Which Scenario Can Be Represented Using The Inequalities Below?$\[1.25 \leq X \leq 1.5\\]A. A Container Of Milk Costs At Least \[$\$1.25\$\] But Less Than \[$\$1.50\$\].B. A Student Spends At Least 1 Hour 15 Minutes, But No

Feb 27, 2025 by ADMIN 224 views

Which Scenario Can Be Represented Using the Inequalities Below?

Understanding the Inequalities

In mathematics, inequalities are used to represent a range of values for a variable. The given inequalities are $1.25 \leq x \leq 1.5$ . These inequalities indicate that the value of $x$ must be greater than or equal to $1.25$ and less than or equal to $1.5$ . This range of values is often represented on a number line, with the lower bound at $1.25$ and the upper bound at $1.5$ .

Scenario A: Container of Milk Costs

Let's analyze the first scenario: A container of milk costs at least $\$1.25$ but less than $\$1.50$ . This scenario can be represented using the given inequalities. The cost of the container of milk is the variable $x$ , and the inequalities indicate that the cost must be within the range of $\$1.25$ to $\$1.50$ . This means that the cost of the container of milk can be any value between $\$1.25$ and $\$1.50$ , including $\$1.25$ and $\$1.50$ .

Scenario B: Student Spends Time

The second scenario is: A student spends at least 1 hour 15 minutes, but no more than 1 hour 30 minutes. This scenario cannot be represented using the given inequalities. The inequalities $1.25 \leq x \leq 1.5$ represent a range of values for the variable $x$ , but the values are in dollars, not minutes. To represent the time spent by the student, we would need inequalities with values in minutes, such as $75 \leq x \leq 90$ .

Key Differences

The key difference between the two scenarios is the unit of measurement. The first scenario uses dollars, while the second scenario uses minutes. This difference in unit of measurement makes the second scenario incompatible with the given inequalities.

Conclusion

Additional Considerations

When working with inequalities, it's essential to consider the unit of measurement and the context in which the inequalities are being used. In this case, the inequalities $1.25 \leq x \leq 1.5$ are specific to a range of values in dollars, and cannot be used to represent a range of values in minutes.

Real-World Applications

Inequalities are used in a wide range of real-world applications, including finance, economics, and science. In finance, inequalities are used to represent the range of values for investments, while in economics, they are used to represent the range of values for prices. In science, inequalities are used to represent the range of values for physical quantities, such as temperature or pressure.

Common Mistakes

When working with inequalities, it's essential to avoid common mistakes, such as:

Using the wrong unit of measurement
Failing to consider the context in which the inequalities are being used
Not properly representing the range of values

Best Practices

To avoid common mistakes and ensure that inequalities are used correctly, follow these best practices:

Clearly define the unit of measurement
Consider the context in which the inequalities are being used
Properly represent the range of values

Conclusion

In conclusion, the scenario that can be represented using the inequalities $1.25 \leq x \leq 1.5$ is the first scenario: A container of milk costs at least $\$1.25$ but less than $\$1.50$ . This scenario is compatible with the given inequalities, while the second scenario is not. By following best practices and avoiding common mistakes, we can ensure that inequalities are used correctly and effectively in a wide range of real-world applications.

References

[1] "Inequalities" by Math Open Reference
[2] "Inequalities" by Khan Academy
[3] "Inequalities" by Wolfram MathWorld

Further Reading

For further reading on inequalities, see:

"Inequalities" by Math Open Reference
"Inequalities" by Khan Academy
"Inequalities" by Wolfram MathWorld

Additional Resources

For additional resources on inequalities, see:

[1] "Inequalities" by Mathway
[2] "Inequalities" by Symbolab
[3] "Inequalities" by Wolfram Alpha
Frequently Asked Questions (FAQs) About Inequalities

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two values or expressions, indicating that one is greater than, less than, or equal to the other. Inequalities are used to represent a range of values for a variable.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and nonlinear inequalities. Linear inequalities are inequalities that can be written in the form $ax + b \leq c$ or $ax + b \geq c$ , where $a$ , $b$ , and $c$ are constants. Nonlinear inequalities are inequalities that cannot be written in this form.

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. This can be done 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 difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $x < 2$ or $x > 3$ . A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as $x \leq 2$ or $x \geq 3$ .

Q: Can I use the same methods to solve strict and non-strict inequalities?

A: Yes, you can use the same methods to solve both strict and non-strict inequalities. However, when solving a strict inequality, you need to be careful not to include the value that makes the inequality false.

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 value of the variable. Then, you need to shade the region on the number line that represents the solution to the inequality.

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, you need to be careful to enter the correct values and to use the correct operations.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

Not isolating the variable on one side of the inequality sign
Not considering the direction of the inequality sign
Not checking for extraneous solutions
Not using the correct operations

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original inequality and check if it is true. If it is not true, then the solution is extraneous.

Q: Can I use inequalities to solve real-world problems?

A: Yes, you can use inequalities to solve real-world problems. Inequalities are used in a wide range of applications, including finance, economics, and science.

Q: What are some examples of real-world problems that can be solved using inequalities?

A: Some examples of real-world problems that can be solved using inequalities include:

Finding the range of values for a product or a quotient
Determining the maximum or minimum value of a function
Solving systems of linear inequalities
Modeling real-world situations using inequalities

Q: How do I determine if an inequality is linear or nonlinear?

A: To determine if an inequality is linear or nonlinear, you need to look at the inequality sign and the coefficients of the variable. If the inequality sign is a linear inequality sign (such as $<$ or $>$ ) and the coefficients of the variable are constants, then the inequality is linear. If the inequality sign is a nonlinear inequality sign (such as $\leq$ or $\geq$ ) or the coefficients of the variable are not constants, then the inequality is nonlinear.

Q: Can I use inequalities to solve systems of equations?

A: Yes, you can use inequalities to solve systems of equations. Inequalities can be used to represent the constraints of a system of equations, and to find the solution to the system.

Q: What are some common applications of inequalities in real-world problems?

A: Some common applications of inequalities in real-world problems include:

Finance: Inequalities are used to represent the range of values for investments, and to determine the maximum or minimum value of a portfolio.
Economics: Inequalities are used to represent the range of values for prices, and to determine the maximum or minimum value of a market.
Science: Inequalities are used to represent the range of values for physical quantities, such as temperature or pressure.

Q: How do I use inequalities to solve optimization problems?

A: To use inequalities to solve optimization problems, you need to identify the objective function and the constraints of the problem. Then, you need to use inequalities to represent the constraints, and to find the maximum or minimum value of the objective function.

Q: What are some common mistakes to avoid when using inequalities to solve optimization problems?

A: Some common mistakes to avoid when using inequalities to solve optimization problems include:

Not identifying the objective function and the constraints of the problem
Not using the correct inequalities to represent the constraints
Not checking for extraneous solutions
Not using the correct operations

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

A: Yes, you can use inequalities to solve problems with multiple variables. Inequalities can be used to represent the constraints of a problem with multiple variables, and to find the solution to the problem.

Q: What are some common applications of inequalities in problems with multiple variables?

A: Some common applications of inequalities in problems with multiple variables include:

Finance: Inequalities are used to represent the range of values for investments, and to determine the maximum or minimum value of a portfolio.
Economics: Inequalities are used to represent the range of values for prices, and to determine the maximum or minimum value of a market.
Science: Inequalities are used to represent the range of values for physical quantities, such as temperature or pressure.

Q: How do I use inequalities to solve problems with multiple constraints?

A: To use inequalities to solve problems with multiple constraints, you need to identify the objective function and the constraints of the problem. Then, you need to use inequalities to represent the constraints, and to find the maximum or minimum value of the objective function.

Q: What are some common mistakes to avoid when using inequalities to solve problems with multiple constraints?

A: Some common mistakes to avoid when using inequalities to solve problems with multiple constraints include:

Not identifying the objective function and the constraints of the problem
Not using the correct inequalities to represent the constraints
Not checking for extraneous solutions
Not using the correct operations