Determine The Region That Satisfies The Inequality: $y \ \textgreater \ -\frac{5}{4}x$

Feb 27, 2025 by ADMIN 89 views

Determine the Region that Satisfies the Inequality: $y \ \textgreater \ -\frac{5}{4}x$

In mathematics, inequalities are used to describe relationships between variables. A linear inequality is an inequality that can be written in the form $ax + by > c$ , where $a$ , $b$ , and $c$ are constants, and $x$ and $y$ are variables. In this article, we will focus on determining the region that satisfies the inequality $y \ \textgreater \ -\frac{5}{4}x$ . This inequality represents a line with a slope of $-\frac{5}{4}$ and a y-intercept of $0$ . We will explore the different regions that satisfy this inequality and provide a detailed explanation of the solution.

The inequality $y \ \textgreater \ -\frac{5}{4}x$ represents a line with a slope of $-\frac{5}{4}$ and a y-intercept of $0$ . The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is $-\frac{5}{4}$ , which means that for every 4 units of horizontal change, the line will drop by 5 units.

To graph the inequality, we need to plot the line $y = -\frac{5}{4}x$ and then determine the region that satisfies the inequality. The line $y = -\frac{5}{4}x$ has a slope of $-\frac{5}{4}$ and a y-intercept of $0$ . We can plot this line by starting at the origin (0, 0) and moving down and to the right.

To determine the region that satisfies the inequality, we need to consider the line $y = -\frac{5}{4}x$ and the inequality $y \ \textgreater \ -\frac{5}{4}x$ . The inequality represents a region above the line $y = -\frac{5}{4}x$ . We can determine this region by plotting the line and then drawing a boundary around the region above the line.

To solve the inequality, we need to find the values of $x$ and $y$ that satisfy the inequality. We can do this by substituting the values of $x$ and $y$ into the inequality and solving for the values that make the inequality true.

The boundary of the region is the line $y = -\frac{5}{4}x$ . This line represents the values of $x$ and $y$ that are on the boundary of the region. We can find the boundary by solving the equation $y = -\frac{5}{4}x$ for $y$ .

To find the region above the line, we need to consider the values of $x$ and $y$ that satisfy the inequality $y \ \textgreater \ -\frac{5}{4}x$ . We can do this by plotting the line and then drawing a boundary around the region above the line.

In conclusion, the inequality $y \ \textgreater \ -\frac{5}{4}x$ represents a line with a slope of $-\frac{5}{4}$ and a y-intercept of $0$ . We have explored the different regions that satisfy this inequality and provided a detailed explanation of the solution. The region that satisfies the inequality is the region above the line $y = -\frac{5}{4}x$ . We have also found the boundary of the region, which is the line $y = -\frac{5}{4}x$ . This line represents the values of $x$ and $y$ that are on the boundary of the region.

The final answer is the region above the line $y = -\frac{5}{4}x$ . This region is represented by the inequality $y \ \textgreater \ -\frac{5}{4}x$ .

[1] "Linear Inequalities" by Math Open Reference
[2] "Graphing Linear Inequalities" by Math Is Fun
[3] "Solving Linear Inequalities" by Khan Academy

[1] "Linear Inequalities" by Wolfram MathWorld
[2] "Graphing Linear Inequalities" by Purplemath
[3] "Solving Linear Inequalities" by MIT OpenCourseWare
Determine the Region that Satisfies the Inequality: $y \ \textgreater \ -\frac{5}{4}x$ - Q&A

In our previous article, we explored the inequality $y \ \textgreater \ -\frac{5}{4}x$ and determined the region that satisfies this inequality. In this article, we will provide a Q&A section to help clarify any questions or doubts that readers may have.

Q: What is the slope of the line $y = -\frac{5}{4}x$ ?

A: The slope of the line $y = -\frac{5}{4}x$ is $-\frac{5}{4}$ . This means that for every 4 units of horizontal change, the line will drop by 5 units.

Q: What is the y-intercept of the line $y = -\frac{5}{4}x$ ?

A: The y-intercept of the line $y = -\frac{5}{4}x$ is 0. This means that the line passes through the origin (0, 0).

Q: How do I graph the line $y = -\frac{5}{4}x$ ?

A: To graph the line $y = -\frac{5}{4}x$ , start at the origin (0, 0) and move down and to the right. The line will have a slope of $-\frac{5}{4}$ and a y-intercept of 0.

Q: What is the region that satisfies the inequality $y \ \textgreater \ -\frac{5}{4}x$ ?

A: The region that satisfies the inequality $y \ \textgreater \ -\frac{5}{4}x$ is the region above the line $y = -\frac{5}{4}x$ . This region includes all points that are above the line and satisfy the inequality.

Q: How do I find the boundary of the region?

A: To find the boundary of the region, solve the equation $y = -\frac{5}{4}x$ for $y$ . This will give you the line that represents the boundary of the region.

Q: What is the significance of the boundary of the region?

A: The boundary of the region represents the values of $x$ and $y$ that are on the boundary of the region. It is an important concept in mathematics and is used to solve linear inequalities.

Q: Can I use the boundary of the region to solve linear inequalities?

A: Yes, you can use the boundary of the region to solve linear inequalities. By finding the boundary of the region, you can determine the values of $x$ and $y$ that satisfy the inequality.

Q: Are there any other ways to solve linear inequalities?

A: Yes, there are other ways to solve linear inequalities. You can use algebraic methods, such as substitution and elimination, to solve linear inequalities.

In conclusion, the inequality $y \ \textgreater \ -\frac{5}{4}x$ represents a line with a slope of $-\frac{5}{4}$ and a y-intercept of 0. We have explored the different regions that satisfy this inequality and provided a detailed explanation of the solution. We have also answered some common questions and provided additional resources for further learning.

The final answer is the region above the line $y = -\frac{5}{4}x$ . This region is represented by the inequality $y \ \textgreater \ -\frac{5}{4}x$ .

[1] "Linear Inequalities" by Math Open Reference
[2] "Graphing Linear Inequalities" by Math Is Fun
[3] "Solving Linear Inequalities" by Khan Academy

[1] "Linear Inequalities" by Wolfram MathWorld
[2] "Graphing Linear Inequalities" by Purplemath
[3] "Solving Linear Inequalities" by MIT OpenCourseWare