Which Number Line Represents The Solution To $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$?

Mar 8, 2025 by ADMIN 88 views

$Which Number Line Represents the Solution to $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$?$

Introduction to Inequalities

Inequalities are mathematical expressions that compare two values using greater than, less than, or equal to symbols. They are used to represent relationships between variables and constants, and are a fundamental concept in mathematics. In this article, we will explore the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ and determine which number line represents the solution.

Understanding the Inequality

The given inequality is $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$. To solve this inequality, we need to isolate the variable x. We can start by adding 1.2x to both sides of the inequality, which gives us:

2.5 \ \textless \ 6.5 - 2x

Next, we can add 2x to both sides of the inequality, which gives us:

2.5 + 2x \ \textless \ 6.5

Simplifying the Inequality

Now, we can simplify the inequality by subtracting 2.5 from both sides, which gives us:

2x \ \textless \ 4

Solving for x

To solve for x, we can divide both sides of the inequality by 2, which gives us:

x \ \textless \ 2

Understanding the Solution

The solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ is x < 2. This means that any value of x that is less than 2 will satisfy the inequality.

Number Line Representation

To represent the solution on a number line, we need to draw a line that includes all values of x that are less than 2. This can be represented by a line that starts at negative infinity and ends at 2, with an open circle at 2 to indicate that 2 is not included in the solution.

Conclusion

Frequently Asked Questions

What is the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$?
How do I represent the solution on a number line?
What is the meaning of the open circle at 2 on the number line?

Answer to Frequently Asked Questions

The solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ is x < 2.
To represent the solution on a number line, draw a line that starts at negative infinity and ends at 2, with an open circle at 2 to indicate that 2 is not included in the solution.
The open circle at 2 on the number line indicates that 2 is not included in the solution.

Step-by-Step Solution

Start with the given inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$.
Add 1.2x to both sides of the inequality to get $2.5 \ \textless \ 6.5 - 2x$.
Add 2x to both sides of the inequality to get $2.5 + 2x \ \textless \ 6.5$.
Subtract 2.5 from both sides of the inequality to get $2x \ \textless \ 4$.
Divide both sides of the inequality by 2 to get $x \ \textless \ 2$.

Final Answer

The final answer is x < 2.

Introduction

In our previous article, we explored the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$. We determined that the solution is x < 2 and represented it on a number line. In this article, we will answer some frequently asked questions about the solution to this inequality.

Q&A

Q: What is the solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$?

A: The solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ is x < 2.

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

A: To represent the solution on a number line, draw a line that starts at negative infinity and ends at 2, with an open circle at 2 to indicate that 2 is not included in the solution.

Q: What is the meaning of the open circle at 2 on the number line?

A: The open circle at 2 on the number line indicates that 2 is not included in the solution.

Q: Can I include 2 in the solution?

A: No, 2 is not included in the solution. The solution is x < 2, which means that any value of x that is less than 2 will satisfy the inequality.

Q: How do I determine if a value of x is included in the solution?

A: To determine if a value of x is included in the solution, you can plug the value into the inequality and see if it is true. If the value satisfies the inequality, then it is included in the solution.

Q: Can I use the solution to solve other inequalities?

A: Yes, you can use the solution to solve other inequalities. The solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ is x < 2, which means that any value of x that is less than 2 will satisfy the inequality.

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

A: To graph the solution on a number line, draw a line that starts at negative infinity and ends at 2, with an open circle at 2 to indicate that 2 is not included in the solution.

Q: Can I use the solution to solve systems of inequalities?

A: Yes, you can use the solution to solve systems of inequalities. The solution to the inequality $2.5 - 1.2x \ \textless \ 6.5 - 3.2x$ is x < 2, which means that any value of x that is less than 2 will satisfy the inequality.