Number Line Problem: Finding Variable 'a' Between 0 and 5

Q: What does 'a is a positive number less than 5' mean exactly?

This means two things at once: $ a > 0 $ (positive) AND \( a (less than 5). The variable must satisfy both conditions simultaneously.

Q: Why is 0 0, a < 5?

Compound inequality notation \( 0 shows the relationship clearly in one expression. It means 'a is between 0 and 5' which is more precise than separate conditions.

Q: Does the solution include the endpoints 0 and 5?

No! The inequality uses strict inequality symbols (), not ≤ or ≥. So a cannot equal 0 or 5, only values strictly between them like 2.5 or 4.9.

Q: What if I picked answer choice with a = 1?

That's too specific! While a = 1 satisfies the conditions, the variable a can be any value between 0 and 5, like 2.7 or 4.999. Don't limit yourself to just one number.

Interval Notation with Number Line Visualization

$a$ is a positive number less than 5.

Select the correct expression regarding the value of $a$ on the number line:

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Understand the problem

$a$ is a positive number less than 5.

Select the correct expression regarding the value of $a$ on the number line:

Step-by-step solution

First, let's mark the expression $a < 5$ on the axis:

Now let's mark the expression $a > 0$ on the axis:

Then we can mark in orange the overlapping segment that represents the expression in question:

$0 < a < 5$

Final Answer

$0 < a < 5$

Key Points to Remember

Essential concepts to master this topic

Compound Inequality: Combine two conditions using logical AND to find overlap
Visualization: Mark $a > 0$ and $a < 5$ separately on number line
Check: Verify solution $0 < a < 5$ includes all valid values ✓

Common Mistakes

Avoid these frequent errors

Writing separate conditions instead of combined inequality
Don't write $a > 0$ , $a < 5$ as two separate statements = incomplete answer! This misses that both conditions must be true simultaneously. Always combine using $0 < a < 5$ to show the intersection.

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 'a is a positive number less than 5' mean exactly?

This means two things at once: $a > 0$ (positive) AND $a < 5$ (less than 5). The variable must satisfy both conditions simultaneously.

Why is 0 < a < 5 better than writing a > 0, a < 5?

Compound inequality notation $0 < a < 5$ shows the relationship clearly in one expression. It means 'a is between 0 and 5' which is more precise than separate conditions.

Does the solution include the endpoints 0 and 5?

No! The inequality uses strict inequality symbols (< and >), not ≤ or ≥. So a cannot equal 0 or 5, only values strictly between them like 2.5 or 4.9.

How do I visualize this on a number line?

Draw two separate rays: one starting from 0 going right (for $a > 0$ ) and one ending at 5 going left (for $a < 5$ ). The overlapping region between 0 and 5 is your answer!

What if I picked answer choice with a = 1?

That's too specific! While a = 1 satisfies the conditions, the variable a can be any value between 0 and 5, like 2.7 or 4.999. Don't limit yourself to just one number.

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