Ceciala Correctly Solved This Inequality:$\[ \begin{aligned} 3x & \ \textgreater \ 102 \\ x & \ \textgreater \ 34 \end{aligned} \\]Which Graph Matches The Inequality?

Feb 26, 2025 by ADMIN 171 views

**Solving Inequalities: A Graphical Approach**

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more values. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $3x > 102$ and $x > 34$ and determine which graph matches the inequality.

Understanding the Inequality

The given inequality is $3x > 102$ . To solve this inequality, we need to isolate the variable $x$ . We can do this by dividing both sides of the inequality by 3. This gives us $x > 34$ . The inequality $x > 34$ means that the value of $x$ must be greater than 34.

Graphical Representation

To represent the inequality $x > 34$ graphically, we can use a number line. A number line is a line that represents all the real numbers. We can mark the point 34 on the number line and shade the region to the right of 34. This represents all the values of $x$ that are greater than 34.

Types of Graphs

There are two types of graphs that can represent the inequality $x > 34$ : a solid line graph and a dashed line graph. A solid line graph represents the inequality $x = 34$ , while a dashed line graph represents the inequality $x > 34$ .

Solid Line Graph

A solid line graph represents the inequality $x = 34$ . This graph is a vertical line that passes through the point 34 on the number line. The line is solid, indicating that the value of $x$ is equal to 34.

Dashed Line Graph

A dashed line graph represents the inequality $x > 34$ . This graph is a vertical line that passes through the point 34 on the number line. The line is dashed, indicating that the value of $x$ is greater than 34.

Which Graph Matches the Inequality?

Based on the inequality $x > 34$ , we can conclude that the dashed line graph is the correct representation of the inequality. The dashed line graph represents all the values of $x$ that are greater than 34, which is the solution to the inequality.

Conclusion

In conclusion, solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we solved the inequality $3x > 102$ and $x > 34$ and determined which graph matches the inequality. The dashed line graph is the correct representation of the inequality $x > 34$ , as it represents all the values of $x$ that are greater than 34.

Example Problems

Here are some example problems that involve solving inequalities and determining which graph matches the inequality:

$2x > 50$
$x > 25$
$3x < 90$
$x < 30$

Tips and Tricks

Here are some tips and tricks for solving inequalities and determining which graph matches the inequality:

Always isolate the variable on one side of the inequality.
Use a number line to represent the inequality graphically.
Use a solid line graph to represent the inequality $x = a$ , where $a$ is a constant.
Use a dashed line graph to represent the inequality $x > a$ or $x < a$ , where $a$ is a constant.

Real-World Applications

Solving inequalities has many real-world applications. Here are a few examples:

In finance, inequalities are used to determine the minimum or maximum value of an investment.
In science, inequalities are used to determine the minimum or maximum value of a physical quantity.
In engineering, inequalities are used to determine the minimum or maximum value of a structural component.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities and determine which graph matches the inequality. In this article, we will provide a Q&A section to help you better understand the concept of solving inequalities and graphing.

Q1: What is an inequality?

A: An inequality is a statement that compares two or more values. It can be written in the form of $a > b$ , $a < b$ , $a \geq b$ , or $a \leq b$ , where $a$ and $b$ are real numbers.

Q2: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q3: What is a number line?

A: A number line is a line that represents all the real numbers. It is used to represent the solution to an inequality graphically.

Q4: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to mark the point that represents the value of the variable on the number line. Then, you need to shade the region to the right or left of the point, depending on the direction of the inequality.

Q5: What is a solid line graph?

A: A solid line graph represents the inequality $x = a$ , where $a$ is a constant. It is a vertical line that passes through the point $a$ on the number line.

Q6: What is a dashed line graph?

A: A dashed line graph represents the inequality $x > a$ or $x < a$ , where $a$ is a constant. It is a vertical line that passes through the point $a$ on the number line, but it is dashed to indicate that the value of $x$ is not equal to $a$ .

Q7: How do I determine which graph matches the inequality?

A: To determine which graph matches the inequality, you need to look at the direction of the inequality. If the inequality is $x > a$ , then the dashed line graph is the correct representation. If the inequality is $x = a$ , then the solid line graph is the correct representation.

Q8: Can I use a number line to represent an inequality with a variable in the denominator?

A: No, you cannot use a number line to represent an inequality with a variable in the denominator. This is because the number line only represents real numbers, and a variable in the denominator can result in a non-real value.

Q9: How do I solve an inequality with a variable in the denominator?

A: To solve an inequality with a variable in the denominator, you need to multiply or divide both sides of the inequality by the denominator. This will eliminate the variable in the denominator and allow you to solve the inequality.

Q10: Can I use a graph to represent an inequality with multiple variables?

A: No, you cannot use a graph to represent an inequality with multiple variables. This is because a graph only represents a single variable, and an inequality with multiple variables requires a more complex representation.

Conclusion

In conclusion, solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we provided a Q&A section to help you better understand the concept of solving inequalities and graphing. We hope this article has been helpful in answering your questions and providing a better understanding of the concept of solving inequalities.