Which Graph Shows All The Values That Satisfy $\frac{2}{9} X + 3 \ \textgreater \ 4 \frac{5}{9}$?

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Introduction

In mathematics, graphing inequalities is a crucial concept that helps us visualize the solution set of an inequality. When dealing with linear inequalities, we can use a graph to represent the solution set. In this article, we will explore how to graph the inequality $\frac{2}{9} x + 3 \ \textgreater \ 4 \frac{5}{9}$ and determine which graph shows all the values that satisfy this inequality.

Understanding the Inequality

To begin, let's simplify the given inequality by converting the mixed number $4 \frac{5}{9}$ to an improper fraction. We can do this by multiplying the whole number part (4) by the denominator (9) and then adding the numerator (5). This gives us:

$$4 \frac{5}{9} = \frac{4 \times 9 + 5}{9} = \frac{36 + 5}{9} = \frac{41}{9}$$

Now, we can rewrite the inequality as:

$$\frac{2}{9} x + 3 \ \textgreater \ \frac{41}{9}$$

Graphing the Inequality

To graph the inequality, we need to find the boundary line and then determine the direction of the inequality. The boundary line is the equation that is equal to zero, which in this case is:

$$\frac{2}{9} x + 3 = \frac{41}{9}$$

We can solve for x by subtracting 3 from both sides and then multiplying both sides by 9 to eliminate the fraction:

$$\frac{2}{9} x = \frac{41}{9} - 3$$

$$\frac{2}{9} x = \frac{41}{9} - \frac{27}{9}$$

$$\frac{2}{9} x = \frac{14}{9}$$

Multiplying both sides by $\frac{9}{2}$ gives us:

$$x = \frac{14}{9} \times \frac{9}{2}$$

$$x = 7$$

So, the boundary line is x = 7.

Determining the Direction of the Inequality

Since the inequality is greater than (>) the boundary line, we need to determine which side of the boundary line satisfies the inequality. To do this, we can choose a test point that is not on the boundary line. Let's choose x = 0 as our test point.

Substituting x = 0 into the inequality gives us:

$$\frac{2}{9} (0) + 3 \ \textgreater \ \frac{41}{9}$$

Simplifying the left-hand side gives us:

$$3 \ \textgreater \ \frac{41}{9}$$

Since 3 is indeed greater than $\frac{41}{9}$, we know that the test point x = 0 satisfies the inequality.

Graphing the Solution Set

Now that we have determined the direction of the inequality, we can graph the solution set. The solution set is the region on one side of the boundary line that satisfies the inequality.

Since the test point x = 0 satisfies the inequality, we know that the solution set is the region to the right of the boundary line x = 7.

Conclusion

In conclusion, the graph that shows all the values that satisfy the inequality $\frac{2}{9} x + 3 \ \textgreater \ 4 \frac{5}{9}$ is the graph with the boundary line x = 7 and the solution set to the right of the boundary line.

Final Answer

The final answer is the graph with the boundary line x = 7 and the solution set to the right of the boundary line.

Graph Representation

Here is a graph representation of the solution set:

### Graph Representation

#### Boundary Line

* x = 7

#### Solution Set

* To the right of the boundary line x = 7

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Simplify the given inequality by converting the mixed number $4 \frac{5}{9}$ to an improper fraction.
2. Rewrite the inequality with the improper fraction.
3. Find the boundary line by solving for x.
4. Determine the direction of the inequality by choosing a test point that is not on the boundary line.
5. Graph the solution set by drawing a line at the boundary line and shading the region to the right of the boundary line.

Frequently Asked Questions

Here are some frequently asked questions about the problem:

• Q: What is the boundary line for the inequality $\frac{2}{9} x + 3 \ \textgreater \ 4 \frac{5}{9}$? A: The boundary line is x = 7.
• Q: Which side of the boundary line satisfies the inequality? A: The region to the right of the boundary line satisfies the inequality.
• Q: How do I graph the solution set? A: Draw a line at the boundary line and shade the region to the right of the boundary line.

Related Topics

Here are some related topics to the problem:

• Graphing linear inequalities
• Solving linear inequalities
• Graphing solution sets

Conclusion

In conclusion, graphing the inequality $\frac{2}{9} x + 3 \ \textgreater \ 4 \frac{5}{9}$ requires finding the boundary line and determining the direction of the inequality. The solution set is the region to the right of the boundary line. By following the step-by-step solution, you can graph the solution set and determine which graph shows all the values that satisfy the inequality.

Introduction

Graphing inequalities is a crucial concept in mathematics that helps us visualize the solution set of an inequality. In our previous article, we explored how to graph the inequality $\frac{2}{9} x + 3 \ \textgreater \ 4 \frac{5}{9}$ and determine which graph shows all the values that satisfy this inequality. In this article, we will answer some frequently asked questions about graphing inequalities.

Q&A

Q: What is the difference between graphing an equation and graphing an inequality?

A: Graphing an equation involves finding the solution set, which is a single point or a line, whereas graphing an inequality involves finding the solution set, which is a region on one side of the boundary line.

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

A: To determine the direction of the inequality, choose a test point that is not on the boundary line. If the test point satisfies the inequality, then the region to the right of the boundary line satisfies the inequality. If the test point does not satisfy the inequality, then the region to the left of the boundary line satisfies the inequality.

Q: What is the boundary line?

A: The boundary line is the equation that is equal to zero. It is the line that separates the solution set from the non-solution set.

Q: How do I graph the solution set?

A: To graph the solution set, draw a line at the boundary line and shade the region to the right of the boundary line if the inequality is greater than (>) or less than (<) the boundary line, and shade the region to the left of the boundary line if the inequality is less than or equal to (≤) or greater than or equal to (≥) the boundary line.

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

A: Yes, you can use a graphing calculator to graph the solution set. However, it is essential to understand the concept of graphing inequalities and how to determine the direction of the inequality.

Q: How do I determine if a point is in the solution set?

A: To determine if a point is in the solution set, substitute the x-coordinate of the point into the inequality and check if it satisfies the inequality.

Q: Can I graph the solution set of a system of inequalities?

A: Yes, you can graph the solution set of a system of inequalities by graphing each inequality separately and finding the intersection of the solution sets.

Examples

Example 1: Graphing the inequality $x - 2 \ \textgreater \ 3$

To graph the inequality $x - 2 \ \textgreater \ 3$, we need to find the boundary line and determine the direction of the inequality.

• The boundary line is x = 5.
• The direction of the inequality is to the right of the boundary line.

So, the graph of the solution set is:

### Graph Representation

#### Boundary Line

* x = 5

#### Solution Set

* To the right of the boundary line x = 5

Example 2: Graphing the inequality $x + 1 \ \textless \ 2$

To graph the inequality $x + 1 \ \textless \ 2$, we need to find the boundary line and determine the direction of the inequality.

• The boundary line is x = 1.
• The direction of the inequality is to the left of the boundary line.

So, the graph of the solution set is:

### Graph Representation

#### Boundary Line

* x = 1

#### Solution Set

* To the left of the boundary line x = 1

Conclusion

In conclusion, graphing inequalities is a crucial concept in mathematics that helps us visualize the solution set of an inequality. By understanding the concept of graphing inequalities and how to determine the direction of the inequality, you can graph the solution set and determine which graph shows all the values that satisfy the inequality.