Which Is The Graph Of The Linear Inequality 1/2X-2Y-6?

November 05, 2023 Posting Komentar

Linear inequalities are mathematical equations that use symbols like <, >, ≤ and ≥ to represent the relationship between two variables. In this article, we will explore the graph of the linear inequality 1/2x-2y-6 and how it can be plotted on a graph in order to understand its solution set.

Understanding Linear Inequalities

Before we dive into the graph of the linear inequality 1/2x-2y-6, let's first understand what a linear inequality is. A linear inequality is an equation that involves two variables and an inequality symbol. For example, 2x + 3y ≤ 10 is a linear inequality. It represents all the points (x,y) that satisfy the inequality.

When we graph a linear inequality, we plot all the points that satisfy the inequality on a coordinate plane. The solution set is the shaded region that includes all the points that satisfy the inequality.

The Graph of the Linear Inequality 1/2x-2y-6

Now, let's look at the graph of the linear inequality 1/2x-2y-6. We can rewrite the inequality as:

1/2x-6 ≤ 2y

y ≥ 1/4x - 3

This inequality represents all the points (x,y) that lie above the line y = 1/4x - 3. In other words, the solution set is the shaded region above the line.

To graph this inequality, we can start by plotting the line y = 1/4x - 3. To do this, we can choose two values of x and solve for y. For example, if we choose x = 0, y = -3. If we choose x = 4, y = -2. This gives us two points on the line: (0,-3) and (4,-2).

Next, we can plot these points on a coordinate plane and draw a line through them. The line represents all the points that satisfy the equation y = 1/4x - 3.

Finally, we can shade the region above the line to represent the solution set of the inequality y ≥ 1/4x - 3.

Interpreting the Graph

Now that we have plotted the graph of the linear inequality 1/2x-2y-6, let's interpret what it means. The shaded region above the line represents all the points (x,y) that satisfy the inequality. In other words, any point above the line is a solution to the inequality.

For example, the point (4,0) is above the line and satisfies the inequality. However, the point (0,-4) is below the line and does not satisfy the inequality.

We can also interpret the slope of the line. The slope of the line is 1/4, which means that for every increase of 1 in x, there is an increase of 1/4 in y. This means that the line is sloping upward to the right.

Solving Linear Inequalities

Now that we know how to graph a linear inequality, let's briefly discuss how to solve one. To solve a linear inequality, we need to find all the values of the variables that satisfy the inequality.

For example, to solve the inequality y ≥ 1/4x - 3, we can start by setting y = 1/4x - 3. Then, we can solve for x. This gives us:

x ≤ 4(y+3)

This means that all values of x that are less than or equal to 4 times the value of (y+3) satisfy the inequality.

Conclusion

In conclusion, the graph of the linear inequality 1/2x-2y-6 represents all the points (x,y) that lie above the line y = 1/4x - 3. To graph the inequality, we can plot the line and shade the region above it. Interpreting the graph helps us understand which points satisfy the inequality. To solve a linear inequality, we need to find all the values of the variables that satisfy the inequality.

Remember that linear inequalities are an important part of algebra and can be used to model real-world situations. Understanding how to graph and solve linear inequalities is an essential skill for any student of mathematics.