Comparing Absolute Values: Are -|-801| and |+801| Opposite Numbers?

Q: Why does the absolute value of -801 equal positive 801?

Absolute value measures distance from zero on the number line. Since -801 is 801 units away from zero, $ \left|-801\right| = 801 $. Distance is always positive!

Q: What's the difference between |-801| and -|-801|?

Great question! $ \left|-801\right| = 801 $ (positive), but $ -\left|-801\right| = -801 $ (negative). The negative sign outside the absolute value bars makes the final result negative.

Q: How do I know if two numbers are opposites?

Two numbers are opposites if they have the same magnitude but different signs. For example, -801 and 801 are opposites because they're both 801 units from zero, just in opposite directions.

Q: What if the numbers were 801 and 801 instead?

If both results were positive 801, they would not be opposites - they'd be equal! Opposites must have different signs, like -801 and +801.

Absolute Value with Negative Signs

Are the numbers opposite?

$-\left|-801\right|,\left|+801\right|$

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Understand the problem

Are the numbers opposite?

$-\left|-801\right|,\left|+801\right|$

Step-by-step solution

To solve this problem, we will evaluate each expression involving absolute value separately and then determine if they are opposites.

Step 1: Evaluate the absolute value of -801:
$\left|-801\right| = 801$ because the absolute value is the positive value or magnitude of the number.
Step 2: Evaluate the absolute value of +801:
$\left|+801\right| = 801$ since the absolute value of a positive number is the number itself.
Step 3: Negate the result of the first step:
$-\left|-801\right| = -801$ .
Step 4: Compare $-801$ and $801$ :
These numbers have the same magnitude (801) but opposite signs.

Therefore, the numbers $-\left|-801\right|$ and $\left|+801\right|$ are indeed opposites.

The correct choice is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value always gives positive result or zero
Technique: Evaluate absolute value first, then apply negative: $-\left|-801\right| = -801$
Check: Opposite numbers have same magnitude but different signs: -801 and 801 ✓

Common Mistakes

Avoid these frequent errors

Applying negative sign before evaluating absolute value
Don't calculate $-\left|-801\right|$ as $\left|(-1)(-801)\right| = \left|801\right| = 801$ ! This ignores proper order of operations and gives the wrong sign. Always evaluate the absolute value first, then apply any operations outside the absolute value bars.

Practice Quiz

Test your knowledge with interactive questions

Determine the absolute value of the following number:

$\left|18\right|=$

FAQ

Everything you need to know about this question

Why does the absolute value of -801 equal positive 801?

Absolute value measures distance from zero on the number line. Since -801 is 801 units away from zero, $\left|-801\right| = 801$ . Distance is always positive!

What's the difference between |-801| and -|-801|?

Great question! $\left|-801\right| = 801$ (positive), but $-\left|-801\right| = -801$ (negative). The negative sign outside the absolute value bars makes the final result negative.

How do I know if two numbers are opposites?

Two numbers are opposites if they have the same magnitude but different signs. For example, -801 and 801 are opposites because they're both 801 units from zero, just in opposite directions.

Does |+801| mean something different from |801|?

No difference at all! The plus sign is optional for positive numbers. $\left|+801\right| = \left|801\right| = 801$ . Both expressions give the same result.

What if the numbers were 801 and 801 instead?

If both results were positive 801, they would not be opposites - they'd be equal! Opposites must have different signs, like -801 and +801.

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