Find Variable 'a': Expressing Numbers Between 3 and 7 on Number Line

Q: What does 3 < a < 7 actually mean in words?

It means 'a is greater than 3 AND less than 7'. The variable a must satisfy both conditions at the same time. So a could be 4, 5, 6, or any decimal between 3 and 7.

Q: Why can't a equal 3 or 7?

Because we use the , not ≤. The not included. If 3 and 7 were included, we'd write 3 ≤ a ≤ 7.

Q: How do I know which answer choice is correct?

Look for the expression that shows both conditions: a > 3 AND a \( 3 . Avoid single equalities like a = 7 or impossible ranges like 7

Q: What would the number line look like for this inequality?

Draw open circles at 3 and 7 (since they're not included), then shade the region between them. This shows all possible values of a that make both 3

Compound Inequalities with Number Line Representation

Select a compatible expression to show that a is greater than 3 but smaller than 7:

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Understand the problem

Select a compatible expression to show that a is greater than 3 but smaller than 7:

Step-by-step solution

First, we'll mark on the axis the expression $a > 3$

Then we'll mark on the axis the expression $a < 7$

We'll mark in orange the segment representing the requested expression:

$3 < a < 7$

Final Answer

$3 < a < 7$

Key Points to Remember

Essential concepts to master this topic

Compound Inequalities: Two conditions connected by 'and' must both be true
Technique: 3 < a < 7 means a > 3 AND a < 7
Check: Test values like a = 5: 3 < 5 < 7 is true ✓

Common Mistakes

Avoid these frequent errors

Reading inequality symbols incorrectly
Don't confuse 3 < a < 7 with 7 < a < 3 = impossible range! The first number must be smaller than the second for a valid interval. Always read left to right: 'a is greater than 3 AND less than 7'.

Practice Quiz

Test your knowledge with interactive questions

All negative numbers appear on the number line to the left of the number 0.

FAQ

Everything you need to know about this question

What does 3 < a < 7 actually mean in words?

It means 'a is greater than 3 AND less than 7'. The variable a must satisfy both conditions at the same time. So a could be 4, 5, 6, or any decimal between 3 and 7.

Why can't a equal 3 or 7?

Because we use the < symbol, not ≤. The < symbol means 'strictly less than' or 'strictly greater than', so the endpoints are not included. If 3 and 7 were included, we'd write 3 ≤ a ≤ 7.

How do I know which answer choice is correct?

Look for the expression that shows both conditions: a > 3 AND a < 7. This is written as $3 < a < 7$ . Avoid single equalities like a = 7 or impossible ranges like 7 < a < 3.

What would the number line look like for this inequality?

Draw open circles at 3 and 7 (since they're not included), then shade the region between them. This shows all possible values of a that make both 3 < a and a < 7 true.

Can I give a specific example of a value that works?

Yes! Try a = 5: Is 3 < 5? Yes. Is 5 < 7? Yes. Since both conditions are true, a = 5 is in our solution set. Values like 2.5 or 8 would fail one of the conditions.

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