To Fit In An Existing Frame, The Length, $x$, Of A Piece Of Glass Must Be Longer Than 12 Cm But Not Longer Than 12.2 Cm. Which Inequality Can Be Used To Represent The Lengths Of The Glass That Will Fit In The Frame?A. $12 \ \textless \ X

Feb 28, 2025 by ADMIN 240 views

**To Fit in an Existing Frame: Understanding Inequalities in Mathematics**

Introduction

In mathematics, inequalities are used to represent relationships between numbers or expressions. They are essential in solving problems that involve constraints or limitations. In this article, we will explore how to represent the lengths of a piece of glass that will fit in an existing frame using an inequality.

Understanding the Problem

The problem states that the length, $x$ , of a piece of glass must be longer than 12 cm but not longer than 12.2 cm. This means that the length of the glass must be within a specific range. To represent this range using an inequality, we need to understand the concept of inequalities and how to write them.

What is an Inequality?

An inequality is a statement that compares two expressions or numbers using a mathematical symbol, such as <, >, ≤, or ≥. Inequalities can be used to represent relationships between numbers or expressions, and they are essential in solving problems that involve constraints or limitations.

Types of Inequalities

There are four main types of inequalities:

Less than (<): This inequality is used to represent a relationship where one expression or number is less than another.
Greater than (>): This inequality is used to represent a relationship where one expression or number is greater than another.
Less than or equal to (≤): This inequality is used to represent a relationship where one expression or number is less than or equal to another.
Greater than or equal to (≥): This inequality is used to represent a relationship where one expression or number is greater than or equal to another.

Representing the Length of the Glass Using an Inequality

To represent the lengths of the glass that will fit in the frame, we need to use an inequality that reflects the given constraints. The length of the glass must be longer than 12 cm, but not longer than 12.2 cm. This can be represented using the following inequality:

12 < x ≤ 12.2

This inequality states that the length of the glass, $x$ , must be greater than 12 cm (12 < x) and less than or equal to 12.2 cm (x ≤ 12.2).

Solving the Inequality

To solve the inequality, we need to find the values of $x$ that satisfy the given constraints. In this case, the inequality is already in its simplest form, and we can see that the length of the glass must be between 12 cm and 12.2 cm.

Conclusion

In conclusion, the inequality that can be used to represent the lengths of the glass that will fit in the frame is:

12 < x ≤ 12.2

This inequality reflects the given constraints and provides a clear representation of the range of lengths that will fit in the frame. By understanding and using inequalities, we can solve problems that involve constraints or limitations and make informed decisions.

Example Problems

Here are some example problems that involve inequalities:

A bookshelf has a maximum height of 1.5 meters. If the height of the bookshelf is represented by $h$ , what is the inequality that represents the height of the bookshelf?
A car can travel a maximum distance of 500 kilometers per hour. If the distance traveled by the car is represented by $d$ , what is the inequality that represents the distance traveled by the car?
A container can hold a maximum volume of 2 liters. If the volume of the container is represented by $v$ , what is the inequality that represents the volume of the container?

Answer Key

Here are the answers to the example problems:

The inequality that represents the height of the bookshelf is: 0 ≤ h ≤ 1.5
The inequality that represents the distance traveled by the car is: 0 ≤ d ≤ 500
The inequality that represents the volume of the container is: 0 ≤ v ≤ 2

Final Thoughts

In conclusion, inequalities are an essential concept in mathematics that can be used to represent relationships between numbers or expressions. By understanding and using inequalities, we can solve problems that involve constraints or limitations and make informed decisions. The inequality that can be used to represent the lengths of the glass that will fit in the frame is:

12 < x ≤ 12.2

Introduction

In our previous article, we explored how to represent the lengths of a piece of glass that will fit in an existing frame using an inequality. In this article, we will answer some frequently asked questions about inequalities and provide additional examples to help you understand this concept better.

Q&A

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

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that compares two expressions using a mathematical symbol, such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. 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.

Q: What is the order of operations for inequalities?

A: The order of operations for inequalities is the same as for equations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Q: Can I use the same methods to solve linear inequalities as I would to solve linear equations?

A: Yes, you can use the same methods to solve linear inequalities as you would to solve linear equations. However, you need to be careful when multiplying or dividing both sides of the inequality by a negative value, as this can change the direction of the inequality.

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

A: To graph an inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality. If the inequality is of the form x < a, you would plot a point to the left of a. If the inequality is of the form x > a, you would plot a point to the right of a.

Q: Can I use inequalities to solve problems that involve multiple variables?

A: Yes, you can use inequalities to solve problems that involve multiple variables. However, you need to be careful when solving inequalities with multiple variables, as the solution may be a range of values rather than a single value.

Example Problems

Here are some example problems that involve inequalities:

A bookshelf has a maximum height of 1.5 meters. If the height of the bookshelf is represented by $h$ , what is the inequality that represents the height of the bookshelf?
A car can travel a maximum distance of 500 kilometers per hour. If the distance traveled by the car is represented by $d$ , what is the inequality that represents the distance traveled by the car?
A container can hold a maximum volume of 2 liters. If the volume of the container is represented by $v$ , what is the inequality that represents the volume of the container?

Answer Key

Here are the answers to the example problems:

The inequality that represents the height of the bookshelf is: 0 ≤ h ≤ 1.5
The inequality that represents the distance traveled by the car is: 0 ≤ d ≤ 500
The inequality that represents the volume of the container is: 0 ≤ v ≤ 2

Additional Examples

Here are some additional examples of inequalities:

A company has a maximum profit of $100,000. If the profit is represented by $p$ , what is the inequality that represents the profit?
A student has a minimum grade of 70%. If the grade is represented by $g$ , what is the inequality that represents the grade?
A car has a maximum speed of 120 kilometers per hour. If the speed is represented by $s$ , what is the inequality that represents the speed?

Answer Key

Here are the answers to the additional examples:

The inequality that represents the profit is: 0 ≤ p ≤ 100,000
The inequality that represents the grade is: 0.7 ≤ g ≤ 1
The inequality that represents the speed is: 0 ≤ s ≤ 120

Conclusion

In conclusion, inequalities are an essential concept in mathematics that can be used to represent relationships between numbers or expressions. By understanding and using inequalities, we can solve problems that involve constraints or limitations and make informed decisions. We hope that this article has helped you understand inequalities better and provided you with the tools you need to solve problems that involve inequalities.