Finding Distance Between Points A and K on a Number Line with Integer Scale

Distance Formula with Negative Coordinates

What is the distance between A and K?

AAA-5-5-5BBB-4-4-4CCC-3-3-3DDD-2-2-2EEE-1-1-1FFF000GGG111HHH222III333JJJ444KKK555

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1

Understand the problem

What is the distance between A and K?

AAA-5-5-5BBB-4-4-4CCC-3-3-3DDD-2-2-2EEE-1-1-1FFF000GGG111HHH222III333JJJ444KKK555

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Step-by-step solution

One might think that because there are numbers on the axis that go into the negative domain, that the result must also negative.

However it is important to keep in mind that here we are asking about distance.

Distance can never be negative.

Even if we move towards or from the domain of negativity, distance is an existing value (absolute value).

We can think of it as if we were counting the number of steps, and it doesn't matter if we start from five or minus five, both are 5 steps away from zero.

3

Final Answer

10

Key Points to Remember

Essential concepts to master this topic
  • Distance Rule: Distance is always positive, never negative
  • Calculation: Count steps from A(-5) to K(5): |5 - (-5)| = 10
  • Verification: Check by counting individual units on the number line ✓

Common Mistakes

Avoid these frequent errors
  • Thinking distance can be negative
    Don't subtract A's position from K's position to get -5 - 5 = -10! Distance measures how far apart points are, which is always positive. Always use absolute value: |5 - (-5)| = 10.

Practice Quiz

Test your knowledge with interactive questions

\( 5 < -5 \)

AAAKKK-5-5-5BBB-4-4-4CCC-3-3-3DDD-2-2-2EEE-1-1-1FFF000GGG111HHH222III333JJJ444555

FAQ

Everything you need to know about this question

Why isn't the answer -10 since we start at -5?

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Distance is always positive! Think of it like counting steps - whether you walk from your house to school or from school to your house, the distance is the same positive number.

How do I calculate distance between any two points on a number line?

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Use the formula: Distance = |point2 - point1|. The absolute value bars ensure your answer is always positive, no matter which point you subtract from which.

What if both points are negative?

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The same rule applies! For example, distance from -8 to -3 is |-3 - (-8)| = |5| = 5. The absolute value always gives you the positive distance.

Can I just count the spaces on the number line instead?

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Absolutely! Counting units between points is a great way to check your work. From A(-5) to K(5), count: -5 to 0 is 5 steps, then 0 to 5 is 5 more steps, total = 10.

Why do we use absolute value in the distance formula?

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Absolute value removes the negative sign and gives us the actual distance. Distance is a measurement of space, not direction, so it's always positive!

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