Solve the Inequality Puzzle: Finding Missing Digits in ?2?0 Between 7,590 and 9,001

Q: Why can't the thousands digit be 7?

If the thousands digit is 7, you get $ 72?0 $. Even with the largest possible digits (7,290), this is still less than 7,590, violating the lower bound requirement.

Q: What if I try 9 as the thousands digit?

With thousands digit 9, you get $ 92?0 $. The smallest possible number is 9,200, which is greater than 9,001, violating the upper bound.

Q: How do I know 8,280 is the only answer?

Once you determine the thousands digit must be 8, the pattern $ ?2?0 $ becomes $ 82?0 $. The given format shows the answer as two digits: 8,8, representing the thousands and hundreds places.

Q: Can there be multiple correct answers?

No! The specific format $ ?2?0 $ with the middle digit fixed as 2 and last digit as 0 creates a unique solution. Only one pair of digits satisfies the inequality.

Q: What does the answer format 8,8 mean?

The answer 8,8 represents the missing digits in order: first 8 goes in the thousands place, second 8 goes in the hundreds place, giving us $ 8,280 $.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Fill in the missing digits ensuring that all numbers lie in the correct range of values.

$7,590 < ?2?0 < 9,001$

2

Step-by-step solution

To solve this problem, let's follow these steps:

Step 1: Examine the tens digit.
Step 2: Determine the thousands digit first.
Step 3: Confirm that the number fits within the specified range.

Now, let's work through each step:

Step 1: Examine the tens-digit placeholder in $?2?0$ . This number must be greater than 7,590. Therefore, the most significant digit, thousands' place, must be 8 or 9, because if it were 7, the last digit would need to be at least 6 (since 7,620 > 7,590, but 7,520 is not).

Step 2: Check thousands place digit choice:
- If the thousands digit is 9, the smallest number possible is 9,020, which is greater than 9,001 which does not fit.
- Thus, the thousands digit must be 8.

Step 3: Complete the number.
Given that the thousands digit is 8, verify:

With the thousands digit as 8, $82?0$ is possible, align with the given range ensuring the complete number satisfies $7,590 < 8,2?0 < 9,001$ .

Continue by recognizing only matching pairs from $8800$ to $8999$ apply to meet $7,590 < 8,2?0 < 9,001$ .

Hence, the complete number is $8,280$ .

Therefore, the solution to this problem is to fill the blanks with $8,8$ .

3

Final Answer

$8,8$

Key Points to Remember

Essential concepts to master this topic

Range Analysis: Check thousands digit first to narrow possibilities quickly
Technique: If thousands = 9, then 9,2?0 > 9,001 fails upper bound
Check: Verify 7,590 < 8,280 < 9,001 satisfies both bounds ✓

Common Mistakes

Avoid these frequent errors

Starting with the units or tens digit first
Don't fill in the smaller place values before determining the thousands digit = wrong range entirely! This leads to numbers outside the bounds. Always analyze the most significant digit first to establish the correct range.

Practice Quiz

Test your knowledge with interactive questions

Identify the missing numbers for the given range:

$$ 2,301 < ?6?2 < 4,503 $$

FAQ

Everything you need to know about this question

Why can't the thousands digit be 7?

+

If the thousands digit is 7, you get $72?0$ . Even with the largest possible digits (7,290), this is still less than 7,590, violating the lower bound requirement.

What if I try 9 as the thousands digit?

+

With thousands digit 9, you get $92?0$ . The smallest possible number is 9,200, which is greater than 9,001, violating the upper bound.

How do I know 8,280 is the only answer?

+

Once you determine the thousands digit must be 8, the pattern $?2?0$ becomes $82?0$ . The given format shows the answer as two digits: 8,8, representing the thousands and hundreds places.

Can there be multiple correct answers?

+

No! The specific format $?2?0$ with the middle digit fixed as 2 and last digit as 0 creates a unique solution. Only one pair of digits satisfies the inequality.

What does the answer format 8,8 mean?

+

The answer 8,8 represents the missing digits in order: first 8 goes in the thousands place, second 8 goes in the hundreds place, giving us $8,280$ .

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Solve the Inequality Puzzle: Finding Missing Digits in ?2?0 Between 7,590 and 9,001

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