For One Photography Session, Dexter Earns No Less Than $ 50 \$50 $50 , But No More Than $ 100 \$100 $100 . Which Inequality Can Be Used To Represent His Earnings, E E E ?A. E ≥ 50 E \geq 50 E ≥ 50 Or E ≤ 100 E \leq 100 E ≤ 100 B. $e \ \textgreater \

Understanding the Problem

Dexter's photography session earnings range from a minimum of $\$50$ to a maximum of $\$100$. We need to represent this range using an inequality to express the possible values of his earnings, denoted as $e$.

Inequality Representation

To represent the range of Dexter's earnings, we can use a compound inequality. A compound inequality is a combination of two or more inequalities joined by the words "and" or "or." In this case, we want to express that Dexter's earnings are either greater than or equal to $\$50$ or less than or equal to $\$100$.

Option A: $e \geq 50$ or $e \leq 100$

This option represents the range of Dexter's earnings using two separate inequalities joined by the word "or." The first inequality, $e \geq 50$, indicates that Dexter's earnings are greater than or equal to $\$50$. The second inequality, $e \leq 100$, indicates that Dexter's earnings are less than or equal to $\$100$.

Option B: $e \ \textgreater \ 50$ or $e \ \textless \ 100$

This option represents the range of Dexter's earnings using two separate inequalities joined by the word "or." The first inequality, $e \ \textgreater \ 50$, indicates that Dexter's earnings are greater than $\$50$. The second inequality, $e \ \textless \ 100$, indicates that Dexter's earnings are less than $\$100$.

Correct Representation

The correct representation of Dexter's earnings is Option A: $e \geq 50$ or $e \leq 100$. This option accurately represents the range of Dexter's earnings, which is from a minimum of $\$50$ to a maximum of $\$100$.

Why Option A is Correct

Option A is correct because it includes the minimum and maximum values of Dexter's earnings. The inequality $e \geq 50$ ensures that Dexter's earnings are at least $\$50$, while the inequality $e \leq 100$ ensures that his earnings are no more than $\$100$. This combination of inequalities accurately represents the range of Dexter's earnings.

Why Option B is Incorrect

Option B is incorrect because it does not include the minimum and maximum values of Dexter's earnings. The inequality $e \ \textgreater \ 50$ only indicates that Dexter's earnings are greater than $\$50$, but it does not account for the possibility that his earnings may be exactly $\$50$. Similarly, the inequality $e \ \textless \ 100$ only indicates that Dexter's earnings are less than $\$100$, but it does not account for the possibility that his earnings may be exactly $\$100$. Therefore, Option B does not accurately represent the range of Dexter's earnings.

Conclusion

Q: What is the minimum and maximum value of Dexter's earnings?

A: The minimum value of Dexter's earnings is $\$50$, and the maximum value is $\$100$.

Q: How can we represent the range of Dexter's earnings using an inequality?

A: We can use a compound inequality to represent the range of Dexter's earnings. A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."

Q: What is the correct representation of Dexter's earnings using an inequality?

A: The correct representation of Dexter's earnings is Option A: $e \geq 50$ or $e \leq 100$. This option accurately represents the range of Dexter's earnings, which is from a minimum of $\$50$ to a maximum of $\$100$.

Q: Why is Option A the correct representation of Dexter's earnings?

A: Option A is correct because it includes the minimum and maximum values of Dexter's earnings. The inequality $e \geq 50$ ensures that Dexter's earnings are at least $\$50$, while the inequality $e \leq 100$ ensures that his earnings are no more than $\$100$. This combination of inequalities accurately represents the range of Dexter's earnings.

Q: Why is Option B an incorrect representation of Dexter's earnings?

A: Option B is incorrect because it does not include the minimum and maximum values of Dexter's earnings. The inequality $e \ \textgreater \ 50$ only indicates that Dexter's earnings are greater than $\$50$, but it does not account for the possibility that his earnings may be exactly $\$50$. Similarly, the inequality $e \ \textless \ 100$ only indicates that Dexter's earnings are less than $\$100$, but it does not account for the possibility that his earnings may be exactly $\$100$. Therefore, Option B does not accurately represent the range of Dexter's earnings.

Q: How can we use inequalities to represent other real-world scenarios?

A: Inequalities can be used to represent a wide range of real-world scenarios, such as:

• Representing the range of scores on a test
• Representing the range of temperatures in a given region
• Representing the range of prices for a product
• Representing the range of values for a variable in a mathematical model

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: These are inequalities that involve a linear expression, such as $e \geq 50$.
• Quadratic inequalities: These are inequalities that involve a quadratic expression, such as $e^2 \geq 100$.
• Rational inequalities: These are inequalities that involve a rational expression, such as $\frac{e}{2} \geq 50$.

Q: How can we solve inequalities?

A: Inequalities can be solved using a variety of methods, including:

• Adding or subtracting the same value to both sides of the inequality
• Multiplying or dividing both sides of the inequality by the same value
• Using inverse operations to isolate the variable

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

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

• Not following the order of operations
• Not checking the direction of the inequality
• Not considering the possibility of multiple solutions

Conclusion

In conclusion, representing Dexter's earnings with inequalities is a useful tool for understanding and working with real-world scenarios. By using compound inequalities, we can accurately represent the range of values for a variable, and by solving inequalities, we can find the specific values that satisfy a given inequality.