Which Answer Shows $y - 3x \ \textless \ -x + 4$, Rewritten To Isolate $y$, And Its Graph?A. $y \ \textless \ 2x + 4$B. $y \ \textless \ 4x + 4$C. $y \ \textless \ 2x + 4$

by ADMIN 180 views

Understanding the Problem

When dealing with linear inequalities, it's essential to isolate the variable we're interested in, which in this case is y. The given inequality is y−3x \textless −x+4y - 3x \ \textless \ -x + 4, and we need to rewrite it to isolate y. This will allow us to understand the relationship between x and y and visualize the graph of the inequality.

Rewriting the Inequality

To isolate y, we need to get all the terms involving y on one side of the inequality. We can start by adding 3x3x to both sides of the inequality:

y−3x+3x \textless −x+4+3xy - 3x + 3x \ \textless \ -x + 4 + 3x

This simplifies to:

y \textless −x+4+3xy \ \textless \ -x + 4 + 3x

Now, we can combine like terms on the right-hand side:

y \textless 2x+4y \ \textless \ 2x + 4

Graphing the Inequality

To graph the inequality, we need to understand that the inequality is a linear inequality in two variables. The graph of the inequality will be a line that divides the coordinate plane into two regions: one where the inequality is true and one where it is false.

The inequality y \textless 2x+4y \ \textless \ 2x + 4 can be rewritten as y−2x−4 \textless 0y - 2x - 4 \ \textless \ 0. This means that the graph of the inequality will be a line with a slope of 2 and a y-intercept of 4.

Analyzing the Graph

To analyze the graph, we need to understand that the line divides the coordinate plane into two regions: one where the inequality is true and one where it is false. The region where the inequality is true is the region below the line, and the region where the inequality is false is the region above the line.

Conclusion

In conclusion, the correct answer is A. y \textless 2x+4y \ \textless \ 2x + 4. This is the rewritten inequality with y isolated, and its graph is a line with a slope of 2 and a y-intercept of 4. The graph of the inequality divides the coordinate plane into two regions: one where the inequality is true and one where it is false.

Key Takeaways

  • To isolate y, we need to get all the terms involving y on one side of the inequality.
  • The graph of the inequality is a line that divides the coordinate plane into two regions: one where the inequality is true and one where it is false.
  • The region where the inequality is true is the region below the line, and the region where the inequality is false is the region above the line.

Common Mistakes

  • Not isolating y correctly
  • Not understanding the graph of the inequality
  • Not analyzing the graph correctly

Real-World Applications

  • In economics, linear inequalities are used to model the relationship between two variables, such as the cost of production and the quantity produced.
  • In finance, linear inequalities are used to model the relationship between two variables, such as the interest rate and the amount borrowed.
  • In engineering, linear inequalities are used to model the relationship between two variables, such as the speed of a machine and the distance traveled.

Tips and Tricks

  • To isolate y, use the distributive property to get all the terms involving y on one side of the inequality.
  • To graph the inequality, use a graphing calculator or draw the line by hand.
  • To analyze the graph, use a coordinate plane and shade the region where the inequality is true.

Practice Problems

  • Rewrite the inequality x+2y \textless 3x + 2y \ \textless \ 3 to isolate y.
  • Graph the inequality y−2x \textless 1y - 2x \ \textless \ 1.
  • Analyze the graph of the inequality y+3x \textless 2y + 3x \ \textless \ 2.

Conclusion

In conclusion, rewriting inequalities to isolate y and graphing the inequality are essential skills in mathematics. By understanding the problem, rewriting the inequality, graphing the inequality, and analyzing the graph, we can solve linear inequalities and apply them to real-world problems.

Q: What is the first step in rewriting an inequality to isolate y?

A: The first step in rewriting an inequality to isolate y is to get all the terms involving y on one side of the inequality. This can be done by adding or subtracting the same value to both sides of the inequality.

Q: How do I graph the inequality y < 2x + 4?

A: To graph the inequality y < 2x + 4, draw a line with a slope of 2 and a y-intercept of 4. The region below the line is where the inequality is true, and the region above the line is where the inequality is false.

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

A: A linear equation is an equation in which the highest power of the variable is 1. A linear inequality is an inequality in which the highest power of the variable is 1. The main difference between a linear equation and a linear inequality is that a linear equation is an equality, while a linear inequality is an inequality.

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

A: To determine the direction of the inequality, look at the sign of the coefficient of x. If the coefficient of x is positive, the inequality is true for values of x greater than the y-intercept. If the coefficient of x is negative, the inequality is true for values of x less than the y-intercept.

Q: Can I use a graphing calculator to graph the inequality?

A: Yes, you can use a graphing calculator to graph the inequality. Enter the inequality into the calculator and use the graphing function to visualize the inequality.

Q: How do I analyze the graph of the inequality?

A: To analyze the graph of the inequality, look at the region where the inequality is true and the region where the inequality is false. The region where the inequality is true is the region below the line, and the region where the inequality is false is the region above the line.

Q: What are some common mistakes to avoid when rewriting inequalities and graphing?

A: Some common mistakes to avoid when rewriting inequalities and graphing include:

  • Not isolating y correctly
  • Not understanding the graph of the inequality
  • Not analyzing the graph correctly

Q: How do I apply rewriting inequalities and graphing to real-world problems?

A: Rewriting inequalities and graphing can be applied to real-world problems in a variety of ways, including:

  • Modeling the relationship between two variables
  • Analyzing data
  • Making predictions

Q: What are some tips and tricks for rewriting inequalities and graphing?

A: Some tips and tricks for rewriting inequalities and graphing include:

  • Using the distributive property to get all the terms involving y on one side of the inequality
  • Using a graphing calculator to visualize the inequality
  • Analyzing the graph of the inequality to determine the region where the inequality is true and the region where the inequality is false

Q: How do I practice rewriting inequalities and graphing?

A: You can practice rewriting inequalities and graphing by:

  • Working through practice problems
  • Using online resources and tutorials
  • Asking a teacher or tutor for help

Q: What are some resources for learning more about rewriting inequalities and graphing?

A: Some resources for learning more about rewriting inequalities and graphing include:

  • Online tutorials and videos
  • Textbooks and workbooks
  • Online communities and forums

Q: How do I know if I am ready to move on to more advanced topics in mathematics?

A: You can determine if you are ready to move on to more advanced topics in mathematics by:

  • Reviewing the material you have covered so far
  • Practicing problems and exercises
  • Asking a teacher or tutor for feedback and guidance