Solve the Equation: -|2z| with Negative Absolute Value

Q: Why is the answer always -2z regardless of whether z is positive or negative?

Great question! The absolute value $ |2z| $ always gives us a positive result (or zero). Whether z is positive or negative, $ |2z| = 2|z| $ is always non-negative. The negative sign outside the absolute value then makes the final result negative.

Q: What's the difference between -|2z| and |-2z|?

They're actually the same thing! Both equal $ -2z $ when z ≥ 0 and $ 2z $ when z

Q: Can -|2z| ever be positive?

Only when z is negative! If z $ |2z| = -2z $ (positive), so $ -|2z| = -(-2z) = 2z $ which is positive since z is negative.

Q: How do I evaluate this when z = 3?

Step by step: $ -|2(3)| = -|6| = -6 $. The absolute value of 6 is 6, then the negative sign outside makes it -6. This matches our general answer of -2z = -2(3) = -6!

Q: What if z = 0?

When z = 0: $ -|2(0)| = -|0| = -0 = 0 $. Zero is neither positive nor negative, so the result is just 0. Our formula -2z still works: -2(0) = 0.

Absolute Value with Negative Signs

$-\left|2z\right|=$

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Step-by-step written solution

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Understand the problem

$-\left|2z\right|=$

Step-by-step solution

The absolute value function $\left|2z\right|$ simply returns $2z$ when $z$ is positive and $-2z$ when $z$ is negative, ensuring the result is non-negative. However, the minus sign outside the absolute value $-\left|2z\right|$ negates the result of the absolute value. Therefore, $-\left|2z\right|$ results in $-2z$ for all $z$ . Hence, the original expression evaluates to $-2z$ .

Final Answer

$-2z$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value always produces non-negative results
Technique: Minus sign outside negates: $-|2z| = -(2z) = -2z$
Check: For any z value, $-|2z|$ always equals $-2z$ ✓

Common Mistakes

Avoid these frequent errors

Confusing the negative sign outside with the absolute value property
Don't think $-|2z|$ changes based on whether z is positive or negative = wrong! The absolute value $|2z|$ is always non-negative, then the minus sign outside negates the whole result. Always remember the negative sign operates on the entire absolute value result.

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 is the answer always -2z regardless of whether z is positive or negative?

Great question! The absolute value $|2z|$ always gives us a positive result (or zero). Whether z is positive or negative, $|2z| = 2|z|$ is always non-negative. The negative sign outside the absolute value then makes the final result negative.

What's the difference between -|2z| and |-2z|?

They're actually the same thing! Both equal $-2z$ when z ≥ 0 and $2z$ when z < 0. The absolute value removes negative signs from inside, while the minus outside negates the whole expression.

Can -|2z| ever be positive?

Only when z is negative! If z < 0, then $|2z| = -2z$ (positive), so $-|2z| = -(-2z) = 2z$ which is positive since z is negative.

How do I evaluate this when z = 3?

Step by step: $-|2(3)| = -|6| = -6$ . The absolute value of 6 is 6, then the negative sign outside makes it -6. This matches our general answer of -2z = -2(3) = -6!

What if z = 0?

When z = 0: $-|2(0)| = -|0| = -0 = 0$ . Zero is neither positive nor negative, so the result is just 0. Our formula -2z still works: -2(0) = 0.

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