Compare Absolute Values: Are -|3 + 5| and |-8| Opposites?

Absolute Value with Opposite Relationships

Are these expressions opposites?

−∣3+5∣  and  ∣−8∣ -\left|3 + 5\right| \; \text{and} \; \left|-8\right|

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Are these expressions opposites?

−∣3+5∣  and  ∣−8∣ -\left|3 + 5\right| \; \text{and} \; \left|-8\right|

2

Step-by-step solution

Let's solve the problem by evaluating the given expressions and checking if they are opposites:

  • First, evaluate −∣3+5∣-\left|3 + 5\right|:
    • Calculate 3+53 + 5: 3+5=83 + 5 = 8.
    • Find ∣8∣\left|8\right|: ∣8∣=8\left|8\right| = 8.
    • Evaluate −∣8∣-\left|8\right|: −∣8∣=−8-\left|8\right| = -8.
  • Next, evaluate ∣−8∣\left|-8\right|:
    • Since −8-8 is negative, ∣−8∣=8\left|-8\right| = 8.
  • Compare the results: −8-8 and 88 are indeed opposites as −8=−1×8-8 = -1 \times 8.

Therefore, the expressions −∣3+5∣-\left|3 + 5\right| and ∣−8∣\left|-8\right| are not opposites according to standard mathematical convention.

No

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Absolute Value Rule: Always gives positive result or zero
  • Technique: Evaluate −∣3+5∣=−∣8∣=−8 -|3 + 5| = -|8| = -8 step by step
  • Check: Opposites sum to zero: -8 + 8 = 0 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing negative signs with absolute value bars
    Don't think |-8| equals -8 = wrong value! The absolute value bars always make the result positive, regardless of what's inside. Always remember |negative| = positive, then apply any outside negative sign.

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 -|8| and |-8|?

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-|8| = -8 (negative of the absolute value), while |-8| = 8 (absolute value makes it positive). The negative sign's position matters!

How do I know if two numbers are opposites?

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Two numbers are opposites if they're the same distance from zero but on opposite sides. They add up to zero: a + (-a) = 0.

Do I solve inside the absolute value bars first?

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Yes! Always follow order of operations. Calculate what's inside the absolute value bars first, then apply the absolute value, then any outside operations.

Can absolute value ever be negative?

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The absolute value itself is never negative. However, you can have a negative sign outside the absolute value bars, like −∣5∣=−5 -|5| = -5 .

What if I get the same number for both expressions?

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If both expressions equal the same number, they are equal, not opposites. For example, if both equal 8, then 8 ≠ -8, so they're not opposites.

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