Comparing Absolute Values: Is -|-123| Equal to |+123|?

Absolute Value with Negative Signs

Are the numbers opposite?

−∣−123∣,∣+123∣ -\left|-123\right|,\left|+123\right|

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Are the numbers opposite?

−∣−123∣,∣+123∣ -\left|-123\right|,\left|+123\right|

2

Step-by-step solution

The opposite of a number is what you add to a number to get zero. The opposite of a positive number is a negative number. And the opposite of a negative number is a positive number.

For instance, the opposite of the number 123123 is −123-123.

In this question, we are asked about −∣−123∣-\left|-123\right| and ∣+123∣\left|+123\right|.

The absolute value of any number, whether positive or negative, is its distance from zero on the number line, without considering the direction. Thus, ∣+123∣=123\left|+123\right| = 123 and ∣−123∣=123\left|-123\right| = 123.

Therefore, −∣−123∣=−123-\left|-123\right| = -123 and +∣123∣=123+\left|123\right| = 123.

Since −123-123 is indeed the opposite of +123+123, the correct answer is Yes.

3

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic
  • Rule: Absolute value is always positive or zero
  • Technique: Calculate −∣−123∣=−123 -|-123| = -123 and ∣+123∣=123 |+123| = 123
  • Check: Verify opposites add to zero: −123+123=0 -123 + 123 = 0 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing the negative sign outside absolute value bars
    Don't think −∣−123∣ -|-123| equals 123 = wrong answer! The negative sign outside affects the final result after taking absolute value. Always evaluate absolute value first, then apply the negative sign outside.

Practice Quiz

Test your knowledge with interactive questions

\( \left|-19\frac{1}{4}\right|= \)

FAQ

Everything you need to know about this question

What's the difference between |-123| and -|-123|?

+

Great question! ∣−123∣=123 |-123| = 123 (absolute value makes it positive), but −∣−123∣=−123 -|-123| = -123 because the negative sign outside the bars makes the result negative.

How do I know if two numbers are opposites?

+

Two numbers are opposites if they add up to zero. For example: 123+(−123)=0 123 + (-123) = 0 , so 123 and -123 are opposites!

Does the plus sign in |+123| matter?

+

No, it doesn't! ∣+123∣ |+123| and ∣123∣ |123| both equal 123. The plus sign is optional for positive numbers inside absolute value bars.

Why is absolute value always positive?

+

Absolute value measures distance from zero on the number line. Since distance is never negative, absolute value is always positive or zero (for |0|).

What if I see multiple negative signs?

+

Work from the inside out! First find the absolute value, then apply any negative signs outside the bars. Remember: each negative sign flips the sign of the result.

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