Inequality For Dexter's Earnings: $50 To $100

Hey guys! Let's break down this math problem about Dexter's photography earnings. We need to figure out which inequality best describes how much he makes in a single session. It's a pretty straightforward question once we understand what each part means.

Understanding the Problem

The problem states that Dexter earns no less than $50 and no more than $100 for each photography session. This means his earnings, which we're representing with the variable 'e', can be $50, can be $100, or can be any amount in between. Keywords here are no less than and no more than. "No less than" implies that $50 is the minimum amount, and "no more than" means that $100 is the maximum amount.

Decoding the Options

Let's look at the inequality options provided and see which one accurately represents this situation:

• Option A: $e \[ge] 50$ or $e \[le] 100$

  This option uses the word "or," which means either condition can be true independently. So, Dexter's earnings could be greater than or equal to $50 or less than or equal to $100. While it's technically true that his earnings fall within these ranges, the "or" makes it too broad. It doesn't capture the fact that his earnings are limited to this range. For example, if e = $200, it does not fulfill the condition.

• Option B: $e > 50$ or $e < 100$

  Similar to Option A, this one also uses "or." This option says Dexter's earnings are either greater than $50 or less than $100. This is even less accurate than option A because it doesn't include the possibility of Dexter earning exactly $50 or $100. If e = $10, it does not fulfill the condition.

• Option C: $50 < e < 100$

  This option looks promising! It states that Dexter's earnings are greater than $50 and less than $100. The implied "and" here is important. However, it's not quite right. This inequality says Dexter's earning is strictly greater than $50 and strictly less than $100, meaning he cannot earn exactly $50 or $100. if e = $50, it does not fulfill the condition.

• Option D: $50 \[le] e \[le] 100$

  This is the winner! This inequality correctly states that Dexter's earnings are greater than or equal to $50 and less than or equal to $100. It includes both the minimum and maximum values, making it the most accurate representation of the problem statement.

Why Option D is the Correct Answer

Option D, $50 \[le] e \[le] 100$, uses the "less than or equal to" ($\[le]$) symbol. This is crucial because it includes the endpoints ($50 and $100) in the possible range of Dexter's earnings. The inequality is read as "50 is less than or equal to e, and e is less than or equal to 100." This perfectly matches the problem's condition that Dexter earns no less than $50 and no more than $100.

Think of it like this: the inequality is a fence, and Dexter's earnings have to stay within that fence. The fence posts are at $50 and $100, and Dexter's earnings can be right on the posts or anywhere in between. Other options don't let Dexter stand right on those posts!

Real-World Application

Understanding inequalities is super useful, not just in math class! Imagine you're planning a budget for a project. You know you need to spend at least $500 but no more than $1000. You can use the inequality $500 \[le] spending \[le] 1000$ to represent your budget range. This helps you stay on track and avoid overspending.

Inequalities are also used in setting speed limits (you can go up to a certain speed), determining age restrictions (you must be at least a certain age to drive), and many other real-life scenarios.

Final Answer

So, the correct answer is Option D: $50 \[le] e \[le] 100$. This inequality accurately represents Dexter's earnings for a photography session, where he makes no less than $50 and no more than $100. Remember to pay close attention to the keywords in the problem and choose the inequality that best reflects those conditions.

Additional Tips for Solving Inequality Problems

• Read Carefully: Always read the problem statement very carefully. Pay attention to keywords like "at least," "no more than," "less than," and "greater than." These words are clues that tell you which inequality symbols to use.
• Define the Variable: Clearly define what the variable represents. In this case, 'e' represents Dexter's earnings. Knowing what the variable stands for will help you write the inequality correctly.
• Test the Inequality: Once you've chosen an inequality, test it with values within the given range. For example, if you think $50 \[le] e \[le] 100$ is correct, try plugging in $e = 60$. Does it make sense? If you plug in $e = 40$, does it violate the condition?
• Visualize: Sometimes it helps to visualize the inequality on a number line. This can make it easier to see which values are included in the solution.
• Practice: Like any math skill, practice makes perfect! The more you work with inequalities, the better you'll become at understanding and solving them.

Common Mistakes to Avoid

• Using the Wrong Symbol: One of the most common mistakes is using the wrong inequality symbol. Remember that "less than" ($<$) and "greater than" ($>$) do not include the endpoint, while "less than or equal to" ($\[le]$) and "greater than or equal to" ($\[ge]$) do.
• Mixing Up "And" and "Or": Be careful when using "and" and "or" in inequalities. "And" means that both conditions must be true, while "or" means that at least one condition must be true. This can significantly change the meaning of the inequality.
• Forgetting to Define the Variable: Always define the variable clearly. This will help you avoid confusion and ensure that you're writing the inequality correctly.
• Not Testing the Inequality: Before submitting your answer, take a few seconds to test the inequality with values within the given range. This can help you catch any mistakes you may have made.

By keeping these tips and common mistakes in mind, you'll be well on your way to mastering inequalities!

Conclusion

So, there you have it! Choosing the right inequality to represent a real-world scenario like Dexter's photography earnings involves understanding the problem's conditions and accurately translating them into mathematical symbols. Remember to focus on the keywords, define your variables, and test your answer. Keep practicing, and you'll become an inequality pro in no time!