equation is 4y - 7x = 0, which represents a line in the coordinate plane.
- Determine the region: The inequality represents the region below the line.
Graphing the Related Equation
To graph the related equation 4y - 7x = 0, we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.
- Slope (m): The slope of the line is -7/4, which is the coefficient of x in the inequality.
- Y-intercept (b): The y-intercept of the line is 0, since the constant term in the inequality is 0.
Graphing the Inequality
To graph the inequality 4y - 7x < 0, we can use the following steps:
- Graph the related equation: The related equation is 4y - 7x = 0, which represents a line in the coordinate plane.
- Determine the region: The inequality represents the region below the line.
Solution Set
The solution set of the inequality 4y - 7x < 0 is the region below the line 4y - 7x = 0. This region includes all the points (x, y) that satisfy the inequality.
Conclusion
In this article, we have solved the inequality 4y - 7x < 0, which is a linear inequality in two variables. We have used the graphical representation of the related equation to visualize the inequality and determine the solution set. The solution set is the region below the line 4y - 7x = 0, which includes all the points (x, y) that satisfy the inequality.
Frequently Asked Questions
- What is the solution set of the inequality 4y - 7x < 0? The solution set is the region below the line 4y - 7x = 0.
- How do I graph the related equation 4y - 7x = 0? You can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.
- How do I determine the region of the inequality 4y - 7x < 0? You can graph the related equation 4y - 7x = 0 and determine the region below the line.
References
- Algebra and Trigonometry by Michael Sullivan
- Calculus by James Stewart
- Linear Algebra and Its Applications by Gilbert Strang
Further Reading
- Solving Linear Inequalities by Math Open Reference
- Graphing Linear Equations by Math Is Fun
- Linear Inequalities by Khan Academy
Introduction
In our previous article, we solved the inequality 4y - 7x < 0, which is a linear inequality in two variables. We used the graphical representation of the related equation to visualize the inequality and determine the solution set. In this article, we will answer some frequently asked questions related to the inequality 4y - 7x < 0.
Q&A
Q: What is the solution set of the inequality 4y - 7x < 0?
A: The solution set is the region below the line 4y - 7x = 0.
Q: How do I graph the related equation 4y - 7x = 0?
A: You can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. The slope of the line is -7/4, and the y-intercept is 0.
Q: How do I determine the region of the inequality 4y - 7x < 0?
A: You can graph the related equation 4y - 7x = 0 and determine the region below the line.
Q: What is the relationship between the inequality 4y - 7x < 0 and the related equation 4y - 7x = 0?
A: The inequality 4y - 7x < 0 represents the region below the line 4y - 7x = 0.
Q: How do I solve the inequality 4y - 7x < 0 using algebraic methods?
A: You can isolate the variable y by rewriting the inequality as y < (7/4)x.
Q: What is the significance of the slope of the line 4y - 7x = 0?
A: The slope of the line represents the rate of change of the variable y with respect to the variable x.
Q: How do I find the y-intercept of the line 4y - 7x = 0?
A: The y-intercept of the line is 0, since the constant term in the inequality is 0.
Q: What is the relationship between the inequality 4y - 7x < 0 and the concept of linear programming?
A: The inequality 4y - 7x < 0 is a linear inequality, which is a fundamental concept in linear programming.
Conclusion
In this article, we have answered some frequently asked questions related to the inequality 4y - 7x < 0. We have provided detailed explanations and examples to help readers understand the concepts and techniques involved in solving linear inequalities.
Frequently Asked Questions
- What is the solution set of the inequality 4y - 7x < 0? The solution set is the region below the line 4y - 7x = 0.
- How do I graph the related equation 4y - 7x = 0? You can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.
- How do I determine the region of the inequality 4y - 7x < 0? You can graph the related equation 4y - 7x = 0 and determine the region below the line.
References
- Algebra and Trigonometry by Michael Sullivan
- Calculus by James Stewart
- Linear Algebra and Its Applications by Gilbert Strang
Further Reading
- Solving Linear Inequalities by Math Open Reference
- Graphing Linear Equations by Math Is Fun
- Linear Inequalities by Khan Academy