Evaluating the Graph Options

Oct 13, 2025 by miodeya.com 112 views

Understanding the Problem

When solving inequalities, it's essential to understand the concept of graphing solutions. Graphing solutions to inequalities involves representing the solution set on a number line or a coordinate plane. In this article, we'll focus on solving the inequality $\frac{1}{3}(9x + 27) \ \textgreater \ x + 33$ and determining which graph represents the solution.

Step 1: Simplify the Inequality

To simplify the inequality, we'll start by distributing the $\frac{1}{3}$ to the terms inside the parentheses:

\frac{1}{3}(9x + 27) \ \textgreater \ x + 33

3x + 9 \ \textgreater \ x + 33

Step 2: Isolate the Variable

Next, we'll isolate the variable $x$ by subtracting $x$ from both sides of the inequality:

3x - x + 9 \ \textgreater \ x - x + 33

2x + 9 \ \textgreater \ 33

Step 3: Subtract 9 from Both Sides

Now, we'll subtract 9 from both sides of the inequality to further isolate the variable:

2x + 9 - 9 \ \textgreater \ 33 - 9

2x \ \textgreater \ 24

Step 4: Divide Both Sides by 2

Finally, we'll divide both sides of the inequality by 2 to solve for $x$ :

\frac{2x}{2} \ \textgreater \ \frac{24}{2}

x \ \textgreater \ 12

Graphing the Solution

Now that we've solved the inequality, we can graph the solution on a number line. The solution set is all real numbers greater than 12, which can be represented as follows:

The open circle at $x = 12$ indicates that 12 is not included in the solution set.
The arrow pointing to the right indicates that all real numbers greater than 12 are included in the solution set.

Evaluating the Graph Options

Given the solution set $x \ \textgreater \ 12$ , we can evaluate the graph options to determine which one represents the solution.

Option A: This graph option includes the point $x = 12$ , which is not part of the solution set.
Option B: This graph option does not include the point $x = 12$ and has an arrow pointing to the right, indicating that all real numbers greater than 12 are included in the solution set.
Option C: This graph option includes the point $x = 12$ and has an arrow pointing to the right, indicating that all real numbers greater than 12 are included in the solution set.
Option D: This graph option does not include the point $x = 12$ and has an arrow pointing to the left, indicating that all real numbers less than 12 are included in the solution set.

Conclusion

Based on the solution set $x \ \textgreater \ 12$ , the correct graph option is Option B. This graph option accurately represents the solution set, including all real numbers greater than 12.

Key Takeaways

When solving inequalities, it's essential to understand the concept of graphing solutions.
Graphing solutions involves representing the solution set on a number line or a coordinate plane.
To solve the inequality $\frac{1}{3}(9x + 27) \ \textgreater \ x + 33$ , we simplified the inequality, isolated the variable, and divided both sides by 2.
The solution set is all real numbers greater than 12, which can be represented on a number line with an open circle at $x = 12$ and an arrow pointing to the right.

Final Thoughts

Solving inequalities and graphing solutions is a crucial concept in mathematics. By understanding how to graph solutions, we can accurately represent the solution set and make informed decisions. In this article, we solved the inequality $\frac{1}{3}(9x + 27) \ \textgreater \ x + 33$ and determined which graph option represents the solution.

Understanding Inequalities

Inequalities are mathematical statements that compare two expressions using greater than, less than, greater than or equal to, or less than or equal to. In this article, we'll focus on solving inequalities and graphing solutions.

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

A: An equation is a mathematical statement that states two expressions are equal, while an inequality is a mathematical statement that compares two expressions using greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you'll need to follow these steps:

Simplify the inequality by combining like terms.
Isolate the variable by adding or subtracting the same value from both sides of the inequality.
Divide both sides of the inequality by a positive value to solve for the variable.

Q: What is the concept of graphing solutions?

A: Graphing solutions involves representing the solution set on a number line or a coordinate plane. The solution set is all real numbers that satisfy the inequality.

Q: How do I graph the solution set on a number line?

A: To graph the solution set on a number line, follow these steps:

Draw a number line and mark the point that corresponds to the value of the variable.
If the inequality is greater than or equal to, draw an open circle at the point and an arrow pointing to the right.
If the inequality is less than or equal to, draw a closed circle at the point and an arrow pointing to the left.

Q: What is the difference between an open circle and a closed circle on a number line?

A: An open circle on a number line indicates that the point is not included in the solution set, while a closed circle indicates that the point is included in the solution set.

Q: How do I determine which graph option represents the solution set?

A: To determine which graph option represents the solution set, follow these steps:

Evaluate each graph option to see if it includes the point that corresponds to the value of the variable.
Check if the graph option has an arrow pointing to the right or left, indicating the direction of the solution set.

Q: What is the importance of graphing solutions?

A: Graphing solutions is an essential concept in mathematics that allows us to visualize the solution set and make informed decisions.

Q: Can you provide an example of solving an inequality and graphing the solution set?

A: Let's consider the inequality $\frac{1}{3}(9x + 27) \ \textgreater \ x + 33$ . To solve this inequality, we'll follow the steps outlined above:

Simplify the inequality by combining like terms.
Isolate the variable by adding or subtracting the same value from both sides of the inequality.
Divide both sides of the inequality by a positive value to solve for the variable.

The solution set is all real numbers greater than 12, which can be represented on a number line with an open circle at $x = 12$ and an arrow pointing to the right.

Conclusion

Solving inequalities and graphing solutions is a crucial concept in mathematics. By understanding how to graph solutions, we can accurately represent the solution set and make informed decisions. In this article, we provided a Q&A guide to help you understand the concept of solving inequalities and graphing solutions.

Key Takeaways

Inequalities are mathematical statements that compare two expressions using greater than, less than, greater than or equal to, or less than or equal to.
To solve an inequality, simplify the inequality, isolate the variable, and divide both sides by a positive value.
Graphing solutions involves representing the solution set on a number line or a coordinate plane.
The solution set is all real numbers that satisfy the inequality.

Final Thoughts

Solving inequalities and graphing solutions is an essential concept in mathematics that allows us to visualize the solution set and make informed decisions. By understanding how to graph solutions, we can accurately represent the solution set and make informed decisions.